types of hypothesis testing in spss

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SPSS Tutorial: General Statistics and Hypothesis Testing

• About This Tutorial
• SPSS Components
• Importing Data
• General Statistics and Hypothesis Testing
• Further Resources

Merging Files based on a shared variable.

This section and the "Graphics" section provide a quick tutorial for a few common functions in SPSS, primarily to provide the reader with a feel for the SPSS user interface. This is not a comprehensive tutorial, but SPSS itself provides comprehensive tutorials and case studies through it's help menu. SPSS's help menu is more than a quick reference. It provides detailed information on how and when to use SPSS's various menu options. See the "Further Resources" section for more information. 

To perform a one sample t-test click "Analyze"→"Compare Means"→"One Sample T-Test" and the following dialog box will appear:

The dialogue allows selection of any scale variable from the box at the left and a test value that represents a hypothetical mean. Select the test variable and set the test value, then press "Ok." Three tables will appear in the Output Viewer:

The first table gives descriptive statistics about the variable. The second shows the results of the t_test, including the "t" statistic, the degrees of freedom ("df") the p-value ("Sig."), the difference of the test value from the variable mean, and the upper and lower bounds for a ninety-five percent confidence interval. The final table shows one-sample effect sizes.

One-Way ANOVA

In the Data Editor, select "Analyze"→"Compare Means"→"One-Way ANOVA..." to open the dialog box shown below.

To generate the ANOVA statistic the variables chosen cannot have a "Nominal" level of measurement; they must be "ordinal." 

Once the nominal variables have been changed to ordinal, select "the dependent variable and  the factor, then click "OK." The following output will appear in the Output Viewer:

Linear Regression

To obtain a linear regression select "Analyze"->"Regression"->"Linear" from the menu, calling up the dialog box shown below:

The output of this most basic case produces a summary chart showing R, R-square, and the Standard error of the prediction; an ANOVA chart; and a chart providing statistics on model coefficients:

For Multiple regression, simply add more independent variables in the "Linear Regression" dialogue box. To plot a regression line see the "Legacy Dialogues" section of the "Graphics" tab.

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Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

• State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a  or H 1 ).
• Collect data in a way designed to test the hypothesis.
• Perform an appropriate statistical test .
• Decide whether to reject or fail to reject your null hypothesis.
• Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

• H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

• an estimate of the difference in average height between the two groups.
• a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

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Statistics By Jim

Making statistics intuitive

Comparing Hypothesis Tests for Continuous, Binary, and Count Data

By Jim Frost 46 Comments

In a previous blog post, I introduced the basic concepts of hypothesis testing and explained the need for performing these tests. In this post, I’ll build on that and compare various types of hypothesis tests that you can use with different types of data, explore some of the options, and explain how to interpret the results. Along the way, I’ll point out important planning considerations, related analyses, and pitfalls to avoid.

A hypothesis test uses sample data to assess two mutually exclusive theories about the properties of a population . Hypothesis tests allow you to use a manageable-sized sample from the process to draw inferences about the entire population.

I’ll cover common hypothesis tests for three types of variables —continuous, binary, and count data. Recognizing the different types of data is crucial because the type of data determines the hypothesis tests you can perform and, critically, the nature of the conclusions that you can draw. If you collect the wrong data, you might not be able to get the answers that you need.

Related posts : Qualitative vs. Quantitative Data , Guide to Data Types and How to Graph Them , Discrete vs. Continuous , and Nominal, Ordinal, Interval, and Ratio Scales

Hypothesis Tests for Continuous Data

Continuous data can take on any numeric value, and it can be meaningfully divided into smaller increments, including fractional and decimal values. There are an infinite number of possible values between any two values. You often measure a continuous variable on a scale. For example, when you measure height, weight, and temperature, you have continuous data . With continuous variables, you can use hypothesis tests to assess the mean, median, and standard deviation.

When you collect continuous data, you usually get more bang for your data buck compared to discrete data. The two key advantages of continuous data are that you can:

• Draw conclusions with a smaller sample size.
• Use a wider variety of analyses, which allows you to learn more.

I’ll cover two of the more common hypothesis tests that you can use with continuous data—t-tests to assess means and variance tests to evaluate dispersion around the mean. Both of these tests come in one-sample and two-sample versions. One-sample tests allow you to compare your sample estimate to a target value. The two-sample tests let you compare the samples to each other. I’ll cover examples of both types.

There is also a group of tests that assess the median rather than the mean. These are known as nonparametric tests and practitioners use them less frequently. However, consider using a nonparametric test if your data are highly skewed and the median better represents the actual center of your data than the mean.

Related posts : Nonparametric vs. Parametric Tests and Determining Which Measure of Central Tendency is Best for Your Data

Graphing the data for the example scenario

Suppose we have two production methods, and our goal is to determine which one produces a stronger product. To evaluate the two methods, we draw a random sample of 30 products from each production line and measure the strength of each unit. Before performing any analyses, it’s always a good idea to graph the data because it provides an excellent overview. Here is the CSV data file in case you want to follow along: Continuous_Data_Examples .

These histograms suggest that Method 2 produces a higher mean strength while Method 1 produces more consistent strength scores. The higher mean strength is good for our product, but the greater variability might produce more defects.

Graphs provide a good picture, but they do not test the data statistically. The differences in the graphs might be caused by random sample error rather than an actual difference between production methods. If the observed differences are due to random error, it would not be surprising if another sample showed different patterns. It can be a costly mistake to base decisions on “results” that vary with each sample. Hypothesis tests factor in random error to improve our chances of making correct decisions.

Keep this graph in mind when we look at binary data because they illustrate how much more information continuous data convey.

Related posts : Using Histograms to Understand Your Data and How Hypothesis Tests Work: Significance Levels and P-values

Two-sample t-test to compare means

The first thing we want to determine is whether one of the methods produces stronger products. We’ll use a two-sample t-test to determine whether the population means are different. The hypotheses for our 2-sample t-test are:

• Null hypothesis: The mean strengths for the two populations are equal.
• Alternative hypothesis : The mean strengths for the two populations are different.

A p-value less than the significance level indicates that you can reject the null hypothesis. In other words, the sample provides sufficient evidence to conclude that the population means are different. Below is the output for the analysis.

The p-value (0.034) is less than 0.05. From the output, we can see that the difference between the mean of Method 2 (98.39) and Method 1 (95.39) is statistically significant. We can conclude that Method 2 produces a stronger product on average.

That sounds great, and it appears that we should use Method 2 to manufacture a stronger product. However, there are other considerations. The t-test tells us that Method 2’s mean strength is greater than Method 1, but it says nothing about the variability of strength values. For that, we need to use another test.

Related posts : How T-Tests Work and How to Interpret P-values Correctly and Step-by-Step Instructions for How to Do t-Tests in Excel .

2-Variances test to compare variability

A production method that has excessive variability creates too many defects. Consequently, we will also assess the standard deviations of both methods. To determine whether either method produces greater variability in the product’s strength, we’ll use the 2 Variances test. The hypotheses for our 2 Variances test are:

• Null hypothesis: The standard deviations for the populations are equal.
• Alternative hypothesis: The standard deviations for the populations are different.

A p-value less than the significance level indicates that you can reject the null hypothesis. In other words, the sample provides sufficient evidence for concluding that the population standard deviations are different. The 2-Variances output for our product is below.

Both of the p-values are less than 0.05. The output indicates that the variability of Method 1 is significantly less than Method 2. We can conclude that Method 1 produces a more consistent product.

Related post : Measures of Variability

What we learned and did not learn with the hypothesis tests

The hypothesis test results confirm the patterns in the graphs. Method 2 produces stronger products on average while Method 1 produces a more consistent product. The statistically significant test results indicate that these results are likely to represent actual differences between the production methods rather than sampling error.

Our example also illustrates how you can assess different properties using continuous data, which can point towards different decisions. We might want the stronger products of Method 2 but the greater consistency of Method 1. To navigate this dilemma, we’ll need to use our process knowledge.

Finally, it’s crucial to note that the tests produce estimates of population parameters—the population means (μ) and the population standard deviations (σ). While these parameters can help us make decisions, they tell us little about where individual values are likely to fall. In certain circumstances, knowing the proportion of values that fall within specified intervals is crucial.

For the examples, the products must fall within spec limits. Even when the mean falls within the spec limit, it’s possible that too many individual items will fall outside the spec limits if the variability is too high.

Other types of analyses

To better understand the distribution of individual values rather than the population parameters, use the following analyses:

Tolerance intervals : A tolerance interval is a range that likely contains a specific proportion of a population. For our example, we might want to know the range where 99% of the population falls for each production method. We can compare the tolerance interval to our requirements to determine whether there is too much variability.

Capability analysis : This type of analysis uses sample data to determine how effectively a process produces output with characteristics that fall within the spec limits. These tools incorporate both the mean and spread of your data to estimate the proportion of defects.

Related post : Confidence Intervals vs. Prediction Intervals vs. Tolerance Intervals

Proportion Hypothesis Tests for Binary Data

Let’s switch gears and move away from continuous data. Suppose we take another random sample of our product from each of the production lines. However, instead of measuring a characteristic, inspectors evaluate each product and either accept or reject it.

Binary data can have only two values. If you can place an observation into only two categories, you have a  binary variable . For example, pass/fail and accept/reject data are binary. Quality improvement practitioners often use binary data to record defective units.

Binary data are useful for calculating proportions or percentages, such as the proportion of defective products in a sample. You simply take the number of defective products and divide by the sample size. Hypothesis tests that assess proportions require binary data and allow you to use sample data to make inferences about the proportions of populations.

2 Proportions test to compare two samples

For our first example, we will make a decision based on the proportions of defective parts. Our goal is to determine whether the two methods produce different proportions of defective parts.

To make this determination, we’ll use the 2 Proportions test. For this test, the hypotheses are as follows:

• Null hypothesis: The proportions of defective parts for the two populations are equal.
• Alternative hypothesis: The proportions of defective parts for the two populations are different.

A p-value less than the significance level indicates that you can reject the null hypothesis. In this case, the sample provides sufficient evidence for concluding that the population proportions are different. The 2 Proportions output for our product is below.

Both p-values are less than 0.05. The output indicates that the difference between the proportion of defective parts for Method 1 (~0.062) and Method 2 (~0.146) is statistically significant. We can conclude that Method 1 produces defective parts less frequently.

1 Proportion test example: comparison to a target

The 1 Proportion test is also handy because you can compare a sample to a target value. Suppose you receive parts from a supplier who guarantees that less than 3% of all parts they produce are defective. You can use the 1 Proportion test to assess this claim.

First, collect a random sample of parts and determine how many are defective. Then, use the 1 Proportion test to compare your sample estimate to the target proportion of 0.03. Because we are interested in detecting only whether the population proportion is greater than 0.03, we’ll use a one-sided test. One-sided tests have greater power to detect differences in one direction, but no ability to detect differences in the other direction. Our one-sided 1 Proportion test has the following hypotheses:

• Null hypothesis: The proportion of defective parts for the population equals 0.03 or less.
• Alternative hypothesis: The proportion of defective parts for the population is greater than 0.03.

For this test, a significant p-value indicates that the supplier is in trouble! The sample provides sufficient evidence to conclude that the proportion of all parts from the supplier’s process is greater than 0.03 despite their assertions to the contrary.

Comparing continuous data to binary data

Think back to the graphs for the continuous data. At a glance, you can see both the central location and spread of the data. If we added spec limits, we could see how many data points are close and far away from them. Is the process centered between the spec limits? Continuous data provide a lot of insight into our processes.

Now, compare that to the binary data that we used in the 2 Proportions test. All we learn from that data is the proportion of defects for Method 1 (0.062) and Method 2 (0.146). There is no distribution to analyze, no indication of how close the items are to the specs, and no indication of how they failed the inspection. We only know the two proportions.

Additionally, the samples sizes are much larger for the binary data than the continuous data (130 vs. 30). When the difference between proportions is smaller, the required sample sizes can become quite large. Had we used a sample size of 30 like before, we almost certainly would not have detected this difference.

In general, binary data provide less information than an equivalent amount of continuous data. If you can collect continuous data, it’s the better route to take!

Related post : Estimating a Good Sample Size for Your Study Using Power Analysis

Poisson Hypothesis Tests for Count Data

Count data can have only non-negative integers (e.g., 0, 1, 2, etc.). In statistics , we often model count data using the Poisson distribution. Poisson data are a count of the presence of a characteristic, result, or activity over a constant amount of time, area, or other length of observation. For example, you can use count data to record the number of defects per item or defective units per batch. With Poisson data, you can assess a rate of occurrence.

For this scenario, we’ll assume that we receive shipments of parts from two different suppliers. Each supplier sends the parts in the same sized batch. We need to determine whether one supplier produces fewer defects per batch than the other supplier.

To perform this analysis, we’ll randomly sample batches of parts from both suppliers. The inspectors examine all parts in each batch and record the count of defective parts. We’ll randomly sample 30 batches from each supplier. Here is the CSV data file for this example: Count_Data_Example .

Performing the Two-Sample Poisson Rate Test

We’ll use the 2-Sample Poisson Rate test. For this test, the hypotheses are as follows:

• Null hypothesis: The rates of defective parts for the two populations are equal.
• Alternative hypothesis: The rates of defective parts for the two populations are different.

A p-value less than the significance level indicates that you can reject the null hypothesis because the sample provides sufficient evidence to conclude that the population rates are different. The 2-Sample Poisson Rate output for our product is below.

Both p-values are less than 0.05. The output indicates that the difference between the rate of defects per batch for Supplier 1 (3.56667) and Supplier 2 (5.36667) is statistically significant. We can conclude that Supplier 1 produces defects at a lower rate than Supplier 2.

Hypothesis tests are a great tool that allow you to take relatively small samples and draw conclusions about entire populations. There is a selection of tests available, and different options within the tests, which make them useful for a wide variety of situations.

To see an alternative approach to these traditional hypothesis testing methods, learn about bootstrapping in statistics !

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Reader Interactions

May 10, 2024 at 11:06 am

Ah that explains why I couldn’t find an R function that returned the same output! Thank you very much for your reply – I can stop looking now!

May 9, 2024 at 11:24 am

Thank you for this article, I’ve learned a lot from reading it.

Your Two-Sample Poisson Rate Test example is very similar in structure to my data so I am trying to follow the same approach. The results pictured look like output from an R function – but I have been unable to find one that outputs results in this way. If these were indeed created by an R library/function, would you mind sharing which one you used, please?

Kind regards, Ash

May 9, 2024 at 4:49 pm

Sorry, I’m not using R. The results are from Minitab statistical software.

September 29, 2023 at 3:21 pm

Hello guys, what is the outcome varaible in independent sample t test? binary or not? Because it compares the means of two independent populations as which is greater or lower

September 29, 2023 at 5:47 pm

The outcome variable for an independent sample t-test is continuous because you’re using it to calculate the means for two groups.

The binary variable is the categorical factor in the design. The binary variable defines the two groups for which you’re calculating the means. For example, your binary variable could be gender (male/female), experimental group (control/treatment), or material type (A/B). But the outcome variable is continuous so you can calculate the means for both groups and compare them. Click the link to learn more about the test.

March 22, 2023 at 3:32 pm

The document can’t be found, is the link still working?

March 22, 2023 at 3:35 pm

You’ll need to specify which link you’re talking about so I can check it. All links should be working.

October 9, 2021 at 3:28 am

Greetings!!! Very intuitive explanation. Liked the way you have explained with sufficient examples.

Jim based on Inferential Statistics, could you include an article on A/B Testing Methodology incorporating from basics like —Data Collection Process, Dataset Splitting Procedures & Duration for carrying out such experiments.

Also if you could incorporate illustrations from different industries viz. Healthcare, Manufacturing, Logistics, Quality, Ecommerce, Marketing, Advertisement Domains, this would indeed be useful.

Nowadays A/B Testing & Multivariate Testing is being incorporated & implemented in a robust manner across Data Science domain. Article or Write-up regarding this would immensely be useful.

Looking forward to a favourable and positive response.

August 21, 2021 at 1:22 pm

The poisson test example has N of 30. I am wondering the appropriate distribution if the sample is lower than 30. Is it a t statistic or chi-square

June 16, 2021 at 12:02 pm

Hi, great post! I have an expected and observed data set and want to do additional testing to see if they differ signficantly from each other. Furthermore, the specific entries that contribute to the most weight in that significance or places that should have special attention. I did chi-square goodness of fit, but want to go further. Just to add, this is count data.

June 19, 2021 at 4:17 pm

I’m not 100% sure what you want to do to go further. Because it’s count data, you could model it with the Poisson or Negative Binomial distribution. If you have other relevant variables, you can fit a Poisson or Negative Binomial regression model to explore relationships in your data. I talk a bit about those types of models in my post about choosing the correct type of regression model . You can also perform hypothesis tests designed for that type of data. The chi-squared test you performed seems like a good one to see how the expected and observed differs!

April 29, 2021 at 11:44 pm

How do you do a independence test in Stata for a categorical variable with 6 levels and a binary variable.

April 30, 2021 at 12:23 am

I’m not a Stata user, but it sounds like you need to perform a chi-square test of independence .

December 13, 2020 at 1:00 pm

Hi Jim – thank you for this great site! I have a situation where there is a reference standard (tells me if there is truly fat in a mass) and I have 2 different methods of detecting if there is (or is not) fat in the mass. My null hypothesis is that there is no difference in detection. I have a set of masses where I know if there is fat in the masses and used the 2 methods to detect whether they were able to detect the fat. Is the 2 proportions test the most appropriate for this question? Thank you so much!

December 3, 2020 at 8:31 am

Thank you Jim for the wonderful post. It was clearly written and I enjoyed reading through it. I have an additional query. I wanted to compare the variances of two methods of measurement applied at each observation point of a field survey. The variables from both methods have binary data type. How can I do the statistical test. Thank you in advance for your help.

December 3, 2020 at 3:02 pm

With binary data, you can’t compare variances. You can compare proportions using a proportions test. I discuss these tests in the binary section of this post. To read an example of a 2-sample proportions test, read my post about flu shot effectiveness . In it, I use 2-proportions tests to evaluate real flu study data. Or read my post about Mythbusters test about whether yawns are contagious , where I use a 2-proportions test. That way you can see what these tests can do. I cover them, and many other tests, in much more detail in my Hypothesis Testing book !

I hope this helps!

August 5, 2020 at 9:36 pm

Hi Jim. Just wanted to follow up and see if you’ve had a chance to review this question yet?

August 6, 2020 at 1:09 am

Hi Jack, thanks for the reminder! I just replied!

August 3, 2020 at 4:45 am

Hi Jim , a green belt has a project on flu vaccinations , with 5 data points, % vaccination rates per year averaging about 36% of staff numbers. Her project was to increase vaccination rates this year , and has accumulated a lot of data points to measure vaccination rates in different office areas as a percent of total staff numbers which have almost doubled. Should she use 2 sample t test to measure difference in means between before and after data ( continuous) or should she use 2 sample test for proportions (attribute). There is small sample size for before data and large sample size for after data

August 5, 2020 at 12:46 am

I see two possibilities. The choice depends on whether she measured the same sites in the before and after. If they’re different sites before and after, she has independent groups and can use the regular 2-sample proportions test.

If they’re the same sites before and after, she can use either the test of marginal homogeneity or McNemar’s test. I have not used these tests myself and cannot provide more information. However, if she used the same sites before and after, she has proportions data for dependent groups (same groups) and should not use the regular 2-sample proportions test. These two tests can handle proportions for dependent groups.

July 19, 2020 at 3:18 am

Hi Jim. Would I be able to use the 2 Proportion Test for comparing 2 proportions from different time periods? Example scenario: I run a satisfaction survey on a MOOC site during Q1 to a random sample of visitors and find that 80% of them were satisfied with their experience. The following quarter I run the same survey and find that 75% were satisfied. Is the 5 percentage point drop statistically significant or just due to random noise?

Sorry about the delay in replying! Sometimes comments slip through the cracks!

Yes, you can do as you suggest assuming the respondents are different in the two quarters and assuming that the data are binary (satisfied/not satisfied). The 2 proportions test is designed for independent groups and binary data.

I hope that helps even belatedly!

