hypothesis test sign

Sign Test: Step by Step Calculation

Hypothesis Testing > Sign Test

What is the Sign Test?

The sign test compares the sizes of two groups. It is a non-parametric or “distribution free” test, which means the test doesn’t assume the data comes from a particular distribution, like the normal distribution . The sign test is an alternative to a one sample t test or a paired t test . It can also be used for ordered (ranked) categorical data.

The null hypothesis for the sign test is that the difference between medians is zero.

For a one sample sign test , where the median for a single sample is analyzed, see: One Sample Median Tests.

How to Calculate a Paired/Matched Sample Sign Test

Assumptions for the test (your data should meet these requirements before running the test) are:

• The data should be from two samples.
• The two dependent samples should be paired or matched. For example, depression scores from before a medical procedure and after.

To set up the test, put your two sets of sample data into a table (I used Excel ). This set of data represents test scores at the end of Spring and the beginning of the Fall semesters. The hypothesis is that the summer break means a significant drop in test scores.

• H 0 : No difference in median of the signed differences.
• H 1 : Median of the signed differences is less than zero.

Step 2: Add a fourth column indicating the sign of the number in column 3.

Step 3: Count the number of positives and negatives.

• 4 positives.
• 12 negatives.

12 negatives seems like a lot , but we can’t say for sure that it’s significant (i.e. that it didn’t happen by chance) until we run the sign test.

Step 3: Add up the number of items in your sample and subtract any you had a difference of zero for (in column 3). The sample size in this question was 17, with one zero, so n = 16.

Step 4: Find the p-value using a binomial distribution table or use a binomial calculator . I used the calculator, putting in:

• .5 for the probability . The null hypothesis is that there are an equal number of signs (i.e. 50/50). Therefore, the test is a simple binomial experiment with a .5 chance of the sign being negative and .5 of it being positive (assuming the null hypothesis is true).
• 16 for the number of trials.
• 4 for the number of successes. “Successes” here is the smaller of either the positive or negative signs from Step 2 .

The p-value is 0.038, which is smaller than the alpha level of 0.05. We can reject the null hypothesis and say there is a significant difference.

Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley. Lindstrom, D. (2010). Schaum’s Easy Outline of Statistics , Second Edition (Schaum’s Easy Outlines) 2nd Edition. McGraw-Hill Education Vogt, W.P. (2005). Dictionary of Statistics & Methodology: A Nontechnical Guide for the Social Sciences . SAGE. Wheelan, C. (2014). Naked Statistics . W. W. Norton & Company

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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 ).

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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

Hypothesis Testing: Uses, Steps & Example

By Jim Frost 4 Comments

What is Hypothesis Testing?

Hypothesis testing in statistics uses sample data to infer the properties of a whole population . These tests determine whether a random sample provides sufficient evidence to conclude an effect or relationship exists in the population. Researchers use them to help separate genuine population-level effects from false effects that random chance can create in samples. These methods are also known as significance testing.

For example, researchers are testing a new medication to see if it lowers blood pressure. They compare a group taking the drug to a control group taking a placebo. If their hypothesis test results are statistically significant, the medication’s effect of lowering blood pressure likely exists in the broader population, not just the sample studied.

Using Hypothesis Tests

A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement the sample data best supports. These two statements are called the null hypothesis and the alternative hypothesis . The following are typical examples:

• Null Hypothesis : The effect does not exist in the population.
• Alternative Hypothesis : The effect does exist in the population.

Hypothesis testing accounts for the inherent uncertainty of using a sample to draw conclusions about a population, which reduces the chances of false discoveries. These procedures determine whether the sample data are sufficiently inconsistent with the null hypothesis that you can reject it. If you can reject the null, your data favor the alternative statement that an effect exists in the population.

Statistical significance in hypothesis testing indicates that an effect you see in sample data also likely exists in the population after accounting for random sampling error , variability, and sample size. Your results are statistically significant when the p-value is less than your significance level or, equivalently, when your confidence interval excludes the null hypothesis value.

Conversely, non-significant results indicate that despite an apparent sample effect, you can’t be sure it exists in the population. It could be chance variation in the sample and not a genuine effect.

Learn more about Failing to Reject the Null .

5 Steps of Significance Testing

Hypothesis testing involves five key steps, each critical to validating a research hypothesis using statistical methods:

• Formulate the Hypotheses : Write your research hypotheses as a null hypothesis (H 0 ) and an alternative hypothesis (H A ).
• Data Collection : Gather data specifically aimed at testing the hypothesis.
• Conduct A Test : Use a suitable statistical test to analyze your data.
• Make a Decision : Based on the statistical test results, decide whether to reject the null hypothesis or fail to reject it.
• Report the Results : Summarize and present the outcomes in your report’s results and discussion sections.