May 19, 2020 at 9:58 pm

Thanks…Let me see that document

May 19, 2020 at 7:04 pm

I would like to ask about 2 sample poisson rate. How do you calculate the 95% CI and test for difference ? Your answer is really appreciated. Thank you so much for giving this tutorial.

May 19, 2020 at 9:17 pm

This document describes the calculations. I hope it helps!

March 6, 2020 at 4:52 am

Thank you so much for your kind support. Esteemed regards.

March 5, 2020 at 9:23 pm

Thanks for your helpful comments. Basically I have developed my research model based on competing theories. For example, I have one IV (Exploitation) and Two DV’s (Incremental innovation and Radical Innovation). Each variable in the model has own indicators. Some researchers claims that exploitation support only incremental innovations. On the other hand there are also studies that claims that in-depth exploitation also support radical innovation. However, these researchers claim that exploitation support radical innovations in a limited capacity as compared to incremental innovation. ON the basis of these competing theories I developed my hypothesis as: Exploitation significantly and positively influences both incremental and radical innovation, however exploitation influence incremental innovation more than radical innovation.

Thank you very much for your quick response. Its really helpful.

March 6, 2020 at 1:46 am

Hi Shabahat,

Thanks you for the additional information. I messed up one thing in my previous reply to you. Because you only have the one IV, you don’t need to worry about standardizing that IV for comparability. However, assuming the DVs use different units of measurement, that’ll make comparisons problematic. Consequently, you probably need to standardize the two dependent variables instead. You’d learn how a one unit change in the IV relates to the standard deviation of each DV. That puts the DVs on the same scale. See how other researchers have handled this situation in your field to be sure that’s an accepted approach.

Additionally, if you find a significant relationship between exploitation and radical innovation, then you’d have good evidence to support that claim you’re making.

March 5, 2020 at 2:37 am

Hi Jim, Its really amazing. However I have a query regarding my analysis. I have one independent variable (Continuous) and Two dependent variables (Continuous). In the linear regression, the Independent variable significantly explains both dependent variables. Problem: Now i want to compare the effect of my Independent variable on both dependent variables. How can I compare?. If the effect is different, how can I test whether the effect difference is statistically significant or not in SPSS.

March 5, 2020 at 4:01 pm

The fact that you’re talking about different DVs complicates things because they’re presumably measuring different things and using different units, which makes comparison difficult. The standardized coefficients can help you get around that but it changes the interpretation of the results.

Assuming that the two DV variables measure different characteristics, you might try standardizing your IVs and fitting the model using the standardized values. This process produces standardized coefficients, which use the same units and allows you to compare–although it does change the interpretation of the coefficients. I write about this in my post about assessing the importance of your predictors . You can also look at the CIs for the standardized coefficients and see if they overlap. If they don’t overlap, you know the difference between the standardized coefficients is statistically significant.

January 20, 2019 at 12:40 pm

Great Post! Can we test proportions from a continuos variable with unknown distribution using the poisson distribution using a cut-off value for good and bad samples and couting them?

November 14, 2018 at 4:05 pm

Hi Jim, I really enjoy reading your posts and they have cleared many stat concepts!

I had a question about the chi square probability distribution. Although it is a non-parametric test, why does it fall into a continuous probability distribution and why can we use the chi square distribution for categorical data if it’s a continuous probability distribution?

November 14, 2018 at 10:53 pm

Hi Sanjana,

That’s a great question! I’m glad you’re thinking about the types of variables and distributions, and how they’re used together.

You’re correct on both counts. Chi-squared test of independent is nonparameteric because it doesn’t assume a particular data distribution. Additionally, analysts use it to test the independence of categorical variables . There are other ways to use this distribution as well.

Now, onto why we use chi-square (a distribution for continuous data) with categorical variables! Yes, it involves categorical variables, but the analysis assesses the observed and expected counts of these variables. For each cell, the analysis takes the squared difference between the observed count and the expected count and then divides that by the expected count. These values are summed acrossed all cells to produce the chi-square value. This process produces a continuous variable that is based on the differences between the observed and expected counts of the categorical variables. When the value of this variable is large enough, we know that the difference between the observed counts and the expected counts is large enough to be unlikely due to chance. And, that’s why we use a continuous distribution to analyze categorical variables.

May 27, 2018 at 12:58 am

This is very helpful! Thank you!

May 26, 2018 at 11:38 pm

Hi Jim, Great post. I was wondering, do you know of any references that discuss the difference in sample size between binary and continuous data? I am looking for a reference to cite in a journal article. Thanks, Amanda.

May 26, 2018 at 11:55 pm

The article I cite below discusses the different sample sizes in terms of observations per model term in order to avoid overfitting your model. I also cover these ideas in my post about how to avoid overfitting regression models . For regression models, this provides a good context for sample size requirements.

Babyak, MA., What You See May Not Be What You Get: A Brief, Nontechnical Introduction to Overfitting in Regression-Type Models, Psychosomatic Medicine 66:411-421 (2004).

February 3, 2018 at 7:04 am

I am totally new to statistics,

Following a small sample from my dataset.

Views PosEmo NegEmo 1650077 2.63 1.27 753826 2.39 0.47 926647 1.71 1.02

Views = Dependent continous Variable PosEmo = Independent Continous Variable NegEmo = Independent Continous Variable

My query : 1. How to run Hypothesis testing on same, Im pretty confused what to use , what to do , I am using SPSS modeler and SPSS statistics tool. 2.I think Multiple Regression is Ok for this . Let me know how to use it in SPSS modeler or stats tool.

Regards Sarika

February 5, 2018 at 1:30 am

Hi Sarika, yes, it sounds like you can use multiple regression for those data. The hypothesis test in this case would be the p-values for the regression coefficients . Click that link to learn more about that. In your stats software, choose multiple linear regression and then specify the dependent variable and the two independent variables. Fit the model and then check the statistical output and the residual plots to see if you have a good model. Be sure to check out my regression tutorial too. That covers many aspects of regression analysis.

November 30, 2017 at 8:45 pm

Thanks for your sharing!

In the binary case (or proportion case), is there any comparison between “two proportion test” and “Chi-square” test? Is there any guideline to choose which test to use?

November 30, 2017 at 9:46 pm

You’re welcome! Regarding your question, a two proportion test requires one categorical variable with two levels. For example, the variable could be “test result” and the two levels are “pass” and “fail.”

A chi-square test of independence requires at least two categorical variables. Those variables can have two or more levels. You can read an example of the chi-square test of independence that I’ve written about. The example is based on the original Star Trek TV series and determines whether the uniform color affects the fatality rate. That analysis has two categorical variables–fatalities and uniform color. Fatalities has two levels that indicate whether a crewmember survived or died. Uniform color has three levels–gold, blue, and red.

As you can see, the data requirements for the two tests are different.

I hope this helps! Jim

November 30, 2017 at 2:18 am

November 29, 2017 at 10:41 pm

Great post. Thanks for sharing your expertise.

November 29, 2017 at 11:38 pm

Thank you! I’m glad it was helpful.

November 29, 2017 at 9:02 pm

Very nice article. Could you explain more on hypothesis testing on median?

November 29, 2017 at 11:39 pm

Thank you! For more information about testing the median, click the link in the article for where I compare parametric vs nonparametric analyses.

November 29, 2017 at 6:51 pm

Please let me know when one can use Probit Analysis. May I know the Procedure in SPSS.

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S.3 hypothesis testing.

In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail.

The general idea of hypothesis testing involves:

• Making an initial assumption.
• Collecting evidence (data).
• Based on the available evidence (data), deciding whether to reject or not reject the initial assumption.

Every hypothesis test — regardless of the population parameter involved — requires the above three steps.

Example S.3.1

Is normal body temperature really 98.6 degrees f section  .

Consider the population of many, many adults. A researcher hypothesized that the average adult body temperature is lower than the often-advertised 98.6 degrees F. That is, the researcher wants an answer to the question: "Is the average adult body temperature 98.6 degrees? Or is it lower?" To answer his research question, the researcher starts by assuming that the average adult body temperature was 98.6 degrees F.

Then, the researcher went out and tried to find evidence that refutes his initial assumption. In doing so, he selects a random sample of 130 adults. The average body temperature of the 130 sampled adults is 98.25 degrees.

Then, the researcher uses the data he collected to make a decision about his initial assumption. It is either likely or unlikely that the researcher would collect the evidence he did given his initial assumption that the average adult body temperature is 98.6 degrees:

• If it is likely , then the researcher does not reject his initial assumption that the average adult body temperature is 98.6 degrees. There is not enough evidence to do otherwise.
• either the researcher's initial assumption is correct and he experienced a very unusual event;
• or the researcher's initial assumption is incorrect.

In statistics, we generally don't make claims that require us to believe that a very unusual event happened. That is, in the practice of statistics, if the evidence (data) we collected is unlikely in light of the initial assumption, then we reject our initial assumption.

Example S.3.2

Criminal trial analogy section  .

One place where you can consistently see the general idea of hypothesis testing in action is in criminal trials held in the United States. Our criminal justice system assumes "the defendant is innocent until proven guilty." That is, our initial assumption is that the defendant is innocent.

In the practice of statistics, we make our initial assumption when we state our two competing hypotheses -- the null hypothesis ( H 0 ) and the alternative hypothesis ( H A ). Here, our hypotheses are:

• H 0 : Defendant is not guilty (innocent)
• H A : Defendant is guilty

In statistics, we always assume the null hypothesis is true . That is, the null hypothesis is always our initial assumption.

The prosecution team then collects evidence — such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, and handwriting samples — with the hopes of finding "sufficient evidence" to make the assumption of innocence refutable.

In statistics, the data are the evidence.

The jury then makes a decision based on the available evidence:

• If the jury finds sufficient evidence — beyond a reasonable doubt — to make the assumption of innocence refutable, the jury rejects the null hypothesis and deems the defendant guilty. We behave as if the defendant is guilty.
• If there is insufficient evidence, then the jury does not reject the null hypothesis . We behave as if the defendant is innocent.

In statistics, we always make one of two decisions. We either "reject the null hypothesis" or we "fail to reject the null hypothesis."

Errors in Hypothesis Testing Section  

Did you notice the use of the phrase "behave as if" in the previous discussion? We "behave as if" the defendant is guilty; we do not "prove" that the defendant is guilty. And, we "behave as if" the defendant is innocent; we do not "prove" that the defendant is innocent.

This is a very important distinction! We make our decision based on evidence not on 100% guaranteed proof. Again:

• If we reject the null hypothesis, we do not prove that the alternative hypothesis is true.
• If we do not reject the null hypothesis, we do not prove that the null hypothesis is true.

We merely state that there is enough evidence to behave one way or the other. This is always true in statistics! Because of this, whatever the decision, there is always a chance that we made an error .

Let's review the two types of errors that can be made in criminal trials:

Table S.3.2 shows how this corresponds to the two types of errors in hypothesis testing.

Note that, in statistics, we call the two types of errors by two different  names -- one is called a "Type I error," and the other is called  a "Type II error." Here are the formal definitions of the two types of errors:

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

Making the Decision Section  

Recall that it is either likely or unlikely that we would observe the evidence we did given our initial assumption. If it is likely , we do not reject the null hypothesis. If it is unlikely , then we reject the null hypothesis in favor of the alternative hypothesis. Effectively, then, making the decision reduces to determining "likely" or "unlikely."

In statistics, there are two ways to determine whether the evidence is likely or unlikely given the initial assumption:

• We could take the " critical value approach " (favored in many of the older textbooks).
• Or, we could take the " P -value approach " (what is used most often in research, journal articles, and statistical software).

In the next two sections, we review the procedures behind each of these two approaches. To make our review concrete, let's imagine that μ is the average grade point average of all American students who major in mathematics. We first review the critical value approach for conducting each of the following three hypothesis tests about the population mean $\mu$:

In Practice

• We would want to conduct the first hypothesis test if we were interested in concluding that the average grade point average of the group is more than 3.
• We would want to conduct the second hypothesis test if we were interested in concluding that the average grade point average of the group is less than 3.
• And, we would want to conduct the third hypothesis test if we were only interested in concluding that the average grade point average of the group differs from 3 (without caring whether it is more or less than 3).

Upon completing the review of the critical value approach, we review the P -value approach for conducting each of the above three hypothesis tests about the population mean \(\mu\). The procedures that we review here for both approaches easily extend to hypothesis tests about any other population parameter.

Introduction to SPSS

• Social Science Data Resources This link opens in a new window
• SPSS Components
• Importing Data and Working with Syntax
• Recode Data
• Wilcoxon Signed Rank Test
• Mann-Whitney U Test
• Further Resources

Merging Files based on a shared variable.

This section and the "Graphics" section provide a quick tutorial for a few common functions in SPSS, primarily to provide the reader with a feel for the SPSS user interface. This is not a comprehensive tutorial, but SPSS itself provides comprehensive tutorials and case studies through it's help menu. SPSS's help menu is more than a quick reference. It provides detailed information on how and when to use SPSS's various menu options. See the "Further Resources" section for more information. 

To perform a one sample t-test click "Analyze"→"Compare Means"→"One Sample T-Test" and the following dialogue box will appear:

The dialogue allows selection of any scale variable from the box at the left and a test value that represents a hypothetical mean. To find out the probability that the mean of "median_income" is fifty thousand, select the variable from the left-hand box, set the test value to fifty thousand and click "OK." Two tables will appear in the Output Viewer:

The first table gives descriptive statistics about the variable "median_income." The second shows the results of the t_test, including the "t" statistic, the degrees of freedom ("df") the p-value ("Sig."), the difference of the test value from the variable mean, and the upper and lower bounds for a ninety-five percent confidence interval.

One-Way ANOVA

In the Data Editor, select "Analyze"→"Compare Means"→"One-Way ANOVA..." to open the dialogue box shown below.

To generate the ANOVA statistic the variables chosen cannot have a "Nominal" level of measurement. If the dataset being used has com from the included CSV, the level of measurement will have to be changed to "Ordinal" for the variables "greater_than_thirty_percent_have_bachelors" and "inc_greater_than_ave." This can be easily accomplished in the "Variable View" tab in the data editor (see "Variables in SPSS" under the "SPSS Anatomy" section).

Once the nominal variables have been changed to ordinal, select "high_school" as the dependent variable and "inc_greater_than_ave" as the factor, then click "OK." The following output will appear in the Output Viewer:

Linear Regression

To obtain a linear regression select "Analyze"->"Regression"->"Linear" from the menu, calling up the dialogue box shown below:

The output of this most basic case produces a summary chart showing R, R-square, and the Standard error of the prediction; an ANOVA chart; and a chart providing statistics on model coefficients:

For Multiple regression, simply add more independent variables in the "Linear Regression" dialogue box. To plot a regression line see the "Legacy Dialogues" section of the "Graphics" tab.

• General Statistics and Hypothesis Testing PDF
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What is Hypothesis Testing in Statistics? Types and Examples

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Hypothesis testing in statistics involves testing an assumption about a population parameter using sample data. Learners can download Hypothesis Testing PDF to get instant access to all information!

What exactly is hypothesis testing, and how does it work in statistics? Can I find practical examples and understand the different types from this blog?

Hypothesis Testing : Ever wonder how researchers determine if a new medicine actually works or if a new marketing campaign effectively drives sales? They use hypothesis testing! It is at the core of how scientific studies, business experiments and surveys determine if their results are statistically significant or just due to chance.

Hypothesis testing allows us to make evidence-based decisions by quantifying uncertainty and providing a structured process to make data-driven conclusions rather than guessing. In this post, we will discuss hypothesis testing types, examples, and processes!

Table of Contents

Hypothesis Testing

Hypothesis testing is a statistical method used to evaluate the validity of a hypothesis using sample data. It involves assessing whether observed data provide enough evidence to reject a specific hypothesis about a population parameter. 

Hypothesis Testing in Data Science

Hypothesis testing in data science is a statistical method used to evaluate two mutually exclusive population statements based on sample data. The primary goal is to determine which statement is more supported by the observed data.

Hypothesis testing assists in supporting the certainty of findings in research and data science projects. This statistical inference aids in making decisions about population parameters using sample data. For those who are looking to deepen their knowledge in data science and expand their skillset, we highly recommend checking out Master Generative AI: Data Science Course by Physics Wallah .

Also Read: What is Encapsulation Explain in Details

What is the Hypothesis Testing Procedure in Data Science?

The hypothesis testing procedure in data science involves a structured approach to evaluating hypotheses using statistical methods. Here’s a step-by-step breakdown of the typical procedure:

1) State the Hypotheses:

• Null Hypothesis (H0): This is the default assumption or a statement of no effect or difference. It represents what you aim to test against.
• Alternative Hypothesis (Ha): This is the opposite of the null hypothesis and represents what you want to prove.

2) Choose a Significance Level (α):

• Decide on a threshold (commonly 0.05) beyond which you will reject the null hypothesis. This is your significance level.

3) Select the Appropriate Test:

• Depending on your data type (e.g., continuous, categorical) and the nature of your research question, choose the appropriate statistical test (e.g., t-test, chi-square test, ANOVA, etc.).

4) Collect Data:

• Gather data from your sample or population, ensuring that it’s representative and sufficiently large (or as per your experimental design).

5)Compute the Test Statistic:

• Using your data and the chosen statistical test, compute the test statistic that summarizes the evidence against the null hypothesis.

6) Determine the Critical Value or P-value:

• Based on your significance level and the test statistic’s distribution, determine the critical value from a statistical table or compute the p-value.

7) Make a Decision:

• If the p-value is less than α: Reject the null hypothesis.
• If the p-value is greater than or equal to α: Fail to reject the null hypothesis.

8) Draw Conclusions:

• Based on your decision, draw conclusions about your research question or hypothesis. Remember, failing to reject the null hypothesis doesn’t prove it true; it merely suggests that you don’t have sufficient evidence to reject it.

9) Report Findings:

• Document your findings, including the test statistic, p-value, conclusion, and any other relevant details. Ensure clarity so that others can understand and potentially replicate your analysis.

Also Read: Binary Search Algorithm

How Hypothesis Testing Works?

Hypothesis testing is a fundamental concept in statistics that aids analysts in making informed decisions based on sample data about a larger population. The process involves setting up two contrasting hypotheses, the null hypothesis and the alternative hypothesis, and then using statistical methods to determine which hypothesis provides a more plausible explanation for the observed data.

The Core Principles:

• The Null Hypothesis (H0): This serves as the default assumption or status quo. Typically, it posits that there is no effect or no difference, often represented by an equality statement regarding population parameters. For instance, it might state that a new drug’s effect is no different from a placebo.
• The Alternative Hypothesis (H1 or Ha): This is the counter assumption or what researchers aim to prove. It’s the opposite of the null hypothesis, indicating that there is an effect, a change, or a difference in the population parameters. Using the drug example, the alternative hypothesis would suggest that the new drug has a different effect than the placebo.

Testing the Hypotheses:

Once these hypotheses are established, analysts gather data from a sample and conduct statistical tests. The objective is to determine whether the observed results are statistically significant enough to reject the null hypothesis in favor of the alternative.

Examples to Clarify the Concept:

• Null Hypothesis (H0): The sanitizer’s average efficacy is 95%.
• By conducting tests, if evidence suggests that the sanitizer’s efficacy is significantly less than 95%, we reject the null hypothesis.
• Null Hypothesis (H0): The coin is fair, meaning the probability of heads and tails is equal.
• Through experimental trials, if results consistently show a skewed outcome, indicating a significantly different probability for heads and tails, the null hypothesis might be rejected.

What are the 3 types of Hypothesis Test?

Hypothesis testing is a cornerstone in statistical analysis, providing a framework to evaluate the validity of assumptions or claims made about a population based on sample data. Within this framework, several specific tests are utilized based on the nature of the data and the question at hand. Here’s a closer look at the three fundamental types of hypothesis tests:

The z-test is a statistical method primarily employed when comparing means from two datasets, particularly when the population standard deviation is known. Its main objective is to ascertain if the means are statistically equivalent. 

A crucial prerequisite for the z-test is that the sample size should be relatively large, typically 30 data points or more. This test aids researchers and analysts in determining the significance of a relationship or discovery, especially in scenarios where the data’s characteristics align with the assumptions of the z-test.

The t-test is a versatile statistical tool used extensively in research and various fields to compare means between two groups. It’s particularly valuable when the population standard deviation is unknown or when dealing with smaller sample sizes. 

By evaluating the means of two groups, the t-test helps ascertain if a particular treatment, intervention, or variable significantly impacts the population under study. Its flexibility and robustness make it a go-to method in scenarios ranging from medical research to business analytics.

3. Chi-Square Test:

The Chi-Square test stands distinct from the previous tests, primarily focusing on categorical data rather than means. This statistical test is instrumental when analyzing categorical variables to determine if observed data aligns with expected outcomes as posited by the null hypothesis. 

By assessing the differences between observed and expected frequencies within categorical data, the Chi-Square test offers insights into whether discrepancies are statistically significant. Whether used in social sciences to evaluate survey responses or in quality control to assess product defects, the Chi-Square test remains pivotal for hypothesis testing in diverse scenarios.

Also Read: Python vs Java: Which is Best for Machine learning algorithm

Hypothesis Testing in Statistics

Hypothesis testing is a fundamental concept in statistics used to make decisions or inferences about a population based on a sample of data. The process involves setting up two competing hypotheses, the null hypothesis H 0​ and the alternative hypothesis H 1​. 

Through various statistical tests, such as the t-test, z-test, or Chi-square test, analysts evaluate sample data to determine whether there’s enough evidence to reject the null hypothesis in favor of the alternative. The aim is to draw conclusions about population parameters or to test theories, claims, or hypotheses.

Hypothesis Testing in Research

In research, hypothesis testing serves as a structured approach to validate or refute theories or claims. Researchers formulate a clear hypothesis based on existing literature or preliminary observations. They then collect data through experiments, surveys, or observational studies. 

Using statistical methods, researchers analyze this data to determine if there’s sufficient evidence to reject the null hypothesis. By doing so, they can draw meaningful conclusions, make predictions, or recommend actions based on empirical evidence rather than mere speculation.

Hypothesis Testing in R

R, a powerful programming language and environment for statistical computing and graphics, offers a wide array of functions and packages specifically designed for hypothesis testing. Here’s how hypothesis testing is conducted in R:

• Data Collection : Before conducting any test, you need to gather your data and ensure it’s appropriately structured in R.
• Choose the Right Test : Depending on your research question and data type, select the appropriate hypothesis test. For instance, use the t.test() function for a t-test or chisq.test() for a Chi-square test.
• Set Hypotheses : Define your null and alternative hypotheses. Using R’s syntax, you can specify these hypotheses and run the corresponding test.
• Execute the Test : Utilize built-in functions in R to perform the hypothesis test on your data. For instance, if you want to compare two means, you can use the t.test() function, providing the necessary arguments like the data vectors and type of t-test (one-sample, two-sample, paired, etc.).
• Interpret Results : Once the test is executed, R will provide output, including test statistics, p-values, and confidence intervals. Based on these results and a predetermined significance level (often 0.05), you can decide whether to reject the null hypothesis.
• Visualization : R’s graphical capabilities allow users to visualize data distributions, confidence intervals, or test statistics, aiding in the interpretation and presentation of results.

Hypothesis testing is an integral part of statistics and research, offering a systematic approach to validate hypotheses. Leveraging R’s capabilities, researchers and analysts can efficiently conduct and interpret various hypothesis tests, ensuring robust and reliable conclusions from their data.

Do Data Scientists do Hypothesis Testing?

Yes, data scientists frequently engage in hypothesis testing as part of their analytical toolkit. Hypothesis testing is a foundational statistical technique used to make data-driven decisions, validate assumptions, and draw conclusions from data. Here’s how data scientists utilize hypothesis testing:

• Validating Assumptions : Before diving into complex analyses or building predictive models, data scientists often need to verify certain assumptions about the data. Hypothesis testing provides a structured approach to test these assumptions, ensuring that subsequent analyses or models are valid.
• Feature Selection : In machine learning and predictive modeling, data scientists use hypothesis tests to determine which features (or variables) are most relevant or significant in predicting a particular outcome. By testing hypotheses related to feature importance or correlation, they can streamline the modeling process and enhance prediction accuracy.
• A/B Testing : A/B testing is a common technique in marketing, product development, and user experience design. Data scientists employ hypothesis testing to compare two versions (A and B) of a product, feature, or marketing strategy to determine which performs better in terms of a specified metric (e.g., conversion rate, user engagement).
• Research and Exploration : In exploratory data analysis (EDA) or when investigating specific research questions, data scientists formulate hypotheses to test certain relationships or patterns within the data. By conducting hypothesis tests, they can validate these relationships, uncover insights, and drive data-driven decision-making.
• Model Evaluation : After building machine learning or statistical models, data scientists use hypothesis testing to evaluate the model’s performance, assess its predictive power, or compare different models. For instance, hypothesis tests like the t-test or F-test can help determine if a new model significantly outperforms an existing one based on certain metrics.
• Business Decision-making : Beyond technical analyses, data scientists employ hypothesis testing to support business decisions. Whether it’s evaluating the effectiveness of a marketing campaign, assessing customer preferences, or optimizing operational processes, hypothesis testing provides a rigorous framework to validate assumptions and guide strategic initiatives.

Hypothesis Testing Examples and Solutions

Let’s delve into some common examples of hypothesis testing and provide solutions or interpretations for each scenario.

Example: Testing the Mean

Scenario : A coffee shop owner believes that the average waiting time for customers during peak hours is 5 minutes. To test this, the owner takes a random sample of 30 customer waiting times and wants to determine if the average waiting time is indeed 5 minutes.

Hypotheses :

• H 0​ (Null Hypothesis): 5 μ =5 minutes (The average waiting time is 5 minutes)
• H 1​ (Alternative Hypothesis): 5 μ =5 minutes (The average waiting time is not 5 minutes)

Solution : Using a t-test (assuming population variance is unknown), calculate the t-statistic based on the sample mean, sample standard deviation, and sample size. Then, determine the p-value and compare it with a significance level (e.g., 0.05) to decide whether to reject the null hypothesis.

Example: A/B Testing in Marketing

Scenario : An e-commerce company wants to determine if changing the color of a “Buy Now” button from blue to green increases the conversion rate.

• H 0​: Changing the button color does not affect the conversion rate.
• H 1​: Changing the button color affects the conversion rate.

Solution : Split website visitors into two groups: one sees the blue button (control group), and the other sees the green button (test group). Track the conversion rates for both groups over a specified period. Then, use a chi-square test or z-test (for large sample sizes) to determine if there’s a statistically significant difference in conversion rates between the two groups.

Hypothesis Testing Formula

The formula for hypothesis testing typically depends on the type of test (e.g., z-test, t-test, chi-square test) and the nature of the data (e.g., mean, proportion, variance). Below are the basic formulas for some common hypothesis tests:

Z-Test for Population Mean :

Z=(σ/n​)(xˉ−μ0​)​

• ˉ x ˉ = Sample mean
• 0 μ 0​ = Population mean under the null hypothesis
• σ = Population standard deviation
• n = Sample size

T-Test for Population Mean :

t= (s/ n ​ ) ( x ˉ −μ 0 ​ ) ​ 

s = Sample standard deviation 

Chi-Square Test for Goodness of Fit :

χ2=∑Ei​(Oi​−Ei​)2​

• Oi ​ = Observed frequency
• Ei ​ = Expected frequency

Also Read: Full Form of OOPS

Hypothesis Testing Calculator

While you can perform hypothesis testing manually using the above formulas and statistical tables, many online tools and software packages simplify this process. Here’s how you might use a calculator or software:

• Z-Test and T-Test Calculators : These tools typically require you to input sample statistics (like sample mean, population mean, standard deviation, and sample size). Once you input these values, the calculator will provide you with the test statistic (Z or t) and a p-value.
• Chi-Square Calculator : For chi-square tests, you’d input observed and expected frequencies for different categories or groups. The calculator then computes the chi-square statistic and provides a p-value.
• Software Packages (e.g., R, Python with libraries like scipy, or statistical software like SPSS) : These platforms offer more comprehensive tools for hypothesis testing. You can run various tests, get detailed outputs, and even perform advanced analyses, including regression models, ANOVA, and more.

When using any calculator or software, always ensure you understand the underlying assumptions of the test, interpret the results correctly, and consider the broader context of your research or analysis.

Hypothesis Testing FAQs

What are the key components of a hypothesis test.

The key components include: Null Hypothesis (H0): A statement of no effect or no difference. Alternative Hypothesis (H1 or Ha): A statement that contradicts the null hypothesis. Test Statistic: A value computed from the sample data to test the null hypothesis. Significance Level (α): The threshold for rejecting the null hypothesis. P-value: The probability of observing the given data, assuming the null hypothesis is true.

What is the significance level in hypothesis testing?

The significance level (often denoted as α) is the probability threshold used to determine whether to reject the null hypothesis. Commonly used values for α include 0.05, 0.01, and 0.10, representing a 5%, 1%, or 10% chance of rejecting the null hypothesis when it's actually true.

How do I choose between a one-tailed and two-tailed test?

The choice between one-tailed and two-tailed tests depends on your research question and hypothesis. Use a one-tailed test when you're specifically interested in one direction of an effect (e.g., greater than or less than). Use a two-tailed test when you want to determine if there's a significant difference in either direction.

What is a p-value, and how is it interpreted?

The p-value is a probability value that helps determine the strength of evidence against the null hypothesis. A low p-value (typically ≤ 0.05) suggests that the observed data is inconsistent with the null hypothesis, leading to its rejection. Conversely, a high p-value suggests that the data is consistent with the null hypothesis, leading to no rejection.

Can hypothesis testing prove a hypothesis true?

No, hypothesis testing cannot prove a hypothesis true. Instead, it helps assess the likelihood of observing a given set of data under the assumption that the null hypothesis is true. Based on this assessment, you either reject or fail to reject the null hypothesis.

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Hypothesis Testing

Hypothesis testing uses statistics to choose between hypotheses regarding whether data is statistically significant or occurred by chance alone. One type of hypothesis tests are t-tests, which are tests that examine whether two means are statistically significantly different from each other or whether the difference between them simply occurred by chance. A  One-Sample T-Test  compares a sample mean to a known population mean. An  Independent Samples T-Test  compares two sample means from different populations regarding the same variable. A  Paired Samples T-Test compares two sample means from the same population regarding the same variable at two different times such as during a pre-test and post-test, or it compares two sample means from different populations whose members have been matched. 

One-Sample T-Test

A  One-Sample T-Test  compares a sample mean and a known population mean to determine whether the difference between the two means is statistically significant or occurred by chance alone.

This example will be comparing the respondents' number of children with the known 2013  United States fertility rate  of 2.06 children per woman. The One-Sample T-Test is examining whether the difference between the sample mean number of children per respondent is significantly different from the known population fertility rate.

To generate a One-Sample T-Test, click 'Analyze' in the top toolbar of the Data Editor window. Click 'Compare Means' in the dropdown menu, and click 'One-Sample T Test...' in the side menu.

In the One-Sample T Test dialog box that pops up, select the variable of interest (Number of Children, childs) from the list of variables and bring it over to the 'Test Variable(s)' field. Then, enter the known population mean in the 'Test Value:' field. Click 'OK.'

The output is displayed in the SPSS Viewer window. The output consists of two tables. The first table, One-Sample Statistics, contains statistical information about the Number of Children variable, such as N, the Mean, the Standard Deviation, and the Standard Error of the Mean. The second table, One-Sample Test, contains information specific to the One-Sample T-Test, such as the Test Value, the t value, the df (degrees of freedom), the alpha 2-tailed Significance value ( when the Sig. value is .05 or less, the probability that the difference between the sample mean and the test value was due to chance is 5% or less ), the Mean difference (the difference between the sample mean and the test value), and a 95% Confidence Interval of the Difference. In this case, the difference between the sample mean number of children (1.89) and the known population mean number of children (2.06) is significant.

Independent-Samples T-Test

An Independent-Samples T-Test  compares two sample means from different populations regarding the same variable  to determine whether the difference between the two means is statistically significant or occurred by chance alone.

This example will be comparing the mean number of hours spent emailing per week (Email Hours Per Week, emailhr) by married respondents and single respondents (Not Married, absingle). The 'Email Hours Per Week, emailhr' variable is the test variable, and the 'Not Married, absingle' variable is the nominal grouping variable. The Independent-Samples T-Test is examining whether the difference between the mean number of hours married respondents spent emailing and the mean number of hours single respondents spent emailing is significantly different or occurred by chance.

To generate a Independent-Samples T-Test, click 'Analyze' in the top toolbar of the Data Editor window. Click 'Compare Means' in the dropdown menu, and click 'Independent-Samples T Test...' in the side menu.

In the Independent-Samples T Test dialog box that pops up, select the variable of interest (Email Hours Per Week, emailhr) from the list of variables and bring it over to the 'Test Variable(s)' field. Then, enter the nominal grouping variable (Not Married, absingle) in the 'Grouping Variable:' field. Then click 'Define Groups...' to identify the two populations being compared.

In the Define Groups dialog box, select 'Use specifiec values' and enter the Value Labels of the nominal grouped variable. In this example, Group 1 will correspond with Value Label 1, which refers to the respondents who indicated they are not married ('Yes'), and Group 2 will correspond with Value Label 2, which refers to the respondents who indicated they are married ('No'). Then, click 'Continue,' and back in the Independent-Samples T Test dialog box, click OK.

The output is displayed in the SPSS Viewer window. The output consists of two tables. The first table, Group Statistics, contains statistical information about the Email Hours Per Week variable, split by whether the respondent is not married (indicated in the chart by Yes) or is married (indicated in the chart by No). For each group of respondents, the N, the Mean, the Standard Deviation, and the Standard Error of the Mean are displayed.

The second table, Independent-Samples Test, contains information specific to the Independent-Samples T-Test, such as information about Levene’s Test for Equality of Variances and the t-test for Equality of Means. Levene’s Test for Equality of Variances tests whether variability within each group (married or not married) is equal. The outputs displays two sets of results: one set in which equal variance is assumed and one set in which equal variance is not assumed. It is up to the user to determine which set of results is appropriate. You can determine which results are appropriate by looking at the Sig., the alpha level of significance. If the alpha level is greater than .05, then group variances are assumed to be equal. In this example, Sig. is greater than .05 so group variances are assumed to be equal, and we read the top line of the table. The portion of the table dedicated to the t test for Equality of Means displays the t value, the df (degrees of freedom), the 2-tailed Sig. value ( when the Sig. value is .05 or less, the probability that the difference between the two means was due to chance is 5% or less ), the Mean difference (the difference between the two means), and a 95% Confidence Interval of the Difference. In this case, the difference between the two means is not significant and could have occured by chance.

Paired Samples T-Test

A  Paired Samples T-Test  compares two sample means from the same population regarding the same variable at two different times such as during a pre-test and post-test, or it compares two sample means from different populations whose members have been matched,  to determine whether the difference between the two means is statistically significant or occurred by chance alone.

This example will be comparing the respondents' mean number of children (Number of Children, childs) with the respondents' mean ideal number of children (Ideal Number of Children, chldidel). The respondents in this example are paired with themselves. The Paired-Samples T-Test is examining whether the difference between the mean number of children and the mean ideal number of children is significantly different or occurred by chance.

To generate a Paired-Samples T-Test, click 'Analyze' in the top toolbar of the Data Editor window. Click 'Compare Means' in the dropdown menu, and click 'Paired-Samples T Test...' in the side menu.

In the Paired-Samples T Test dialog box that pops up, select the variables of interest (Number of Children, childs, and Ideal Number of Children, chldidel) from the list of variables and bring them over to the 'Paired Variables:' field. Then, click 'OK.'

The output is displayed in the SPSS Viewer window. The output consists of three tables. The first table, Paired Samples Statistics, contains statistical information about the Number of Children and Ideal Number of Children variables. For each variable, the Mean, the N, the Standard Deviation, and the Standard Error of the Mean are displayed. The second table, Paired Samples Correlations, contains a correlation value measuring how closely related the two variables are to each other. The correlation value is  the  correlation coefficient  of the two variables and measures the strength and direction of the linear relationship between the two variables. Specifically,  the closer the correlation value to 1 or -1, the more strongly linearly related the variables. In this example, the correlation value is not close to 1 so the variables do not have a strong linear relationship.

The third table, Paired Samples Test, displays the Mean (referring to the difference between the two means), the Standard Deviation of the Mean, the Standard Error of the Mean, and a 95% Confidence Interval of the Difference. The table also displays the t value, the df (degrees of freedom), and the 2-tailed Sig. value ( when the Sig. value is .05 or less, the probability that the difference between the two means was due to chance is 5% or less ). In this case, the difference between the two means is significant.

ANOVA Tests

Hypothesis Testing uses statistics to choose between hypotheses regarding whether data is statistically significant or occurred by chance alone. One type of hypothesis tests are ANOVA tests, which are tests that examine whether two or more means are statistically significantly different from each other or whether the difference between them simply occurred by chance. ANOVA stands for Analysis of Variance. A One-Way ANOVA compares the means of two or more groups. A Factorial ANOVA compares the means of two or more groups while examining the interaction of and between two independent variables. (However, the ANOVA tests do not specify which groups differ significantly, and since there are more than two groups, in order to determine which groups differ, further statistical analyses and Post Hoc tests must be done and can be added to the ANOVA procedure in SPSS.)

One-Way ANOVA

A One-Way ANOVA compares the means of two or more groups. A One-Way ANOVA thus requires one categorical variable consisting of two or more groups, serving as the independent variable, and one continuous variable, serving as the dependent variable.

In this example, the variable 'Subjective Class Identification, class' will be serving as the categorical variable with 4 groups, and the variable 'Number of College-Level Sci Courses R Have Taken, colscinm' will be serving as the continuous variable. The One-Way ANOVA is specifically looking at whether respondents of different subjective class identifications differ significantly in the mean number of college-level science classes taken.

To generate a One-Way ANOVA, click 'Analyze' in the top toolbar of the Data Editor window. Click 'Compare Means' in the dropdown menu, and click 'One-Way ANOVA...' in the side menu.

In the One-Way ANOVA dialog box that pops up, select the dependent variable of interest (Number of College-Level Sci Courses R Have Taken, colscinm) from the list of variables and bring it over to the 'Dependent List:' field. Then, select the nominal grouping variable of interest (Subjective Class Identification, class) from the list of variables and bring it over to the 'Factor' field. To include Post Hoc tests in the ANOVA output, click 'Post Hoc...'

In the One-Way ANOVA: Post Hoc Multiple Comparisons dialog box that pops up, select the desired Post Hoc test. In this example, we will be using Least Significant Difference (LSD) tests. Then, click 'Continue.'

Back in the One-Way ANOVA dialog box, click 'Options' if you would like to add any other statistics to the ANOVA output. Here, we selected to include a Descriptives table and a Means plot in the output. Click 'Continue.' Then, back in the One-Way ANOVA dialog box, click 'OK.'

The output is displayed in the SPSS Viewer window. The output consists of four parts. The first table, Descriptives, contains statistical information about the dependent variable, Number of College-Level Sci Courses R Have Taken, split by the independent variable groupings, the respondents' Subjective Class Identification. For each group of respondents, the N, the Mean, the Standard Deviation, the Standard Error of the Mean, a 95% confidence Interval for the Mean, the Minimum value, and the Maximum value are displayed. The second table, ANOVA, contains information about the ANOVA test comparing means both between and within groups and includes the Sum of Squares (a measure of variance), df (degrees of freedom), Mean Square, the F value, and the Sig. value (when the Sig. value is .05 or less, the probability that the difference between the groups was due to chance is 5% or less). In this case, some of the means are significantly different from each other. However, the results of the ANOVA alone do not indicate which groups differ significantly. Thus, the Multiple Comparisons Output which displays the results of the Post Hoc LSD test is necessary.

The next table, Multiple Comparisons Output, displays the results of the LSD test. The LSD test compares each group (class category) to all other groups (class categories). Thus, please note that this table displays some comparisons more than once, since, in every row, each group is compared to all other groups. Each comparison is denoted by a differnet color, and lines of the same color represent repeated comparisons. For each comparison, the table displays the Mean Difference (the difference between the groups' mean number of college-level science classes taken), the Standard Error, the Sig. value (when the Sig. value is .05 or less, the probability that the difference between the groups was due to chance is 5% or less), and a 95% Confidence Interval for the Mean Difference. In this example, the Upper Class differs significantly from the Lower, Working, and Middle Classes, and the Middle Class differs significantly from the Working Class in the number of college-level science courses taken.

The last section of the output, the Means Plot, is a graphical display of how the mean number of college-level science courses the respondents have taken depends on subjective class identification.

Factorial ANOVA

A Factorial ANOVA compares the means of two or more groups while examining the interaction of and between two independent variables. A Factorial ANOVA thus requires one continuous variable to serve as the dependent variable and more than one categorical variable (each consisting of two or more groups) to serve as the independent variables.

In this example, the variable 'Subjective Class Identification, class' will be serving as the first categorical variable with 4 groups, and the variable 'Not Married, absingle' will be serving as the second categorical variable with 2 groups. The variable 'Hours Per Day Watching TV, tvhours' will serve as the continuous variable. The Factorial ANOVA is specifically looking at whether respondents of different subjective class identifications and of different marital statuses differ significantly in the mean number hours spent watching tv per day.

To generate a Factorial ANOVA, click 'Analyze' in the top toolbar of the Data Editor window. Click 'General Linear Model' in the dropdown menu, and click 'Univariate...' in the side menu.

In the Univariate dialog box that pops up, select the dependent variable of interest (Hours Per Day Watching TV, tvhours) from the list of variables and bring it over to the 'Dependent Variable:' field. Then, select the nominal independent grouping variables of interest (Subjective Class Identification, class, and Not Married, absingle) from the list of variables and bring them over to the 'Fixed Factor(s):' field. To include Post Hoc tests in the ANOVA output, click 'Post Hoc...'

In the Univariate: Post Hoc Multiple Comparisons for Observed Means dialog box that pops up, select the desired variable for which you wish to run a Post Hoc Test (any variable with more than two groups, which, in this case, is the 'class' variable). In this example, we will be using Least Significant Difference (LSD) tests. Then, click 'Continue.'

Back in the Univariate dialog box, click 'Options' if you would like to add any other statistics to the ANOVA output. Here, we selected to include a Descriptives table in the output. Click 'Continue.' Then, back in the One-Way ANOVA dialog box, click 'OK.'

The output is displayed in the SPSS Viewer window. The output consists of four parts. The first table, Between-Subjects Factors, lists the independent variables Value Labels and N, the group size, for each group. The second table, Descriptive Statistics, contains statistical information about the dependent variable, Hours Per Day Watching TV, displayed corresponding with each of the indepedent variable groups: the respondents' Subjective Class Identification and marital status. For each group of respondents, the Mean, the Standard Deviation, and the N are displayed.

The third table, Tests of Between-Subjects Effects, contains information about the Factorial ANOVA test and includes the Sum of Squares (a measure of variance), df (degrees of freedom), Mean Square, the F value, and the Sig. (when the Sig. value is .05 or less, the probability that the difference between the means was due to chance is 5% or less). The first two rows of the table, Corrected Model and Intercept are advanced statistics and will not be addressed in this tutorial. The next rows correspond with the independent variables, and their significance levels are measured separately, demonstrating the influence of each independent variable individually on the number of hours per day spent watching TV. There is also a row showing the interaction between the two independent variables, demonstrating whether the two independent variables interacted to significantly impact the results. In this case, class influenced the number of hours respondents spent watching TV but marital status did not. Additionally, there was a significant interaction between class and marital status.

The next table, Multiple Comparisons, displays the results of the LSD test. The results of the ANOVA alone did not indicate which class groups differ significantly. Thus, the Multiple Comparisons table, which displays the results of the Post Hoc LSD test, is necessary. The LSD test compares each group (class category) to all other groups (class categories). Thus, please note that this table displays some comparisons more than once, since, in every row, each group is compared to all other groups. Each comparison is denoted by a differnet color, and lines of the same color represent repeated comparisons. For each comparison, the table displays the Mean Difference (the difference between the groups' mean number of hours spent watching TV), the Standard Error, the Sig. value (when the Sig. value is .05 or less, the probability that the difference between the means was due to chance is 5% or less), and a 95% Confidence Interval for the Mean Difference. In this example, the Lower Class differs significantly from the Working, Middle, and Upper Classes in the amount of time spent watching TV per day.

Chi-Square Tests

Hypothesis Testing uses statistics to choose between hypotheses regarding whether data is statistically significant or occurred by chance alone. One type of hypothesis tests are Chi-Square tests, which are tests that examine whether the frequency of certain categorical values, such as the number of individuals in a group, differs from the frequency distribution expected from random chance alone. The Chi-Square Goodness of Fit Test does this for one variable at a time, and the Test of Independence: Pearson's Chi-Square does this as well but can also test whether multiple categorical variables are significantly associated.

Goodness of Fit Test

A Chi-Square Goodness of Fit Test examines whether the frequency of certain categorical values differs from the frequency distribution expected from random chance alone. The Chi-Square Goodness of Fit Test does this for one variable at a time and requires a categorical variable consisting of two or more groups as input.

In this example, the variable 'Respondents Astrological Sign, zodiac' will be serving as the categorical variable with 12 groups. The Chi-Square Goodness of Fit Test is specifically looking at whether the respondents' astrological signs differ significantly in distribution from one that is expected by random chance. (This example is operating under the assumption that the frequency distribution expected from random chance alone should have all of the categories equally represented with an equal number of people with each zodiac.)

To generate a Chi-Square Goodness of Fit Test, click 'Analyze' in the top toolbar of the Data Editor window. Click 'Nonparametric Tests' in the dropdown menu, and click 'Legacy Dialogs...' in the first side menu and 'Chi Square' in the second side menu.

In the Chi Square dialog box that pops up, select the variable of interest (Respondents Astrological Sign, zodiac) from the list of variables and bring it over to the 'Test Variable List:' field. Then, in the 'Expected Values' field, select 'All categories equal' if all groups would be expected to have equal frequencies, or enter an exact value if a specific frequency is expected. In this case, we selected 'All categories equal' because we are operating under the assumption that the frequency distribution expected from random chance alone should have all of the categories equally represented with an equal number of people with each zodiac. Then, click 'OK.'

The output is displayed in the SPSS Viewer window. The output consists of two tables. The first table, Respondents Astrological Sign, displays the Observed N, the Expected N, and the Residual (the difference between the Observed N and the Expected N) for each of the categories within the categorical variable (for each of the astrological signs). The second table, Test Statistics, displays information specific to the Chi-Square Goodness of Fit Test, including the Chi-Square Value, the df (degrees of freedom), and Asymp. Sig. (when the Sig. value is .05 or less, the probability that the difference between the Observed N and Expected N value was due to chance is 5% or less). In this case, the Sig. level is above .05, so the difference between the Observed and Expected N values is not significant and the sample is an accurate representation of the population.

Test of Independence: Pearson's Chi-Square

Test of Independence: Pearson's Chi Square examines whether the frequency of certain categorical values differs from the frequency distribution expected from random chance alone. The Test of Independence: Pearson's Chi Square does this for one variable at a time but it also tests whether the multiple categorical variables are significantly associated. It requires two categorical variables consisting of two or more groups as input, .

In this example, the variable 'Subjective Class Identification, class' with 4 groups, and the variable 'R Has Given Money To A Charity, givchrty' with 6 groups will be serving as the categorical variables. The Test of Independence: Pearson's Chi Square is specifically looking at whether the two variables are significantly associated, with the distribution of the respondents' subjective class identification and frequency of giving money to a charity differing from the distribution expected by random chance.

To generate a Test of Independence: Pearson's Chi Square click 'Analyze' in the top toolbar of the Data Editor window. Click 'Descriptive Statistics' in the dropdown menu, and click Cross Tabs...' in the side menu.

In the Crosstabs dialog box that pops up, select one of the variables of interest (R Has Given Money To A Charity, givchrty) from the list of variables and bring it over to the 'Row(s):' field, and select the othe variable of interest (Subjective Class Identification, class) from the list of variables and bring it over to the 'Column(s):' field. (The placement of the individual variables into the Row and Column fields is arbitrary.) Then, click 'Statistics...'

In the Crosstabs: Statistics dialog box, click 'Chi Square' and click 'Continue.'

Then, back in the the Crosstabs dialog box, click 'Cells...' In the cells dialog box, in the 'Counts' field, click to select both Observed and Expected, and in the 'Percents' field, select the desired percent values you would like to be displayed in the output. Additionally, select the desired residuals output in the residuals field. Then, click 'Continue, and back in the Crosstabs dialog box, click 'OK.'

The output is displayed in the SPSS Viewer window. The output consists of three tables. The first table, Case Processing Summary, displays the number N and percent of valid cases, missing cases, and total cases in the Chi-Square analysis.

The second table, the Crosstabulation matrix, presents the counts, expected counts, and the adjusted residual (the standardized difference between the observed values and the expected values), as well as percents for each group. 

The third table, Chi-Square Tests,  presents the df (degrees of freedom), Asymp. Sig. (2-sided) for the Pearson Chi-Square value as well as other statistics. If the level of significance for the Pearson Chi-Square statistic is above .05, the probability that the difference between the Observed N and Expected N value was due to chance is 5% or less. In this case, the  Sig. value is below .05. However, the output does not explicitly indicate how the relationship exhibited in the sample data differs from a random expected pattern. The adjusted residuals in the above chart are important in determining how the relationship exhibited in the sample data differs from that of a random pattern: When the adjusted residual is greater than or equal to 1.96 (or less than or equal to -1.96), then the observed frequency value is significantly different from the expected frequency. Furthermore, if the adjusted residual value is positive, then the group is over-represented, and if the value is negative, then the group is under-represented. For example, the lower class giving money to charity once a month, a group which has an adjusted residual of -2.6, is significantly under-represented.

Next: Regressions and Correlations

Previous: Frequency Analysis

Which Statistical Test Should I Use?

• Univariate Tests
• Within-Subjects Tests
• Between-Subjects Tests
• Association Measures
• Prediction Analyses
• Classification Analyses

Finding the appropriate statistical test is easy if you're aware of

• the basic type of test you're looking for and
• the measurement levels of the variables involved.

For each type and measurement level, this tutorial immediately points out the right statistical test. We'll also briefly define the 6 basic types of tests and illustrate them with simple examples.

1. Overview Univariate Tests

Univariate Tests - Quick Definition

Univariate tests are tests that involve only 1 variable. Univariate tests either test if

• some population parameter -usually a mean or median - is equal to some hypothesized value or
• some population distribution is equal to some function, often the normal distribution .

A textbook example is a one sample t-test : it tests if a population mean -a parameter- is equal to some value x . This test involves only 1 variable (even if there's many more in your data file).

2. Overview Within-Subjects Tests

Within-Subjects Tests - Quick Definition

Within-subjects tests compare 2+ variables measured on the same subjects (often people). An example is repeated measures ANOVA : it tests if 3+ variables measured on the same subjects have equal population means.

Within-subjects tests are also known as

• paired samples tests (as in a paired samples t-test ) or
• related samples tests.

3. Overview Between-Subjects Tests

Between-Subjects Tests - Quick Definition

Between-subjects tests examine if 2+ subpopulations are identical with regard to

• a parameter (population mean, standard deviation or proportion) or
• a distribution .

The best known example is a one-way ANOVA as illustrated below. Note that the subpopulations are represented by subsamples -groups of observations indicated by some categorical variable.

“Between-subjects” tests are also known as “ independent samples ” tests, such as the independent samples t-test . “Independent samples” means that subsamples don't overlap: each observation belongs to only 1 subsample.

4. Overview Association Measures

Association Measures - Quick Definition

Association measures are numbers that indicate to what extent 2 variables are associated. The best known association measure is the Pearson correlation : a number that tells us to what extent 2 quantitative variables are linearly related. The illustration below visualizes correlations as scatterplots.

5. Overview Prediction Analyses

Prediction Analyses - Quick Definition

Prediction tests examine how and to what extent a variable can be predicted from 1+ other variables. The simplest example is simple linear regression as illustrated below.

Prediction analyses sometimes quietly assume causality : whatever predicts some variable is often thought to affect this variable. Depending on the contents of an analysis, causality may or may not be plausible. Keep in mind, however, that the analyses listed below don't prove causality.

6. Classification Analyses

Classification analyses attempt to identify and describe groups of observations or variables. The 2 main types of classification analysis are

• factor analysis for finding groups of variables (“factors”) and
• cluster analysis for finding groups of observations (“clusters”).

Factor analysis is based on correlations or covariances . Groups of variables that correlate strongly are assumed to measure similar underlying factors -sometimes called “constructs”. The basic idea is illustrated below.

Cluster analysis is based on distances among observations -often people. Groups of observations with small distances among them are assumed to represent clusters such as market segments.

Right. So that'll do for a basic overview. Hope you found this guide helpful! And last but not least,

thanks for reading!

Tell us what you think!

This tutorial has 18 comments:.

By Aidarous on October 12th, 2022

By ozman on July 29th, 2023

I enjoy reading your tutorials and always look forward to the latest instalment.

I have one small (trivial) comment on the tutorial titled "which statistical test...".

In the last 2 tables (Factor, Cluster) you have "Madelyn" listed twice as a first name. I realize that there may be two Madelyns with different last names, but in the absence of information on last name, one concludes that this is the same person (a repeated measure), which may cause some confusion for the reader. Perhaps a different first name is warranted.

Thanks and keep up the good work.

By Ruben Geert van den Berg on July 29th, 2023

Thanks for your suggestion, I'll implement it in the next update.

These are simulated data but "Madelyn" appearing twice is indeed somewhat unfortunate.

Great catch, really. If all analysts had this level of "eye for detail", there wouldn't be much of a replication crisis in the social sciences.

Kind regards from Amsterdam!

Ruben SPSS tutorials

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Statistical Methods and Data Analytics

What statistical analysis should I use? Statistical analyses using SPSS

Introduction.

This page shows how to perform a number of statistical tests using SPSS.  Each section gives a brief description of the aim of the statistical test, when it is used, an example showing the SPSS commands and SPSS (often abbreviated) output with a brief interpretation of the output. You can see the page Choosing the Correct Statistical Test for a table that shows an overview of when each test is appropriate to use.  In deciding which test is appropriate to use, it is important to consider the type of variables that you have (i.e., whether your variables are categorical, ordinal or interval and whether they are normally distributed), see What is the difference between categorical, ordinal and interval variables? for more information on this.

About the hsb data file

Most of the examples in this page will use a data file called hsb2, high school and beyond.  This data file contains 200 observations from a sample of high school students with demographic information about the students, such as their gender ( female ), socio-economic status ( ses ) and ethnic background ( race ). It also contains a number of scores on standardized tests, including tests of reading ( read ), writing ( write ), mathematics ( math ) and social studies ( socst ). You can get the hsb data file by clicking on hsb2 .

One sample t-test

A one sample t-test allows us to test whether a sample mean (of a normally distributed interval variable) significantly differs from a hypothesized value.  For example, using the hsb2 data file , say we wish to test whether the average writing score ( write ) differs significantly from 50.  We can do this as shown below. t-test  /testval = 50  /variable = write. The mean of the variable write for this particular sample of students is 52.775, which is statistically significantly different from the test value of 50.  We would conclude that this group of students has a significantly higher mean on the writing test than 50.

One sample median test

A one sample median test allows us to test whether a sample median differs significantly from a hypothesized value.  We will use the same variable, write , as we did in the one sample t-test example above, but we do not need to assume that it is interval and normally distributed (we only need to assume that write is an ordinal variable). nptests /onesample test (write) wilcoxon(testvalue = 50).

Binomial test

A one sample binomial test allows us to test whether the proportion of successes on a two-level categorical dependent variable significantly differs from a hypothesized value.  For example, using the hsb2 data file , say we wish to test whether the proportion of females ( female ) differs significantly from 50%, i.e., from .5.  We can do this as shown below. npar tests  /binomial (.5) = female. The results indicate that there is no statistically significant difference (p = .229).  In other words, the proportion of females in this sample does not significantly differ from the hypothesized value of 50%.

Chi-square goodness of fit

A chi-square goodness of fit test allows us to test whether the observed proportions for a categorical variable differ from hypothesized proportions.  For example, let’s suppose that we believe that the general population consists of 10% Hispanic, 10% Asian, 10% African American and 70% White folks.  We want to test whether the observed proportions from our sample differ significantly from these hypothesized proportions. npar test   /chisquare = race  /expected = 10 10 10 70. These results show that racial composition in our sample does not differ significantly from the hypothesized values that we supplied (chi-square with three degrees of freedom = 5.029, p = .170).

Two independent samples t-test

An independent samples t-test is used when you want to compare the means of a normally distributed interval dependent variable for two independent groups.  For example, using the hsb2 data file , say we wish to test whether the mean for write is the same for males and females. t-test groups = female(0 1)   /variables = write. Because the standard deviations for the two groups are similar (10.3 and 8.1), we will use the “equal variances assumed” test.  The results indicate that there is a statistically significant difference between the mean writing score for males and females (t = -3.734, p = .000).  In other words, females have a statistically significantly higher mean score on writing (54.99) than males (50.12). See also SPSS Learning Module: An overview of statistical tests in SPSS

Wilcoxon-Mann-Whitney test

The Wilcoxon-Mann-Whitney test is a non-parametric analog to the independent samples t-test and can be used when you do not assume that the dependent variable is a normally distributed interval variable (you only assume that the variable is at least ordinal).  You will notice that the SPSS syntax for the Wilcoxon-Mann-Whitney test is almost identical to that of the independent samples t-test.  We will use the same data file (the hsb2 data file ) and the same variables in this example as we did in the independent t-test example above and will not assume that write , our dependent variable, is normally distributed.

npar test /m-w = write by female(0 1). The results suggest that there is a statistically significant difference between the underlying distributions of the write scores of males and the write scores of females (z = -3.329, p = 0.001). See also FAQ: Why is the Mann-Whitney significant when the medians are equal?

Chi-square test

A chi-square test is used when you want to see if there is a relationship between two categorical variables.  In SPSS, the chisq option is used on the statistics subcommand of the crosstabs command to obtain the test statistic and its associated p-value.  Using the hsb2 data file , let’s see if there is a relationship between the type of school attended ( schtyp ) and students’ gender ( female ).  Remember that the chi-square test assumes that the expected value for each cell is five or higher. This assumption is easily met in the examples below.  However, if this assumption is not met in your data, please see the section on Fisher’s exact test below. crosstabs /tables = schtyp by female /statistic = chisq. These results indicate that there is no statistically significant relationship between the type of school attended and gender (chi-square with one degree of freedom = 0.047, p = 0.828). Let’s look at another example, this time looking at the linear relationship between gender ( female ) and socio-economic status ( ses ).  The point of this example is that one (or both) variables may have more than two levels, and that the variables do not have to have the same number of levels.  In this example, female has two levels (male and female) and ses has three levels (low, medium and high). crosstabs /tables = female by ses /statistic = chisq. Again we find that there is no statistically significant relationship between the variables (chi-square with two degrees of freedom = 4.577, p = 0.101). See also SPSS Learning Module: An Overview of Statistical Tests in SPSS

Fisher’s exact test

The Fisher’s exact test is used when you want to conduct a chi-square test but one or more of your cells has an expected frequency of five or less.  Remember that the chi-square test assumes that each cell has an expected frequency of five or more, but the Fisher’s exact test has no such assumption and can be used regardless of how small the expected frequency is. In SPSS unless you have the SPSS Exact Test Module, you can only perform a Fisher’s exact test on a 2×2 table, and these results are presented by default.  Please see the results from the chi squared example above.

One-way ANOVA

A one-way analysis of variance (ANOVA) is used when you have a categorical independent variable (with two or more categories) and a normally distributed interval dependent variable and you wish to test for differences in the means of the dependent variable broken down by the levels of the independent variable.  For example, using the hsb2 data file , say we wish to test whether the mean of write differs between the three program types ( prog ).  The command for this test would be: oneway write by prog. The mean of the dependent variable differs significantly among the levels of program type.  However, we do not know if the difference is between only two of the levels or all three of the levels.  (The F test for the Model is the same as the F test for prog because prog was the only variable entered into the model.  If other variables had also been entered, the F test for the Model would have been different from prog .)  To see the mean of write for each level of program type, means tables = write by prog. From this we can see that the students in the academic program have the highest mean writing score, while students in the vocational program have the lowest. See also SPSS Textbook Examples: Design and Analysis, Chapter 7 SPSS Textbook Examples: Applied Regression Analysis, Chapter 8 SPSS FAQ: How can I do ANOVA contrasts in SPSS? SPSS Library: Understanding and Interpreting Parameter Estimates in Regression and ANOVA

Kruskal Wallis test

The Kruskal Wallis test is used when you have one independent variable with two or more levels and an ordinal dependent variable. In other words, it is the non-parametric version of ANOVA and a generalized form of the Mann-Whitney test method since it permits two or more groups.  We will use the same data file as the one way ANOVA example above (the hsb2 data file ) and the same variables as in the example above, but we will not assume that write is a normally distributed interval variable. npar tests /k-w = write by prog (1,3). If some of the scores receive tied ranks, then a correction factor is used, yielding a slightly different value of chi-squared.  With or without ties, the results indicate that there is a statistically significant difference among the three type of programs.

Paired t-test

A paired (samples) t-test is used when you have two related observations (i.e., two observations per subject) and you want to see if the means on these two normally distributed interval variables differ from one another.  For example, using the hsb2 data file we will test whether the mean of read is equal to the mean of write . t-test pairs = read with write (paired). These results indicate that the mean of read is not statistically significantly different from the mean of write (t = -0.867, p = 0.387).

Wilcoxon signed rank sum test

The Wilcoxon signed rank sum test is the non-parametric version of a paired samples t-test.  You use the Wilcoxon signed rank sum test when you do not wish to assume that the difference between the two variables is interval and normally distributed (but you do assume the difference is ordinal). We will use the same example as above, but we will not assume that the difference between read and write is interval and normally distributed. npar test /wilcoxon = write with read (paired). The results suggest that there is not a statistically significant difference between read and write . If you believe the differences between read and write were not ordinal but could merely be classified as positive and negative, then you may want to consider a sign test in lieu of sign rank test.  Again, we will use the same variables in this example and assume that this difference is not ordinal. npar test /sign = read with write (paired). We conclude that no statistically significant difference was found (p=.556).

McNemar test

You would perform McNemar’s test if you were interested in the marginal frequencies of two binary outcomes. These binary outcomes may be the same outcome variable on matched pairs (like a case-control study) or two outcome variables from a single group.  Continuing with the hsb2 dataset used in several above examples, let us create two binary outcomes in our dataset: himath and hiread . These outcomes can be considered in a two-way contingency table.  The null hypothesis is that the proportion of students in the himath group is the same as the proportion of students in hiread group (i.e., that the contingency table is symmetric). compute himath = (math>60). compute hiread = (read>60). execute. crosstabs /tables=himath BY hiread /statistic=mcnemar /cells=count. McNemar’s chi-square statistic suggests that there is not a statistically significant difference in the proportion of students in the himath group and the proportion of students in the hiread group.

One-way repeated measures ANOVA

You would perform a one-way repeated measures analysis of variance if you had one categorical independent variable and a normally distributed interval dependent variable that was repeated at least twice for each subject.  This is the equivalent of the paired samples t-test, but allows for two or more levels of the categorical variable. This tests whether the mean of the dependent variable differs by the categorical variable.  We have an example data set called rb4wide , which is used in Kirk’s book Experimental Design.  In this data set, y is the dependent variable, a is the repeated measure and s is the variable that indicates the subject number. glm y1 y2 y3 y4 /wsfactor a(4). You will notice that this output gives four different p-values.  The output labeled “sphericity assumed”  is the p-value (0.000) that you would get if you assumed compound symmetry in the variance-covariance matrix.  Because that assumption is often not valid, the three other p-values offer various corrections (the Huynh-Feldt, H-F, Greenhouse-Geisser, G-G and Lower-bound).  No matter which p-value you use, our results indicate that we have a statistically significant effect of a at the .05 level. See also SPSS Textbook Examples from Design and Analysis: Chapter 16 SPSS Library: Advanced Issues in Using and Understanding SPSS MANOVA SPSS Code Fragment: Repeated Measures ANOVA

Repeated measures logistic regression

If you have a binary outcome measured repeatedly for each subject and you wish to run a logistic regression that accounts for the effect of multiple measures from single subjects, you can perform a repeated measures logistic regression.  In SPSS, this can be done using the GENLIN command and indicating binomial as the probability distribution and logit as the link function to be used in the model. The exercise data file contains 3 pulse measurements from each of 30 people assigned to 2 different diet regiments and 3 different exercise regiments. If we define a “high” pulse as being over 100, we can then predict the probability of a high pulse using diet regiment. GET FILE='C:mydatahttps://stats.idre.ucla.edu/wp-content/uploads/2016/02/exercise.sav'. GENLIN highpulse (REFERENCE=LAST) BY diet (order = DESCENDING) /MODEL diet DISTRIBUTION=BINOMIAL LINK=LOGIT /REPEATED SUBJECT=id CORRTYPE = EXCHANGEABLE. These results indicate that diet is not statistically significant (Wald Chi-Square = 1.562, p = 0.211).

Factorial ANOVA

A factorial ANOVA has two or more categorical independent variables (either with or without the interactions) and a single normally distributed interval dependent variable.  For example, using the hsb2 data file we will look at writing scores ( write ) as the dependent variable and gender ( female ) and socio-economic status ( ses ) as independent variables, and we will include an interaction of female by ses .  Note that in SPSS, you do not need to have the interaction term(s) in your data set.  Rather, you can have SPSS create it/them temporarily by placing an asterisk between the variables that will make up the interaction term(s). glm write by female ses. These results indicate that the overall model is statistically significant (F = 5.666, p = 0.00).  The variables female and ses are also statistically significant (F = 16.595, p = 0.000 and F = 6.611, p = 0.002, respectively).  However, that interaction between female and ses is not statistically significant (F = 0.133, p = 0.875). See also SPSS Textbook Examples from Design and Analysis: Chapter 10 SPSS FAQ: How can I do tests of simple main effects in SPSS? SPSS FAQ: How do I plot ANOVA cell means in SPSS? SPSS Library: An Overview of SPSS GLM

Friedman test

You perform a Friedman test when you have one within-subjects independent variable with two or more levels and a dependent variable that is not interval and normally distributed (but at least ordinal).  We will use this test to determine if there is a difference in the reading, writing and math scores.  The null hypothesis in this test is that the distribution of the ranks of each type of score (i.e., reading, writing and math) are the same.  To conduct a Friedman test, the data need to be in a long format.  SPSS handles this for you, but in other statistical packages you will have to reshape the data before you can conduct this test. npar tests /friedman = read write math. Friedman’s chi-square has a value of 0.645 and a p-value of 0.724 and is not statistically significant.  Hence, there is no evidence that the distributions of the three types of scores are different.

Ordered logistic regression

Ordered logistic regression is used when the dependent variable is ordered, but not continuous.  For example, using the hsb2 data file we will create an ordered variable called write3 .  This variable will have the values 1, 2 and 3, indicating a low, medium or high writing score.  We do not generally recommend categorizing a continuous variable in this way; we are simply creating a variable to use for this example.  We will use gender ( female ), reading score ( read ) and social studies score ( socst ) as predictor variables in this model.  We will use a logit link and on the print subcommand we have requested the parameter estimates, the (model) summary statistics and the test of the parallel lines assumption. if write ge 30 and write le 48 write3 = 1. if write ge 49 and write le 57 write3 = 2. if write ge 58 and write le 70 write3 = 3. execute. plum write3 with female read socst /link = logit /print = parameter summary tparallel. The results indicate that the overall model is statistically significant (p < .000), as are each of the predictor variables (p < .000).  There are two thresholds for this model because there are three levels of the outcome variable.  We also see that the test of the proportional odds assumption is non-significant (p = .563).  One of the assumptions underlying ordinal logistic (and ordinal probit) regression is that the relationship between each pair of outcome groups is the same.  In other words, ordinal logistic regression assumes that the coefficients that describe the relationship between, say, the lowest versus all higher categories of the response variable are the same as those that describe the relationship between the next lowest category and all higher categories, etc.  This is called the proportional odds assumption or the parallel regression assumption.  Because the relationship between all pairs of groups is the same, there is only one set of coefficients (only one model).  If this was not the case, we would need different models (such as a generalized ordered logit model) to describe the relationship between each pair of outcome groups. See also SPSS Data Analysis Examples: Ordered logistic regression SPSS Annotated Output:  Ordinal Logistic Regression

Factorial logistic regression

A factorial logistic regression is used when you have two or more categorical independent variables but a dichotomous dependent variable.  For example, using the hsb2 data file we will use female as our dependent variable, because it is the only dichotomous variable in our data set; certainly not because it common practice to use gender as an outcome variable.  We will use type of program ( prog ) and school type ( schtyp ) as our predictor variables.  Because prog is a categorical variable (it has three levels), we need to create dummy codes for it. SPSS will do this for you by making dummy codes for all variables listed after the keyword with .  SPSS will also create the interaction term; simply list the two variables that will make up the interaction separated by the keyword by . logistic regression female with prog schtyp prog by schtyp /contrast(prog) = indicator(1). The results indicate that the overall model is not statistically significant (LR chi2 = 3.147, p = 0.677).  Furthermore, none of the coefficients are statistically significant either.  This shows that the overall effect of prog is not significant. See also Annotated output for logistic regression

Correlation

A correlation is useful when you want to see the relationship between two (or more) normally distributed interval variables.  For example, using the hsb2 data file we can run a correlation between two continuous variables, read and write . correlations /variables = read write. In the second example, we will run a correlation between a dichotomous variable, female , and a continuous variable, write . Although it is assumed that the variables are interval and normally distributed, we can include dummy variables when performing correlations. correlations /variables = female write. In the first example above, we see that the correlation between read and write is 0.597.  By squaring the correlation and then multiplying by 100, you can determine what percentage of the variability is shared.  Let’s round 0.597 to be 0.6, which when squared would be .36, multiplied by 100 would be 36%.  Hence read shares about 36% of its variability with write .  In the output for the second example, we can see the correlation between write and female is 0.256.  Squaring this number yields .065536, meaning that female shares approximately 6.5% of its variability with write . See also Annotated output for correlation SPSS Learning Module: An Overview of Statistical Tests in SPSS SPSS FAQ: How can I analyze my data by categories? Missing Data in SPSS

Simple linear regression

Simple linear regression allows us to look at the linear relationship between one normally distributed interval predictor and one normally distributed interval outcome variable.  For example, using the hsb2 data file , say we wish to look at the relationship between writing scores ( write ) and reading scores ( read ); in other words, predicting write from read . regression variables = write read /dependent = write /method = enter. We see that the relationship between write and read is positive (.552) and based on the t-value (10.47) and p-value (0.000), we would conclude this relationship is statistically significant.  Hence, we would say there is a statistically significant positive linear relationship between reading and writing. See also Regression With SPSS: Chapter 1 – Simple and Multiple Regression Annotated output for regression SPSS Textbook Examples: Introduction to the Practice of Statistics, Chapter 10 SPSS Textbook Examples: Regression with Graphics, Chapter 2 SPSS Textbook Examples: Applied Regression Analysis, Chapter 5

Non-parametric correlation

A Spearman correlation is used when one or both of the variables are not assumed to be normally distributed and interval (but are assumed to be ordinal). The values of the variables are converted in ranks and then correlated.  In our example, we will look for a relationship between read and write .  We will not assume that both of these variables are normal and interval. nonpar corr /variables = read write /print = spearman. The results suggest that the relationship between read and write (rho = 0.617, p = 0.000) is statistically significant.

Simple logistic regression

Logistic regression assumes that the outcome variable is binary (i.e., coded as 0 and 1).  We have only one variable in the hsb2 data file that is coded 0 and 1, and that is female .  We understand that female is a silly outcome variable (it would make more sense to use it as a predictor variable), but we can use female as the outcome variable to illustrate how the code for this command is structured and how to interpret the output.  The first variable listed after the logistic command is the outcome (or dependent) variable, and all of the rest of the variables are predictor (or independent) variables.  In our example, female will be the outcome variable, and read will be the predictor variable.  As with OLS regression, the predictor variables must be either dichotomous or continuous; they cannot be categorical. logistic regression female with read. The results indicate that reading score ( read ) is not a statistically significant predictor of gender (i.e., being female), Wald = .562, p = 0.453. Likewise, the test of the overall model is not statistically significant, LR chi-squared – 0.56, p = 0.453. See also Annotated output for logistic regression SPSS Library: What kind of contrasts are these?

Multiple regression

Multiple regression is very similar to simple regression, except that in multiple regression you have more than one predictor variable in the equation.  For example, using the hsb2 data file we will predict writing score from gender ( female ), reading, math, science and social studies ( socst ) scores. regression variable = write female read math science socst /dependent = write /method = enter. The results indicate that the overall model is statistically significant (F = 58.60, p = 0.000).  Furthermore, all of the predictor variables are statistically significant except for read . See also Regression with SPSS: Chapter 1 – Simple and Multiple Regression Annotated output for regression SPSS Frequently Asked Questions SPSS Textbook Examples: Regression with Graphics, Chapter 3 SPSS Textbook Examples: Applied Regression Analysis

Analysis of covariance

Analysis of covariance is like ANOVA, except in addition to the categorical predictors you also have continuous predictors as well.  For example, the one way ANOVA example used write as the dependent variable and prog as the independent variable.  Let’s add read as a continuous variable to this model, as shown below. glm write with read by prog. The results indicate that even after adjusting for reading score ( read ), writing scores still significantly differ by program type ( prog ), F = 5.867, p = 0.003. See also SPSS Textbook Examples from Design and Analysis: Chapter 14 SPSS Library: An Overview of SPSS GLM SPSS Library: How do I handle interactions of continuous and categorical variables?

Multiple logistic regression

Multiple logistic regression is like simple logistic regression, except that there are two or more predictors.  The predictors can be interval variables or dummy variables, but cannot be categorical variables.  If you have categorical predictors, they should be coded into one or more dummy variables. We have only one variable in our data set that is coded 0 and 1, and that is female .  We understand that female is a silly outcome variable (it would make more sense to use it as a predictor variable), but we can use female as the outcome variable to illustrate how the code for this command is structured and how to interpret the output.  The first variable listed after the logistic regression command is the outcome (or dependent) variable, and all of the rest of the variables are predictor (or independent) variables (listed after the keyword with ).  In our example, female will be the outcome variable, and read and write will be the predictor variables. logistic regression female with read write. These results show that both read and write are significant predictors of female . See also Annotated output for logistic regression SPSS Textbook Examples: Applied Logistic Regression, Chapter 2 SPSS Code Fragments: Graphing Results in Logistic Regression

Discriminant analysis

Discriminant analysis is used when you have one or more normally distributed interval independent variables and a categorical dependent variable.  It is a multivariate technique that considers the latent dimensions in the independent variables for predicting group membership in the categorical dependent variable.  For example, using the hsb2 data file , say we wish to use read , write and math scores to predict the type of program a student belongs to ( prog ). discriminate groups = prog(1, 3) /variables = read write math. Clearly, the SPSS output for this procedure is quite lengthy, and it is beyond the scope of this page to explain all of it.  However, the main point is that two canonical variables are identified by the analysis, the first of which seems to be more related to program type than the second. See also discriminant function analysis SPSS Library: A History of SPSS Statistical Features

One-way MANOVA

MANOVA (multivariate analysis of variance) is like ANOVA, except that there are two or more dependent variables. In a one-way MANOVA, there is one categorical independent variable and two or more dependent variables. For example, using the hsb2 data file , say we wish to examine the differences in read , write and math broken down by program type ( prog ). glm read write math by prog. The students in the different programs differ in their joint distribution of read , write and math . See also SPSS Library: Advanced Issues in Using and Understanding SPSS MANOVA GLM: MANOVA and MANCOVA SPSS Library: MANOVA and GLM

Multivariate multiple regression

Multivariate multiple regression is used when you have two or more dependent variables that are to be predicted from two or more independent variables.  In our example using the hsb2 data file , we will predict write and read from female , math , science and social studies ( socst ) scores. glm write read with female math science socst. These results show that all of  the variables in the model have a statistically significant relationship with the joint distribution of write and read .

Canonical correlation

Canonical correlation is a multivariate technique used to examine the relationship between two groups of variables.  For each set of variables, it creates latent variables and looks at the relationships among the latent variables. It assumes that all variables in the model are interval and normally distributed.  SPSS requires that each of the two groups of variables be separated by the keyword with .  There need not be an equal number of variables in the two groups (before and after the with ). manova read write with math science /discrim. * * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * * EFFECT .. WITHIN CELLS Regression Multivariate Tests of Significance (S = 2, M = -1/2, N = 97 ) Test Name Value Approx. F Hypoth. DF Error DF Sig. of F Pillais .59783 41.99694 4.00 394.00 .000 Hotellings 1.48369 72.32964 4.00 390.00 .000 Wilks .40249 56.47060 4.00 392.00 .000 Roys .59728 Note.. F statistic for WILKS' Lambda is exact. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - EFFECT .. WITHIN CELLS Regression (Cont.) Univariate F-tests with (2,197) D. F. Variable Sq. Mul. R Adj. R-sq. Hypoth. MS Error MS F READ .51356 .50862 5371.66966 51.65523 103.99081 WRITE .43565 .42992 3894.42594 51.21839 76.03569 Variable Sig. of F READ .000 WRITE .000 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Raw canonical coefficients for DEPENDENT variables Function No. Variable 1 READ .063 WRITE .049 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Standardized canonical coefficients for DEPENDENT variables Function No. Variable 1 READ .649 WRITE .467 * * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * * Correlations between DEPENDENT and canonical variables Function No. Variable 1 READ .927 WRITE .854 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Variance in dependent variables explained by canonical variables CAN. VAR. Pct Var DE Cum Pct DE Pct Var CO Cum Pct CO 1 79.441 79.441 47.449 47.449 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Raw canonical coefficients for COVARIATES Function No. COVARIATE 1 MATH .067 SCIENCE .048 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Standardized canonical coefficients for COVARIATES CAN. VAR. COVARIATE 1 MATH .628 SCIENCE .478 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Correlations between COVARIATES and canonical variables CAN. VAR. Covariate 1 MATH .929 SCIENCE .873 * * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * * Variance in covariates explained by canonical variables CAN. VAR. Pct Var DE Cum Pct DE Pct Var CO Cum Pct CO 1 48.544 48.544 81.275 81.275 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Regression analysis for WITHIN CELLS error term --- Individual Univariate .9500 confidence intervals Dependent variable .. READ reading score COVARIATE B Beta Std. Err. t-Value Sig. of t MATH .48129 .43977 .070 6.868 .000 SCIENCE .36532 .35278 .066 5.509 .000 COVARIATE Lower -95% CL- Upper MATH .343 .619 SCIENCE .235 .496 Dependent variable .. WRITE writing score COVARIATE B Beta Std. Err. t-Value Sig. of t MATH .43290 .42787 .070 6.203 .000 SCIENCE .28775 .30057 .066 4.358 .000 COVARIATE Lower -95% CL- Upper MATH .295 .571 SCIENCE .158 .418 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - * * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * * EFFECT .. CONSTANT Multivariate Tests of Significance (S = 1, M = 0, N = 97 ) Test Name Value Exact F Hypoth. DF Error DF Sig. of F Pillais .11544 12.78959 2.00 196.00 .000 Hotellings .13051 12.78959 2.00 196.00 .000 Wilks .88456 12.78959 2.00 196.00 .000 Roys .11544 Note.. F statistics are exact. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - EFFECT .. CONSTANT (Cont.) Univariate F-tests with (1,197) D. F. Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig. of F READ 336.96220 10176.0807 336.96220 51.65523 6.52329 .011 WRITE 1209.88188 10090.0231 1209.88188 51.21839 23.62202 .000 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - EFFECT .. CONSTANT (Cont.) Raw discriminant function coefficients Function No. Variable 1 READ .041 WRITE .124 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Standardized discriminant function coefficients Function No. Variable 1 READ .293 WRITE .889 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Estimates of effects for canonical variables Canonical Variable Parameter 1 1 2.196 * * * * * * A n a l y s i s o f V a r i a n c e -- design 1 * * * * * * EFFECT .. CONSTANT (Cont.) Correlations between DEPENDENT and canonical variables Canonical Variable Variable 1 READ .504 WRITE .959 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - The output above shows the linear combinations corresponding to the first canonical correlation.  At the bottom of the output are the two canonical correlations. These results indicate that the first canonical correlation is .7728.  The F-test in this output tests the hypothesis that the first canonical correlation is equal to zero.  Clearly, F = 56.4706 is statistically significant.  However, the second canonical correlation of .0235 is not statistically significantly different from zero (F = 0.1087, p = 0.7420).

Factor analysis

Factor analysis is a form of exploratory multivariate analysis that is used to either reduce the number of variables in a model or to detect relationships among variables.  All variables involved in the factor analysis need to be interval and are assumed to be normally distributed.  The goal of the analysis is to try to identify factors which underlie the variables.  There may be fewer factors than variables, but there may not be more factors than variables.  For our example using the hsb2 data file , let’s suppose that we think that there are some common factors underlying the various test scores.  We will include subcommands for varimax rotation and a plot of the eigenvalues.  We will use a principal components extraction and will retain two factors. (Using these options will make our results compatible with those from SAS and Stata and are not necessarily the options that you will want to use.) factor /variables read write math science socst /criteria factors(2) /extraction pc /rotation varimax /plot eigen. Communality (which is the opposite of uniqueness) is the proportion of variance of the variable (i.e., read ) that is accounted for by all of the factors taken together, and a very low communality can indicate that a variable may not belong with any of the factors.  The scree plot may be useful in determining how many factors to retain.  From the component matrix table, we can see that all five of the test scores load onto the first factor, while all five tend to load not so heavily on the second factor.  The purpose of rotating the factors is to get the variables to load either very high or very low on each factor.  In this example, because all of the variables loaded onto factor 1 and not on factor 2, the rotation did not aid in the interpretation. Instead, it made the results even more difficult to interpret. See also SPSS FAQ: What does Cronbach’s alpha mean?

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ANOVA Test: Definition, Types, Examples, SPSS

Statistics Definitions > ANOVA Contents :

The ANOVA Test

• How to Run a One Way ANOVA in SPSS

Two Way ANOVA

What is manova, what is factorial anova, how to run an anova, anova vs. t test.

• Repeated Measures ANOVA in SPSS: Steps

Related Articles

Watch the video for an introduction to ANOVA.

Can’t see the video? Click here to watch it on YouTube.

An ANOVA test is a way to find out if survey or experiment results are significant . In other words, they help you to figure out if you need to reject the null hypothesis or accept the alternate hypothesis .

Basically, you’re testing groups to see if there’s a difference between them. Examples of when you might want to test different groups:

• A group of psychiatric patients are trying three different therapies: counseling, medication and biofeedback. You want to see if one therapy is better than the others.
• A manufacturer has two different processes to make light bulbs. They want to know if one process is better than the other.
• Students from different colleges take the same exam. You want to see if one college outperforms the other.

What Does “One-Way” or “Two-Way Mean?

One-way or two-way refers to the number of independent variables (IVs) in your Analysis of Variance test.

• One-way has one independent variable (with 2 levels ). For example: brand of cereal ,
• Two-way has two independent variables (it can have multiple levels). For example: brand of cereal, calories .

What are “Groups” or “Levels”?

Groups or levels are different groups within the same independent variable . In the above example, your levels for “brand of cereal” might be Lucky Charms, Raisin Bran, Cornflakes — a total of three levels. Your levels for “Calories” might be: sweetened, unsweetened — a total of two levels.

Let’s say you are studying if an alcoholic support group and individual counseling combined is the most effective treatment for lowering alcohol consumption. You might split the study participants into three groups or levels:

• Medication only,
• Medication and counseling,
• Counseling only.

Your dependent variable would be the number of alcoholic beverages consumed per day.

If your groups or levels have a hierarchical structure (each level has unique subgroups), then use a nested ANOVA for the analysis.

What Does “Replication” Mean?

It’s whether you are replicating (i.e. duplicating) your test(s) with multiple groups. With a two way ANOVA with replication , you have two groups and individuals within that group are doing more than one thing (i.e. two groups of students from two colleges taking two tests). If you only have one group taking two tests, you would use without replication.

Types of Tests.

There are two main types: one-way and two-way. Two-way tests can be with or without replication.

• One-way ANOVA between groups: used when you want to test two groups to see if there’s a difference between them.
• Two way ANOVA without replication: used when you have one group and you’re double-testing that same group. For example, you’re testing one set of individuals before and after they take a medication to see if it works or not.
• Two way ANOVA with replication: Two groups , and the members of those groups are doing more than one thing . For example, two groups of patients from different hospitals trying two different therapies.

Back to Top

One Way ANOVA

A one way ANOVA is used to compare two means from two independent (unrelated) groups using the F-distribution . The null hypothesis for the test is that the two means are equal. Therefore, a significant result means that the two means are unequal.

Examples of when to use a one way ANOVA

Situation 1: You have a group of individuals randomly split into smaller groups and completing different tasks. For example, you might be studying the effects of tea on weight loss and form three groups: green tea, black tea, and no tea. Situation 2: Similar to situation 1, but in this case the individuals are split into groups based on an attribute they possess. For example, you might be studying leg strength of people according to weight. You could split participants into weight categories (obese, overweight and normal) and measure their leg strength on a weight machine.

Limitations of the One Way ANOVA

A one way ANOVA will tell you that at least two groups were different from each other. But it won’t tell you which groups were different. If your test returns a significant f-statistic, you may need to run an ad hoc test (like the Least Significant Difference test) to tell you exactly which groups had a difference in means . Back to Top

How to run a One Way ANOVA in SPSS

A Two Way ANOVA is an extension of the One Way ANOVA. With a One Way, you have one independent variable affecting a dependent variable . With a Two Way ANOVA, there are two independents. Use a two way ANOVA when you have one measurement variable (i.e. a quantitative variable ) and two nominal variables . In other words, if your experiment has a quantitative outcome and you have two categorical explanatory variables , a two way ANOVA is appropriate.

For example, you might want to find out if there is an interaction between income and gender for anxiety level at job interviews. The anxiety level is the outcome, or the variable that can be measured. Gender and Income are the two categorical variables . These categorical variables are also the independent variables, which are called factors in a Two Way ANOVA.

The factors can be split into levels . In the above example, income level could be split into three levels: low, middle and high income. Gender could be split into three levels: male, female, and transgender. Treatment groups are all possible combinations of the factors. In this example there would be 3 x 3 = 9 treatment groups.

Main Effect and Interaction Effect

The results from a Two Way ANOVA will calculate a main effect and an interaction effect . The main effect is similar to a One Way ANOVA: each factor’s effect is considered separately. With the interaction effect, all factors are considered at the same time. Interaction effects between factors are easier to test if there is more than one observation in each cell. For the above example, multiple stress scores could be entered into cells. If you do enter multiple observations into cells, the number in each cell must be equal.

Two null hypotheses are tested if you are placing one observation in each cell. For this example, those hypotheses would be: H 01 : All the income groups have equal mean stress. H 02 : All the gender groups have equal mean stress.

For multiple observations in cells, you would also be testing a third hypothesis: H 03 : The factors are independent or the interaction effect does not exist.

An F-statistic is computed for each hypothesis you are testing.

Assumptions for Two Way ANOVA

• The population must be close to a normal distribution .
• Samples must be independent.
• Population variances must be equal (i.e. homoscedastic ).
• Groups must have equal sample sizes .

MANOVA is just an ANOVA with several dependent variables. It’s similar to many other tests and experiments in that it’s purpose is to find out if the response variable (i.e. your dependent variable) is changed by manipulating the independent variable. The test helps to answer many research questions, including:

• Do changes to the independent variables have statistically significant effects on dependent variables?
• What are the interactions among dependent variables?
• What are the interactions among independent variables?

MANOVA Example

Suppose you wanted to find out if a difference in textbooks affected students’ scores in math and science. Improvements in math and science means that there are two dependent variables, so a MANOVA is appropriate.

An ANOVA will give you a single ( univariate ) f-value while a MANOVA will give you a multivariate F value. MANOVA tests the multiple dependent variables by creating new, artificial, dependent variables that maximize group differences. These new dependent variables are linear combinations of the measured dependent variables.

Interpreting the MANOVA results

If the multivariate F value indicates the test is statistically significant , this means that something is significant. In the above example, you would not know if math scores have improved, science scores have improved (or both). Once you have a significant result, you would then have to look at each individual component (the univariate F tests) to see which dependent variable(s) contributed to the statistically significant result.

Advantages and Disadvantages of MANOVA vs. ANOVA

• MANOVA enables you to test multiple dependent variables.
• MANOVA can protect against Type I errors.

Disadvantages

• MANOVA is many times more complicated than ANOVA, making it a challenge to see which independent variables are affecting dependent variables.
• One degree of freedom is lost with the addition of each new variable .
• The dependent variables should be uncorrelated as much as possible. If they are correlated, the loss in degrees of freedom means that there isn’t much advantages in including more than one dependent variable on the test.

Reference : SFSU. Retrieved April 18, 2022 from: http://online.sfsu.edu/efc/classes/biol710/manova/MANOVAnewest.pdf

A factorial ANOVA is an Analysis of Variance test with more than one independent variable , or “ factor “. It can also refer to more than one Level of Independent Variable . For example, an experiment with a treatment group and a control group has one factor (the treatment) but two levels (the treatment and the control). The terms “two-way” and “three-way” refer to the number of factors or the number of levels in your test. Four-way ANOVA and above are rarely used because the results of the test are complex and difficult to interpret.

• A two-way ANOVA has two factors ( independent variables ) and one dependent variable . For example, time spent studying and prior knowledge are factors that affect how well you do on a test.
• A three-way ANOVA has three factors (independent variables) and one dependent variable. For example, time spent studying, prior knowledge, and hours of sleep are factors that affect how well you do on a test

Factorial ANOVA is an efficient way of conducting a test. Instead of performing a series of experiments where you test one independent variable against one dependent variable, you can test all independent variables at the same time.

Variability

In a one-way ANOVA, variability is due to the differences between groups and the differences within groups. In factorial ANOVA, each level and factor are paired up with each other (“crossed”). This helps you to see what interactions are going on between the levels and factors. If there is an interaction then the differences in one factor depend on the differences in another.

Let’s say you were running a two-way ANOVA to test male/female performance on a final exam. The subjects had either had 4, 6, or 8 hours of sleep.

• IV1: SEX (Male/Female)
• IV2: SLEEP (4/6/8)
• DV: Final Exam Score

A two-way factorial ANOVA would help you answer the following questions:

• Is sex a main effect? In other words, do men and women differ significantly on their exam performance?
• Is sleep a main effect? In other words, do people who have had 4,6, or 8 hours of sleep differ significantly in their performance?
• Is there a significant interaction between factors? In other words, how do hours of sleep and sex interact with regards to exam performance?
• Can any differences in sex and exam performance be found in the different levels of sleep?

Assumptions of Factorial ANOVA

• Normality: the dependent variable is normally distributed.
• Independence: Observations and groups are independent from each other.
• Equality of Variance: the population variances are equal across factors/levels.

These tests are very time-consuming by hand. In nearly every case you’ll want to use software. For example, several options are available in Excel :

• Two way ANOVA in Excel with replication and without replication.
• One way ANOVA in Excel 2013 .

ANOVA tests in statistics packages are run on parametric data. If you have rank or ordered data, you’ll want to run a non-parametric ANOVA (usually found under a different heading in the software, like “ nonparametric tests “).

It is unlikely you’ll want to do this test by hand, but if you must, these are the steps you’ll want to take:

• Find the mean for each of the groups.
• Find the overall mean (the mean of the groups combined).
• Find the Within Group Variation ; the total deviation of each member’s score from the Group Mean.
• Find the Between Group Variation : the deviation of each Group Mean from the Overall Mean.
• Find the F statistic: the ratio of Between Group Variation to Within Group Variation.

A Student’s t-test will tell you if there is a significant variation between groups. A t-test compares means, while the ANOVA compares variances between populations. You could technically perform a series of t-tests on your data. However, as the groups grow in number, you may end up with a lot of pair comparisons that you need to run. ANOVA will give you a single number (the f-statistic ) and one p-value to help you support or reject the null hypothesis . Back to Top

Repeated Measures (Within Subjects) ANOVA

A repeated measures ANOVA is almost the same as one-way ANOVA, with one main difference: you test related groups, not independent ones.

It’s called Repeated Measures because the same group of participants is being measured over and over again. For example, you could be studying the cholesterol levels of the same group of patients at 1, 3, and 6 months after changing their diet. For this example, the independent variable is “time” and the dependent variable is “cholesterol.” The independent variable is usually called the within-subjects factor .

Repeated measures ANOVA is similar to a simple multivariate design. In both tests, the same participants are measured over and over. However, with repeated measures the same characteristic is measured with a different condition. For example, blood pressure is measured over the condition “time”. For simple multivariate design it is the characteristic that changes. For example, you could measure blood pressure, heart rate and respiration rate over time.

Reasons to use Repeated Measures ANOVA

• When you collect data from the same participants over a period of time, individual differences (a source of between group differences) are reduced or eliminated.
• Testing is more powerful because the sample size isn’t divided between groups.
• The test can be economical, as you’re using the same participants.

Assumptions for Repeated Measures ANOVA

The results from your repeated measures ANOVA will be valid only if the following assumptions haven’t been violated:

• There must be one independent variable and one dependent variable.
• The dependent variable must be a continuous variable , on an interval scale or a ratio scale .
• The independent variable must be categorical , either on the nominal scale or ordinal scale.
• Ideally, levels of dependence between pairs of groups is equal (“sphericity”). Corrections are possible if this assumption is violated.

One Way Repeated Measures ANOVA in SPSS: Steps

Watch the video for the steps:

Step 2: Replace the “factor1” name with something that represents your independent variable. For example, you could put “age” or “time.”

Step 3: Enter the “Number of Levels.” This is how many times the dependent variable has been measured. For example, if you took measurements every week for a total of 4 weeks, this number would be 4.

Step 4: Click the “Add” button and then give your dependent variable a name.

Step 7: Click “Plots” and use the arrow keys to transfer the factor from the left box onto the Horizontal Axis box.

Step 9: Click “Options”, then transfer your factors from the left box to the Display Means for box on the right.

Step 10: Click the following check boxes:

• Compare main effects.
• Descriptive Statistics.
• Estimates of Effect Size .

Step 11: Select “Bonferroni” from the drop down menu under Confidence Interval Adjustment . Step 12: Click “Continue” and then click “OK” to run the test. Back to Top

In statistics, sphericity (ε) refers to Mauchly’s sphericity test , which was developed in 1940 by John W. Mauchly , who co-developed the first general-purpose electronic computer.

Sphericity is used as an assumption in repeated measures ANOVA. The assumption states that the variances of the differences between all possible group pairs are equal. If your data violates this assumption, it can result in an increase in a Type I error (the incorrect rejection of the null hypothesis) .

It’s very common for repeated measures ANOVA to result in a violation of the assumption. If the assumption has been violated, corrections have been developed that can avoid increases in the type I error rate. The correction is applied to the degrees of freedom in the F-distribution .

Mauchly’s Sphericity Test

Mauchly’s test for sphericity can be run in the majority of statistical software, where it tends to be the default test for sphericity. Mauchly’s test is ideal for mid-size samples. It may fail to detect sphericity in small samples and it may over-detect in large samples. If the test returns a small p-value (p ≤.05), this is an indication that your data has violated the assumption. The following picture of SPSS output for ANOVA shows that the significance “sig” attached to Mauchly’s is .274. This means that the assumption has not been violated for this set of data.

You would report the above result as “Mauchly’s Test indicated that the assumption of sphericity had not been violated, χ 2 (2) = 2.588, p = .274.”

If your test returned a small p-value , you should apply a correction, usually either the:

• Greehouse-Geisser correction.
• Huynh-Feldt correction .

When ε ≤ 0.75 (or you don’t know what the value for the statistic is), use the Greenhouse-Geisser correction. When ε > .75, use the Huynh-Feldt correction .

Grand mean ANOVA vs Regression

Blokdyk, B. (2018). Ad Hoc Testing . 5STARCooks Miller, R. G. Beyond ANOVA: Basics of Applied Statistics . Boca Raton, FL: Chapman & Hall, 1997 Image: UVM. Retrieved December 4, 2020 from: https://www.uvm.edu/~dhowell/gradstat/psych341/lectures/RepeatedMeasures/repeated1.html

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Lesson 10 of 24 By Avijeet Biswal

Table of Contents

In today’s data-driven world, decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis and hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life - 

• A teacher assumes that 60% of his college's students come from lower-middle-class families.
• A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

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Importance of Hypothesis Testing in Data Analysis

Here is what makes hypothesis testing so important in data analysis and why it is key to making better decisions:

Avoiding Misleading Conclusions (Type I and Type II Errors)

One of the biggest benefits of hypothesis testing is that it helps you avoid jumping to the wrong conclusions. For instance, a Type I error could occur if a company launches a new product thinking it will be a hit, only to find out later that the data misled them. A Type II error might happen when a company overlooks a potentially successful product because their testing wasn’t thorough enough. By setting up the right significance level and carefully calculating the p-value, hypothesis testing minimizes the chances of these errors, leading to more accurate results.

Making Smarter Choices

Hypothesis testing is key to making smarter, evidence-based decisions. Let’s say a city planner wants to determine if building a new park will increase community engagement. By testing the hypothesis using data from similar projects, they can make an informed choice. Similarly, a teacher might use hypothesis testing to see if a new teaching method actually improves student performance. It’s about taking the guesswork out of decisions and relying on solid evidence instead.

Optimizing Business Tactics

In business, hypothesis testing is invaluable for testing new ideas and strategies before fully committing to them. For example, an e-commerce company might want to test whether offering free shipping increases sales. By using hypothesis testing, they can compare sales data from customers who received free shipping offers and those who didn’t. This allows them to base their business decisions on data, not hunches, reducing the risk of costly mistakes.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

• Here, x̅ is the sample mean,
• μ0 is the population mean,
• σ is the standard deviation,
• n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

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Null Hypothesis and Alternative Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average. 

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

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Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps in Hypothesis Testing

Hypothesis testing is a statistical method to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. Here’s a breakdown of the typical steps involved in hypothesis testing:

Formulate Hypotheses

• Null Hypothesis (H0): This hypothesis states that there is no effect or difference, and it is the hypothesis you attempt to reject with your test.
• Alternative Hypothesis (H1 or Ha): This hypothesis is what you might believe to be true or hope to prove true. It is usually considered the opposite of the null hypothesis.

Choose the Significance Level (α)

The significance level, often denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. Common choices for α are 0.05 (5%), 0.01 (1%), and 0.10 (10%).

Select the Appropriate Test

Choose a statistical test based on the type of data and the hypothesis. Common tests include t-tests, chi-square tests, ANOVA, and regression analysis. The selection depends on data type, distribution, sample size, and whether the hypothesis is one-tailed or two-tailed.

Collect Data

Gather the data that will be analyzed in the test. To infer conclusions accurately, this data should be representative of the population.

Calculate the Test Statistic

Based on the collected data and the chosen test, calculate a test statistic that reflects how much the observed data deviates from the null hypothesis.

Determine the p-value

The p-value is the probability of observing test results at least as extreme as the results observed, assuming the null hypothesis is correct. It helps determine the strength of the evidence against the null hypothesis.

Make a Decision

Compare the p-value to the chosen significance level:

• If the p-value ≤ α: Reject the null hypothesis, suggesting sufficient evidence in the data supports the alternative hypothesis.
• If the p-value > α: Do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Report the Results

Present the findings from the hypothesis test, including the test statistic, p-value, and the conclusion about the hypotheses.

Perform Post-hoc Analysis (if necessary)

Depending on the results and the study design, further analysis may be needed to explore the data more deeply or to address multiple comparisons if several hypotheses were tested simultaneously.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

3. Chi-Square 

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

ANOVA , or Analysis of Variance, is a statistical method used to compare the means of three or more groups. It’s particularly useful when you want to see if there are significant differences between multiple groups. For instance, in business, a company might use ANOVA to analyze whether three different stores are performing differently in terms of sales. It’s also widely used in fields like medical research and social sciences, where comparing group differences can provide valuable insights.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

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Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

• The null hypothesis is (H0 <= 90) or less change.
• A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true]. 

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

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Practice Problems on Hypothesis Testing

Here are the practice problems on hypothesis testing that will help you understand how to apply these concepts in real-world scenarios:

A telecom service provider claims that customers spend an average of ₹400 per month, with a standard deviation of ₹25. However, a random sample of 50 customer bills shows a mean of ₹250 and a standard deviation of ₹15. Does this sample data support the service provider’s claim?

Solution: Let’s break this down:

• Null Hypothesis (H0): The average amount spent per month is ₹400.
• Alternate Hypothesis (H1): The average amount spent per month is not ₹400.
• Population Standard Deviation (σ): ₹25
• Sample Size (n): 50
• Sample Mean (x̄): ₹250

1. Calculate the z-value:

z=250-40025/50 −42.42

2. Compare with critical z-values: For a 5% significance level, critical z-values are -1.96 and +1.96. Since -42.42 is far outside this range, we reject the null hypothesis. The sample data suggests that the average amount spent is significantly different from ₹400.

Out of 850 customers, 400 made online grocery purchases. Can we conclude that more than 50% of customers are moving towards online grocery shopping?

Solution: Here’s how to approach it:

• Proportion of customers who shopped online (p): 400 / 850 = 0.47
• Null Hypothesis (H0): The proportion of online shoppers is 50% or more.
• Alternate Hypothesis (H1): The proportion of online shoppers is less than 50%.
• Sample Size (n): 850
• Significance Level (α): 5%

z=p-PP(1-P)/n

z=0.47-0.500.50.5/850  −1.74

2. Compare with the critical z-value: For a 5% significance level (one-tailed test), the critical z-value is -1.645. Since -1.74 is less than -1.645, we reject the null hypothesis. This means the data does not support the idea that most customers are moving towards online grocery shopping.

In a study of code quality, Team A has 250 errors in 1000 lines of code, and Team B has 300 errors in 800 lines of code. Can we say Team B performs worse than Team A?

Solution: Let’s analyze it:

• Proportion of errors for Team A (pA): 250 / 1000 = 0.25
• Proportion of errors for Team B (pB): 300 / 800 = 0.375
• Null Hypothesis (H0): Team B’s error rate is less than or equal to Team A’s.
• Alternate Hypothesis (H1): Team B’s error rate is greater than Team A’s.
• Sample Size for Team A (nA): 1000
• Sample Size for Team B (nB): 800

p=nApA+nBpBnA+nB

p=10000.25+8000.3751000+800 ≈ 0.305

z=​pA−pB​p(1-p)(1nA+1nB)

z=​0.25−0.375​0.305(1-0.305) (11000+1800) ≈ −5.72

2. Compare with the critical z-value: For a 5% significance level (one-tailed test), the critical z-value is +1.645. Since -5.72 is far less than +1.645, we reject the null hypothesis. The data indicates that Team B’s performance is significantly worse than Team A’s.

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Applications of Hypothesis Testing

Apart from the practical problems, let's look at the real-world applications of hypothesis testing across various fields:

Medicine and Healthcare

In medicine, hypothesis testing plays a pivotal role in assessing the success of new treatments. For example, researchers may want to find out if a new exercise regimen improves heart health. By comparing data from patients who followed the program to those who didn’t, they can determine if the exercise significantly improves health outcomes. Such rigorous testing allows medical professionals to rely on proven methods rather than assumptions.

Quality Control and Manufacturing

In manufacturing, ensuring product quality is vital, and hypothesis testing helps maintain those standards. Suppose a beverage company introduces a new bottling process and wants to verify if it reduces contamination. By analyzing samples from the new and old processes, hypothesis testing can reveal whether the new method reduces the risk of contamination. This allows manufacturers to implement improvements that enhance product safety and quality confidently.

Education and Learning

In education and learning, hypothesis testing is a tool to evaluate the impact of innovative teaching techniques. Imagine a situation where teachers introduce project-based learning to boost critical thinking skills. By comparing the performance of students who engaged in project-based learning with those in traditional settings, educators can test their hypothesis. The results can help educators make informed choices about adopting new teaching strategies.

Environmental Science

Hypothesis testing is essential in environmental science for evaluating the effectiveness of conservation measures. For example, scientists might explore whether a new water management strategy improves river health. By collecting and comparing data on water quality before and after the implementation of the strategy, they can determine whether the intervention leads to positive changes. Such findings are crucial for guiding environmental decisions that have long-term impacts.

Marketing and Advertising

In marketing, businesses use hypothesis testing to refine their approaches. For instance, a clothing brand might test if offering limited-time discounts increases customer loyalty. By running campaigns with and without the discount and analyzing the outcomes, they can assess if the strategy boosts customer retention. Data-driven insights from hypothesis testing enable companies to design marketing strategies that resonate with their audience and drive growth.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

• It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
• Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
• Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
• Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

Learn All The Tricks Of The BI Trade

After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore the Post Graduate Program in Data Science.

1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is H0 and H1 in statistics?

In statistics, H0​ and H1​ represent the null and alternative hypotheses. The null hypothesis, H0​, is the default assumption that no effect or difference exists between groups or conditions. The alternative hypothesis, H1​, is the competing claim suggesting an effect or a difference. Statistical tests determine whether to reject the null hypothesis in favor of the alternative hypothesis based on the data.

3. What is a simple hypothesis with an example?

A simple hypothesis is a specific statement predicting a single relationship between two variables. It posits a direct and uncomplicated outcome. For example, a simple hypothesis might state, "Increased sunlight exposure increases the growth rate of sunflowers." Here, the hypothesis suggests a direct relationship between the amount of sunlight (independent variable) and the growth rate of sunflowers (dependent variable), with no additional variables considered.

4. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

• Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
• Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
• Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

5. What software tools can assist with hypothesis testing?

Several software tools offering distinct features can help with hypothesis testing. R and RStudio are popular for their advanced statistical capabilities. The Python ecosystem, including libraries like SciPy and Statsmodels, also supports hypothesis testing. SAS and SPSS are well-established tools for comprehensive statistical analysis. For basic testing, Excel offers simple built-in functions.

6. How do I interpret the results of a hypothesis test?

Interpreting hypothesis test results involves comparing the p-value to the significance level (alpha). If the p-value is less than or equal to alpha, you can reject the null hypothesis, indicating statistical significance. This suggests that the observed effect is unlikely to have occurred by chance, validating your analysis findings.

7. Why is sample size important in hypothesis testing?

Sample size is crucial in hypothesis testing as it affects the test’s power. A larger sample size increases the likelihood of detecting a true effect, reducing the risk of Type II errors. Conversely, a small sample may lack the statistical power needed to identify differences, potentially leading to inaccurate conclusions.

8. Can hypothesis testing be used for non-numerical data?

Yes, hypothesis testing can be applied to non-numerical data through non-parametric tests. These tests are ideal when data doesn't meet parametric assumptions or when dealing with categorical data. Non-parametric tests, like the Chi-square or Mann-Whitney U test, provide robust methods for analyzing non-numerical data and drawing meaningful conclusions.

9. How do I choose the proper hypothesis test?

Selecting the right hypothesis test depends on several factors: the objective of your analysis, the type of data (numerical or categorical), and the sample size. Consider whether you're comparing means, proportions, or associations, and whether your data follows a normal distribution. The correct choice ensures accurate results tailored to your research question.

Find our PL-300 Microsoft Power BI Certification Training Online Classroom training classes in top cities:

About the Author

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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• SPSS Tutorials

Independent Samples t Test

Spss tutorials: independent samples t test.

• The SPSS Environment
• The Data View Window
• Using SPSS Syntax
• Data Creation in SPSS
• Importing Data into SPSS
• Variable Types
• Date-Time Variables in SPSS
• Defining Variables
• Creating a Codebook
• Computing Variables
• Computing Variables: Mean Centering
• Computing Variables: Recoding Categorical Variables
• Computing Variables: Recoding String Variables into Coded Categories (Automatic Recode)
• rank transform converts a set of data values by ordering them from smallest to largest, and then assigning a rank to each value. In SPSS, the Rank Cases procedure can be used to compute the rank transform of a variable." href="https://libguides.library.kent.edu/SPSS/RankCases" style="" >Computing Variables: Rank Transforms (Rank Cases)
• Weighting Cases
• Sorting Data
• Grouping Data
• Descriptive Stats for One Numeric Variable (Explore)
• Descriptive Stats for One Numeric Variable (Frequencies)
• Descriptive Stats for Many Numeric Variables (Descriptives)
• Descriptive Stats by Group (Compare Means)
• Frequency Tables
• Working with "Check All That Apply" Survey Data (Multiple Response Sets)
• Chi-Square Test of Independence
• Pearson Correlation
• One Sample t Test
• Paired Samples t Test
• One-Way ANOVA
• How to Cite the Tutorials

Sample Data Files

Our tutorials reference a dataset called "sample" in many examples. If you'd like to download the sample dataset to work through the examples, choose one of the files below:

• Data definitions (*.pdf)
• Data - Comma delimited (*.csv)
• Data - Tab delimited (*.txt)
• Data - Excel format (*.xlsx)
• Data - SAS format (*.sas7bdat)
• Data - SPSS format (*.sav)

The Independent Samples t Test compares the means of two independent groups in order to determine whether there is statistical evidence that the associated population means are significantly different. The Independent Samples t Test is a parametric test.

This test is also known as:

• Independent t Test
• Independent Measures t Test
• Independent Two-sample t Test
• Student t Test
• Two-Sample t Test
• Uncorrelated Scores t Test
• Unpaired t Test
• Unrelated t Test

The variables used in this test are known as:

• Dependent variable, or test variable
• Independent variable, or grouping variable

Common Uses

The Independent Samples t Test is commonly used to test the following:

• Statistical differences between the means of two groups
• Statistical differences between the means of two interventions
• Statistical differences between the means of two change scores

Note:  The Independent Samples  t  Test can only compare the means for two (and only two) groups. It cannot make comparisons among more than two groups. If you wish to compare the means across more than two groups, you will likely want to run an ANOVA.

Data Requirements

Your data must meet the following requirements:

• Dependent variable that is continuous (i.e., interval or ratio level)
• Independent variable that is categorical (i.e., nominal or ordinal) and has exactly two categories
• Cases that have nonmissing values for both the dependent and independent variables
• Subjects in the first group cannot also be in the second group
• No subject in either group can influence subjects in the other group
• No group can influence the other group
• Violation of this assumption will yield an inaccurate p value
• Random sample of data from the population
• Non-normal population distributions, especially those that are thick-tailed or heavily skewed, considerably reduce the power of the test
• Among moderate or large samples, a violation of normality may still yield accurate p values
• When this assumption is violated and the sample sizes for each group differ, the p value is not trustworthy. However, the Independent Samples t Test output also includes an approximate t statistic that is not based on assuming equal population variances. This alternative statistic, called the Welch t Test statistic 1 , may be used when equal variances among populations cannot be assumed. The Welch t Test is also known an Unequal Variance t Test or Separate Variances t Test.
• No outliers

Note: When one or more of the assumptions for the Independent Samples t Test are not met, you may want to run the nonparametric Mann-Whitney U Test instead.

Researchers often follow several rules of thumb:

• Each group should have at least 6 subjects, ideally more. Inferences for the population will be more tenuous with too few subjects.
• A balanced design (i.e., same number of subjects in each group) is ideal. Extremely unbalanced designs increase the possibility that violating any of the requirements/assumptions will threaten the validity of the Independent Samples t Test.

1  Welch, B. L. (1947). The generalization of "Student's" problem when several different population variances are involved. Biometrika , 34 (1–2), 28–35.

The null hypothesis ( H 0 ) and alternative hypothesis ( H 1 ) of the Independent Samples t Test can be expressed in two different but equivalent ways:

H 0 : µ 1  = µ 2 ("the two population means are equal") H 1 : µ 1  ≠ µ 2 ("the two population means are not equal")

H 0 : µ 1  - µ 2  = 0 ("the difference between the two population means is equal to 0") H 1 :  µ 1  - µ 2  ≠ 0 ("the difference between the two population means is not 0")

where µ 1 and µ 2 are the population means for group 1 and group 2, respectively. Notice that the second set of hypotheses can be derived from the first set by simply subtracting µ 2 from both sides of the equation.

Levene’s Test for Equality of Variances

Recall that the Independent Samples t Test requires the assumption of homogeneity of variance -- i.e., both groups have the same variance. SPSS conveniently includes a test for the homogeneity of variance, called Levene's Test , whenever you run an independent samples t test.

The hypotheses for Levene’s test are: 

H 0 : σ 1 2 - σ 2 2 = 0 ("the population variances of group 1 and 2 are equal") H 1 : σ 1 2 - σ 2 2 ≠ 0 ("the population variances of group 1 and 2 are not equal")

This implies that if we reject the null hypothesis of Levene's Test, it suggests that the variances of the two groups are not equal; i.e., that the homogeneity of variances assumption is violated.

The output in the Independent Samples Test table includes two rows: Equal variances assumed and Equal variances not assumed . If Levene’s test indicates that the variances are equal across the two groups (i.e., p -value large), you will rely on the first row of output, Equal variances assumed , when you look at the results for the actual Independent Samples t Test (under the heading t -test for Equality of Means). If Levene’s test indicates that the variances are not equal across the two groups (i.e., p -value small), you will need to rely on the second row of output, Equal variances not assumed , when you look at the results of the Independent Samples t Test (under the heading t -test for Equality of Means). 

The difference between these two rows of output lies in the way the independent samples t test statistic is calculated. When equal variances are assumed, the calculation uses pooled variances; when equal variances cannot be assumed, the calculation utilizes un-pooled variances and a correction to the degrees of freedom.

Test Statistic

The test statistic for an Independent Samples t Test is denoted t . There are actually two forms of the test statistic for this test, depending on whether or not equal variances are assumed. SPSS produces both forms of the test, so both forms of the test are described here. Note that the null and alternative hypotheses are identical for both forms of the test statistic.

Equal variances assumed

When the two independent samples are assumed to be drawn from populations with identical population variances (i.e., σ 1 2 = σ 2 2 ) , the test statistic t is computed as:

$$ t = \frac{\overline{x}_{1} - \overline{x}_{2}}{s_{p}\sqrt{\frac{1}{n_{1}} + \frac{1}{n_{2}}}} $$

$$ s_{p} = \sqrt{\frac{(n_{1} - 1)s_{1}^{2} + (n_{2} - 1)s_{2}^{2}}{n_{1} + n_{2} - 2}} $$

\(\bar{x}_{1}\) = Mean of first sample \(\bar{x}_{2}\) = Mean of second sample \(n_{1}\) = Sample size (i.e., number of observations) of first sample \(n_{2}\) = Sample size (i.e., number of observations) of second sample \(s_{1}\) = Standard deviation of first sample \(s_{2}\) = Standard deviation of second sample \(s_{p}\) = Pooled standard deviation

The calculated t value is then compared to the critical t value from the t distribution table with degrees of freedom df = n 1 + n 2 - 2 and chosen confidence level. If the calculated t value is greater than the critical t value, then we reject the null hypothesis.

Note that this form of the independent samples t test statistic assumes equal variances.

Because we assume equal population variances, it is OK to "pool" the sample variances ( s p ). However, if this assumption is violated, the pooled variance estimate may not be accurate, which would affect the accuracy of our test statistic (and hence, the p-value).

Equal variances not assumed

When the two independent samples are assumed to be drawn from populations with unequal variances (i.e., σ 1 2  ≠ σ 2 2 ), the test statistic t is computed as:

$$ t = \frac{\overline{x}_{1} - \overline{x}_{2}}{\sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}} $$

\(\bar{x}_{1}\) = Mean of first sample \(\bar{x}_{2}\) = Mean of second sample \(n_{1}\) = Sample size (i.e., number of observations) of first sample \(n_{2}\) = Sample size (i.e., number of observations) of second sample \(s_{1}\) = Standard deviation of first sample \(s_{2}\) = Standard deviation of second sample

The calculated t value is then compared to the critical t value from the t distribution table with degrees of freedom

$$ df = \frac{ \left ( \frac{s_{1}^2}{n_{1}} + \frac{s_{2}^2}{n_{2}} \right ) ^{2} }{ \frac{1}{n_{1}-1} \left ( \frac{s_{1}^2}{n_{1}} \right ) ^{2} + \frac{1}{n_{2}-1} \left ( \frac{s_{2}^2}{n_{2}} \right ) ^{2}} $$

and chosen confidence level. If the calculated t value > critical t value, then we reject the null hypothesis.

Note that this form of the independent samples t test statistic does not assume equal variances. This is why both the denominator of the test statistic and the degrees of freedom of the critical value of  t are different than the equal variances form of the test statistic.

Data Set-Up

Your data should include two variables (represented in columns) that will be used in the analysis. The independent variable should be categorical and include exactly two groups. (Note that SPSS restricts categorical indicators to numeric or short string values only.) The dependent variable should be continuous (i.e., interval or ratio). SPSS can only make use of cases that have nonmissing values for the independent and the dependent variables, so if a case has a missing value for either variable, it cannot be included in the test.

The number of rows in the dataset should correspond to the number of subjects in the study. Each row of the dataset should represent a unique subject, person, or unit, and all of the measurements taken on that person or unit should appear in that row.

Run an Independent Samples t Test

To run an Independent Samples t Test in SPSS, click  Analyze > Compare Means > Independent-Samples T Test .

The Independent-Samples T Test window opens where you will specify the variables to be used in the analysis. All of the variables in your dataset appear in the list on the left side. Move variables to the right by selecting them in the list and clicking the blue arrow buttons. You can move a variable(s) to either of two areas: Grouping Variable or Test Variable(s) .

A Test Variable(s): The dependent variable(s). This is the continuous variable whose means will be compared between the two groups. You may run multiple t tests simultaneously by selecting more than one test variable.

B Grouping Variable: The independent variable. The categories (or groups) of the independent variable will define which samples will be compared in the t test. The grouping variable must have at least two categories (groups); it may have more than two categories but a t test can only compare two groups, so you will need to specify which two groups to compare. You can also use a continuous variable by specifying a cut point to create two groups (i.e., values at or above the cut point and values below the cut point).

C Define Groups : Click Define Groups to define the category indicators (groups) to use in the t test. If the button is not active, make sure that you have already moved your independent variable to the right in the Grouping Variable field. You must define the categories of your grouping variable before you can run the Independent Samples t Test procedure.

You will not be able to run the Independent Samples t Test until the levels (or cut points) of the grouping variable have been defined. The OK and Paste buttons will be unclickable until the levels have been defined. You can tell if the levels of the grouping variable have not been defined by looking at the Grouping Variable box: if a variable appears in the box but has two question marks next to it, then the levels are not defined:

D Options: The Options section is where you can set your desired confidence level for the confidence interval for the mean difference, and specify how SPSS should handle missing values.

When finished, click OK to run the Independent Samples t Test, or click Paste to have the syntax corresponding to your specified settings written to an open syntax window. (If you do not have a syntax window open, a new window will open for you.)

Define Groups

Clicking the Define Groups button (C) opens the Define Groups window:

1 Use specified values: If your grouping variable is categorical, select Use specified values . Enter the values for the categories you wish to compare in the Group 1 and Group 2 fields. If your categories are numerically coded, you will enter the numeric codes. If your group variable is string, you will enter the exact text strings representing the two categories. If your grouping variable has more than two categories (e.g., takes on values of 1, 2, 3, 4), you can specify two of the categories to be compared (SPSS will disregard the other categories in this case).

Note that when computing the test statistic, SPSS will subtract the mean of the Group 2 from the mean of Group 1. Changing the order of the subtraction affects the sign of the results, but does not affect the magnitude of the results.

2 Cut point: If your grouping variable is numeric and continuous, you can designate a cut point for dichotomizing the variable. This will separate the cases into two categories based on the cut point. Specifically, for a given cut point x , the new categories will be:

• Group 1: All cases where grouping variable > x
• Group 2: All cases where grouping variable < x

Note that this implies that cases where the grouping variable is equal to the cut point itself will be included in the "greater than or equal to" category. (If you want your cut point to be included in a "less than or equal to" group, then you will need to use Recode into Different Variables or use DO IF syntax to create this grouping variable yourself.) Also note that while you can use cut points on any variable that has a numeric type, it may not make practical sense depending on the actual measurement level of the variable (e.g., nominal categorical variables coded numerically). Additionally, using a dichotomized variable created via a cut point generally reduces the power of the test compared to using a non-dichotomized variable.

Clicking the Options button (D) opens the Options window:

The Confidence Interval Percentage box allows you to specify the confidence level for a confidence interval. Note that this setting does NOT affect the test statistic or p-value or standard error; it only affects the computed upper and lower bounds of the confidence interval. You can enter any value between 1 and 99 in this box (although in practice, it only makes sense to enter numbers between 90 and 99).

The Missing Values section allows you to choose if cases should be excluded "analysis by analysis" (i.e. pairwise deletion) or excluded listwise. This setting is not relevant if you have only specified one dependent variable; it only matters if you are entering more than one dependent (continuous numeric) variable. In that case, excluding "analysis by analysis" will use all nonmissing values for a given variable. If you exclude "listwise", it will only use the cases with nonmissing values for all of the variables entered. Depending on the amount of missing data you have, listwise deletion could greatly reduce your sample size.

Example: Independent samples T test when variances are not equal

Problem statement.

In our sample dataset, students reported their typical time to run a mile, and whether or not they were an athlete. Suppose we want to know if the average time to run a mile is different for athletes versus non-athletes. This involves testing whether the sample means for mile time among athletes and non-athletes in your sample are statistically different (and by extension, inferring whether the means for mile times in the population are significantly different between these two groups). You can use an Independent Samples t Test to compare the mean mile time for athletes and non-athletes.

The hypotheses for this example can be expressed as:

H 0 : µ non-athlete  − µ athlete  = 0 ("the difference of the means is equal to zero") H 1 : µ non-athlete  − µ athlete  ≠ 0 ("the difference of the means is not equal to zero")

where µ athlete and µ non-athlete are the population means for athletes and non-athletes, respectively.

In the sample data, we will use two variables: Athlete and MileMinDur . The variable Athlete has values of either “0” (non-athlete) or "1" (athlete). It will function as the independent variable in this T test. The variable MileMinDur is a numeric duration variable (h:mm:ss), and it will function as the dependent variable. In SPSS, the first few rows of data look like this:

Before the Test

Before running the Independent Samples t Test, it is a good idea to look at descriptive statistics and graphs to get an idea of what to expect. Running Compare Means ( Analyze > Compare Means > Means ) to get descriptive statistics by group tells us that the standard deviation in mile time for non-athletes is about 2 minutes; for athletes, it is about 49 seconds. This corresponds to a variance of 14803 seconds for non-athletes, and a variance of 2447 seconds for athletes 1 . Running the Explore procedure ( Analyze > Descriptives > Explore ) to obtain a comparative boxplot yields the following graph:

If the variances were indeed equal, we would expect the total length of the boxplots to be about the same for both groups. However, from this boxplot, it is clear that the spread of observations for non-athletes is much greater than the spread of observations for athletes. Already, we can estimate that the variances for these two groups are quite different. It should not come as a surprise if we run the Independent Samples t Test and see that Levene's Test is significant.

Additionally, we should also decide on a significance level (typically denoted using the Greek letter alpha, α ) before we perform our hypothesis tests. The significance level is the threshold we use to decide whether a test result is significant. For this example, let's use α = 0.05.

1 When computing the variance of a duration variable (formatted as hh:mm:ss or mm:ss or mm:ss.s), SPSS converts the standard deviation value to seconds before squaring.

Running the Test

To run the Independent Samples t Test:

• Click  Analyze > Compare Means > Independent-Samples T Test .
• Move the variable Athlete to the Grouping Variable field, and move the variable MileMinDur to the Test Variable(s) area. Now Athlete is defined as the independent variable and MileMinDur is defined as the dependent variable.
• Click Define Groups , which opens a new window. Use specified values is selected by default. Since our grouping variable is numerically coded (0 = "Non-athlete", 1 = "Athlete"), type “0” in the first text box, and “1” in the second text box. This indicates that we will compare groups 0 and 1, which correspond to non-athletes and athletes, respectively. Click Continue when finished.
• Click OK to run the Independent Samples t Test. Output for the analysis will display in the Output Viewer window. 

Two sections (boxes) appear in the output: Group Statistics and Independent Samples Test . The first section, Group Statistics , provides basic information about the group comparisons, including the sample size ( n ), mean, standard deviation, and standard error for mile times by group. In this example, there are 166 athletes and 226 non-athletes. The mean mile time for athletes is 6 minutes 51 seconds, and the mean mile time for non-athletes is 9 minutes 6 seconds.

The second section, Independent Samples Test , displays the results most relevant to the Independent Samples t Test. There are two parts that provide different pieces of information: (A) Levene’s Test for Equality of Variances and (B) t-test for Equality of Means.

A Levene's Test for Equality of of Variances : This section has the test results for Levene's Test. From left to right:

• F is the test statistic of Levene's test
• Sig. is the p-value corresponding to this test statistic.

The p -value of Levene's test is printed as ".000" (but should be read as p < 0.001 -- i.e., p very small), so we we reject the null of Levene's test and conclude that the variance in mile time of athletes is significantly different than that of non-athletes. This tells us that we should look at the "Equal variances not assumed" row for the t test (and corresponding confidence interval) results . (If this test result had not been significant -- that is, if we had observed p > α -- then we would have used the "Equal variances assumed" output.)

B t-test for Equality of Means provides the results for the actual Independent Samples t Test. From left to right:

• t is the computed test statistic, using the formula for the equal-variances-assumed test statistic (first row of table) or the formula for the equal-variances-not-assumed test statistic (second row of table)
• df is the degrees of freedom, using the equal-variances-assumed degrees of freedom formula (first row of table) or the equal-variances-not-assumed degrees of freedom formula (second row of table)
• Sig (2-tailed) is the p-value corresponding to the given test statistic and degrees of freedom
• Mean Difference is the difference between the sample means, i.e. x 1  − x 2 ; it also corresponds to the numerator of the test statistic for that test
• Std. Error Difference is the standard error of the mean difference estimate; it also corresponds to the denominator of the test statistic for that test

Note that the mean difference is calculated by subtracting the mean of the second group from the mean of the first group. In this example, the mean mile time for athletes was subtracted from the mean mile time for non-athletes (9:06 minus 6:51 = 02:14). The sign of the mean difference corresponds to the sign of the t value. The positive t value in this example indicates that the mean mile time for the first group, non-athletes, is significantly greater than the mean for the second group, athletes.

The associated p value is printed as ".000"; double-clicking on the p-value will reveal the un-rounded number. SPSS rounds p-values to three decimal places, so any p-value too small to round up to .001 will print as .000. (In this particular example, the p-values are on the order of 10 -40 .)

C Confidence Interval of the Difference : This part of the t -test output complements the significance test results. Typically, if the CI for the mean difference contains 0 within the interval -- i.e., if the lower boundary of the CI is a negative number and the upper boundary of the CI is a positive number -- the results are not significant at the chosen significance level. In this example, the 95% CI is [01:57, 02:32], which does not contain zero; this agrees with the small p -value of the significance test.

Decision and Conclusions

Since p < .001 is less than our chosen significance level α = 0.05, we can reject the null hypothesis, and conclude that the that the mean mile time for athletes and non-athletes is significantly different.

Based on the results, we can state the following:

• There was a significant difference in mean mile time between non-athletes and athletes ( t 315.846 = 15.047, p < .001).
• The average mile time for athletes was 2 minutes and 14 seconds lower than the average mile time for non-athletes.
• << Previous: Paired Samples t Test
• Next: One-Way ANOVA >>
• Last Updated: Jul 10, 2024 11:08 AM
• URL: https://libguides.library.kent.edu/SPSS

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Which statistical test to use for Likert scale questionaire

I need help on determining what tests to use in SPSS. Let me provide some background on my research:

• I am doing a research on system automation of leave management in a company (which is still using a manual paper form to apply for leave).
• My hypothesis validates the automation of the different processes (independent variables) on the overall improvement of the leave management process (dependent). My control variables are age, gender and dept.
• I have developed a prototype of a system and provided a questionnaire in the form of a 5-point Likert scale to the participant. (Strongly Agree, Agree, Neutral, Disagree, Strongly Disagree).
• The questions measures their response on the statement provided on the test evaluation of the system. These questions are categorised based on the hypothesis.

Where do I start?

• hypothesis-testing

3 Answers 3

It sounds like your response (or "dependent") variable is perception of improvement eg individuals' response to a statement "leave management is better now than it was" (agree/disagree etc). Assuming this is the case, you first need to clarify that you don't really test the hypothesis of improvement of the system, only retrospective perceptions of it.

Putting that aside and given the data that you apparently have, it sounds like the technique you need is ordinal regression . A quick google suggests this is possible in SPSS (I'm not an SPSS user so I won't try to recommend any of the links).

• $\begingroup$ Yes, you are right - I would like to test the users response to the statements as the perception of the improvement. Is there any other tests that I can use? Which test to use to to know whether there is a relationship between the statements and the age, gender, dept the users belongs to? $\endgroup$ –  youngmoon Commented Mar 27, 2013 at 9:01

"Where do I start?"

You should always start a research study by clearly defining your objective (supported by literature review) and then identify your hypothesis to be tested, if any. Based on these, you should collect data to analyze. What's your objective?

• To study the factors affecting the perceptions of users about overall improvement of the leave management process by automation of the different processes.
• To study the impact of demographic characteristics on these factors

If these are your objectives, then you can use the following techniques:

Factor Analysis to identify the factors (for objective -1)

Three ANOVA tests corresponding to each demographic characteristic (age, gender, dept) [Note: for categories of input variable like gender (M/F), t- test can also be used, the results would be the same.

Try Fishers exact test to assess the association between the dependent and independent variables; whenever there is an association go for Contingency coefficient to see the strength of association between the variables. Go for correlation coefficient to see the relationship between demographic variables and perception improvement. But you have to do scoring for perception improvement

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• How to Conduct Hypothesis Testing in Statistics

The Fundamentals of Hypothesis Testing: What Every Student Should Know

Hypothesis testing is a fundamental statistical technique used to make inferences about populations based on sample data. This blog will guide you through the process of hypothesis testing, helping you understand and apply the concepts to solve similar assignments efficiently. By following this structured approach, you'll be able to solve your hypothesis testing homework problem with confidence.

Understanding the Basics of Hypothesis Testing

Hypothesis testing involves making a decision about the validity of a hypothesis based on sample data. It comprises four key steps: defining hypotheses, calculating the test statistic, determining the p-value, and drawing conclusions. Let's explore each of these steps in detail.

Defining Hypotheses

The first step in hypothesis testing is to define the null and alternative hypotheses. These hypotheses represent the statements we want to test.

Null Hypothesis (H0)

The null hypothesis (H0) is a statement that there is no effect or difference. It serves as the default assumption that we aim to test against.

Alternative Hypothesis (Ha or H1)

The alternative hypothesis (Ha or H1) is a statement that indicates the presence of an effect or difference. It represents what we want to prove.

Types of Tests

Depending on the direction of the hypothesis, we have three types of tests: left-tailed, right-tailed, and two-tailed tests.

Left-Tailed Test

A left-tailed test is used when we want to determine if the population mean is less than a specified value.

Right-Tailed Test

A right-tailed test is used when we want to determine if the population mean is greater than a specified value.

Two-Tailed Test

A two-tailed test is used when we want to determine if the population mean is different from a specified value, either higher or lower.

Example Scenario

Consider a scenario where we want to test if the average vehicle price from a sample is less than $27,000. We would set up our hypotheses as follows:

• Null Hypothesis (H0): μ = 27,000
• Alternative Hypothesis (Ha): μ < 27,000

Calculating the Test Statistic

Once the hypotheses are defined, the next step is to calculate the test statistic. The test statistic helps us determine the likelihood of observing the sample data under the null hypothesis.

Formula for the T-Test Statistic

The t-test statistic is calculated using the formula:

[ t = \frac{\bar{X} - \mu}{S / \sqrt{n}} ]

• (\bar{X}) is the sample mean
• (S) is the sample standard deviation
• (n) is the sample size
• (\mu) is the population mean defined in the null hypothesis

Standard Error

The denominator of the t-test formula, (S / \sqrt{n}), is known as the standard error (SE). It measures the variability of the sample mean.

Example Calculation

Let's calculate the test statistic for our vehicle price example. Given:

• Sample mean ((\bar{X})) = 25,650
• Sample standard deviation (S) = 3,488
• Sample size (n) = 10
• Population mean ((\mu)) = 27,000

First, we calculate the standard error (SE):

[ SE = \frac{S}{\sqrt{n}} = \frac{3488}{\sqrt{10}} \approx 1103 ]

Next, we calculate the test statistic (t):

[ t = \frac{25650 - 27000}{1103} \approx -1.2238 ]

Determining the P-Value

The p-value is a critical component of hypothesis testing. It indicates the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true.

Calculating the P-Value

The method to calculate the p-value depends on the type of test (left-tailed, right-tailed, or two-tailed) and the direction of the alternative hypothesis.

For a left-tailed test, the p-value is calculated using the T.DIST() function in Excel.

For a right-tailed test, the p-value is calculated using the T.DIST.RT() function in Excel.

For a two-tailed test, the p-value is calculated using the T.DIST.2T() function in Excel. When the test statistic is negative, use the absolute value function (ABS()) to remove the negative sign before calculating the p-value.

For our vehicle price example with a left-tailed test, we calculate the p-value using the T.DIST() function in Excel:

[ \text{p-value} = T.DIST(-1.2238, 9, TRUE) \approx 0.1261 ]

Drawing Conclusions

The final step in hypothesis testing is to draw a conclusion based on the p-value and a pre-determined significance level ((\alpha)).

Significance Level ((\alpha))

The significance level ((\alpha)) is the threshold for deciding whether to reject the null hypothesis. Common values for (\alpha) are 0.05, 0.01, 0.10, and 0.005.

Decision Rule

• If the p-value is less than (\alpha), we reject the null hypothesis.
• If the p-value is greater than (\alpha), we fail to reject the null hypothesis.

Example Conclusion

For our vehicle price example with (\alpha = 0.05):

• p-value = 0.1261
• (\alpha) = 0.05

Since 0.1261 > 0.05, we fail to reject the null hypothesis. There is not enough evidence to suggest that the average vehicle price is less than $27,000.

Practical Examples of Hypothesis Testing

To further illustrate hypothesis testing, let's explore three different scenarios: left-tailed test, right-tailed test, and two-tailed test.

Left-Tailed Test Example

In this example, we test if the average vehicle price is less than $27,000.

Step-by-Step Process

Define Hypotheses:

Calculate Test Statistic:

• Standard error (SE) = 1103
• Test statistic (t) = -1.2238

Determine P-Value:

Draw Conclusion:

• Since 0.1261 > 0.05, fail to reject the null hypothesis.
• Conclusion: There is not enough evidence to suggest that the average vehicle price is less than $27,000.

Right-Tailed Test Example

In this example, we test if the average vehicle price is greater than $23,500.

• Null Hypothesis (H0): μ = 23,500
• Alternative Hypothesis (Ha): μ > 23,500
• Population mean ((\mu)) = 23,500
• Test statistic (t) = 1.9490
• p-value = 0.0416
• Since 0.0416 < 0.05, reject the null hypothesis.
• Conclusion: There is enough evidence to suggest that the average vehicle price is greater than $23,500.

Two-Tailed Test Example

In this example, we test if the average vehicle price is different from $23,500.

• Alternative Hypothesis (Ha): μ ≠ 23,500
• p-value = 0.0831
• Since 0.0831 > 0.05, fail to reject the null hypothesis.
• Conclusion: There is not enough evidence to suggest that the average vehicle price is different from $23,500.

Tips for Conducting Hypothesis Testing

Successfully conducting hypothesis testing involves several critical steps. Here are some tips to help you perform hypothesis testing effectively.

Proper Data Collection

Accurate and reliable data collection is crucial for hypothesis testing. Ensure that your sample is representative of the population and collected using appropriate methods.

Random Sampling

Use random sampling techniques to avoid bias and ensure that your sample accurately represents the population.

Sample Size

Ensure that your sample size is large enough to provide reliable results. Larger sample sizes reduce the margin of error and increase the power of the test.

Verify Assumptions

Hypothesis tests often rely on certain assumptions about the data. Verify these assumptions before proceeding with the test.

Many hypothesis tests, including the t-test, assume that the data follows a normal distribution. Use graphical methods (e.g., histograms, Q-Q plots) or statistical tests (e.g., Shapiro-Wilk test) to check for normality.

Independence

Ensure that the observations in your sample are independent of each other. Independence is a key assumption for most hypothesis tests.

Utilize Software Tools

Software tools like Excel , R , and SPSS can simplify the calculations involved in hypothesis testing and reduce the risk of errors.

Excel provides several functions for hypothesis testing, such as T.DIST(), T.DIST.RT(), and T.DIST.2T(). Use these functions to calculate p-values and make decisions based on your test statistics.

R is a powerful statistical software that offers various packages for hypothesis testing. Use functions like t.test() to perform t-tests and obtain p-values and confidence intervals.

Interpret Results Carefully

Proper interpretation of the results is crucial for drawing accurate conclusions from hypothesis testing.

Statistical Significance

A statistically significant result (p-value < (\alpha)) indicates that there is strong evidence against the null hypothesis. However, it does not imply practical significance. Consider the context and the practical implications of the results.

Type I and Type II Errors

Be aware of the potential for Type I and Type II errors. A Type I error occurs when the null hypothesis is incorrectly rejected, while a Type II error occurs when the null hypothesis is not rejected despite being false. The significance level ((\alpha)) affects the probability of Type I errors, while the sample size and effect size influence the probability of Type II errors.

Report Results Transparently

When reporting the results of hypothesis testing, include all relevant information to ensure transparency and reproducibility.

Detailed Description

Provide a detailed description of the hypotheses, test statistic, p-value, significance level, and the conclusion. This information helps others understand and evaluate your analysis.

Confidence Intervals

Include confidence intervals for the estimated parameters. Confidence intervals provide a range of plausible values for the population parameter and offer additional context for interpreting the results.

Common Pitfalls in Hypothesis Testing

Hypothesis testing is a powerful tool, but it is essential to be aware of common pitfalls to avoid incorrect conclusions.

Misinterpreting P-Values

P-values indicate the probability of obtaining a test statistic as extreme as the one observed, assuming the null hypothesis is true. A small p-value suggests strong evidence against the null hypothesis, but it does not provide a measure of the effect size or practical significance.

P-Value Misconceptions

Avoid common misconceptions about p-values, such as believing that a p-value of 0.05 means there is a 5% chance that the null hypothesis is true. P-values do not measure the probability that the null hypothesis is true or false.

Ignoring Assumptions

Ignoring the assumptions underlying hypothesis tests can lead to incorrect conclusions. Always verify the assumptions before proceeding with the test.

Assumption Violations

If the assumptions are violated, consider using alternative tests that do not rely on those assumptions. For example, if the data is not normally distributed, use non-parametric tests like the Wilcoxon rank-sum test or the Mann-Whitney U test.

Overemphasizing Statistical Significance

Statistical significance does not imply practical significance. A result can be statistically significant but have a negligible practical effect. Always consider the context and practical implications of the results.

Effect Size

Report and interpret effect sizes alongside p-values. Effect sizes provide a measure of the magnitude of the observed effect and offer valuable context for interpreting the results.

Hypothesis testing is a critical tool in statistics for making inferences about populations based on sample data. By understanding the steps involved—defining hypotheses, calculating the test statistic, determining the p-value, and drawing conclusions—you can approach hypothesis testing with confidence.

Ensure proper data collection, verify assumptions, utilize software tools, interpret results carefully, and report findings transparently to enhance the reliability and validity of your hypothesis tests. By avoiding common pitfalls and considering both statistical and practical significance, you'll be well-equipped to tackle statistics homework and research projects effectively.

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