While the specifics of these steps can vary depending on the research context and the data type, the fundamental process of hypothesis testing remains consistent across different studies.

Let’s work through these steps in an example!

Hypothesis Testing Example

Researchers want to determine if a new educational program improves student performance on standardized tests. They randomly assign 30 students to a control group , which follows the standard curriculum, and another 30 students to a treatment group, which participates in the new educational program. After a semester, they compare the test scores of both groups.

Download the CSV data file to perform the hypothesis testing yourself: Hypothesis_Testing .

The researchers write their hypotheses. These statements apply to the population, so they use the mu (μ) symbol for the population mean parameter .

• Null Hypothesis (H 0 ) : The population means of the test scores for the two groups are equal (μ 1 = μ 2 ).
• Alternative Hypothesis (H A ) : The population means of the test scores for the two groups are unequal (μ 1 ≠ μ 2 ).

Choosing the correct hypothesis test depends on attributes such as data type and number of groups. Because they’re using continuous data and comparing two means, the researchers use a 2-sample t-test .

Here are the results.

The treatment group’s mean is 58.70, compared to the control group’s mean of 48.12. The mean difference is 10.67 points. Use the test’s p-value and significance level to determine whether this difference is likely a product of random fluctuation in the sample or a genuine population effect.

Because the p-value (0.000) is less than the standard significance level of 0.05, the results are statistically significant, and we can reject the null hypothesis. The sample data provides sufficient evidence to conclude that the new program’s effect exists in the population.

Limitations

Hypothesis testing improves your effectiveness in making data-driven decisions. However, it is not 100% accurate because random samples occasionally produce fluky results. Hypothesis tests have two types of errors, both relating to drawing incorrect conclusions.

• Type I error: The test rejects a true null hypothesis—a false positive.
• Type II error: The test fails to reject a false null hypothesis—a false negative.

Learn more about Type I and Type II Errors .

Our exploration of hypothesis testing using a practical example of an educational program reveals its powerful ability to guide decisions based on statistical evidence. Whether you’re a student, researcher, or professional, understanding and applying these procedures can open new doors to discovering insights and making informed decisions. Let this tool empower your analytical endeavors as you navigate through the vast seas of data.

Learn more about the Hypothesis Tests for Various Data Types .

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Reader Interactions

June 10, 2024 at 10:51 am

Thank you, Jim, for another helpful article; timely too since I have started reading your new book on hypothesis testing and, now that we are at the end of the school year, my district is asking me to perform a number of evaluations on instructional programs. This is where my question/concern comes in. You mention that hypothesis testing is all about testing samples. However, I use all the students in my district when I make these comparisons. Since I am using the entire “population” in my evaluations (I don’t select a sample of third grade students, for example, but I use all 700 third graders), am I somehow misusing the tests? Or can I rest assured that my district’s student population is only a sample of the universal population of students?

June 10, 2024 at 1:50 pm

I hope you are finding the book helpful!

Yes, the purpose of hypothesis testing is to infer the properties of a population while accounting for random sampling error.

In your case, it comes down to how you want to use the results. Who do you want the results to apply to?

If you’re summarizing the sample, looking for trends and patterns, or evaluating those students and don’t plan to apply those results to other students, you don’t need hypothesis testing because there is no sampling error. They are the population and you can just use descriptive statistics. In this case, you’d only need to focus on the practical significance of the effect sizes.

On the other hand, if you want to apply the results from this group to other students, you’ll need hypothesis testing. However, there is the complicating issue of what population your sample of students represent. I’m sure your district has its own unique characteristics, demographics, etc. Your district’s students probably don’t adequately represent a universal population. At the very least, you’d need to recognize any special attributes of your district and how they could bias the results when trying to apply them outside the district. Or they might apply to similar districts in your region.

However, I’d imagine your 3rd graders probably adequately represent future classes of 3rd graders in your district. You need to be alert to changing demographics. At least in the short run I’d imagine they’d be representative of future classes.

Think about how these results will be used. Do they just apply to the students you measured? Then you don’t need hypothesis tests. However, if the results are being used to infer things about other students outside of the sample, you’ll need hypothesis testing along with considering how well your students represent the other students and how they differ.

I hope that helps!

June 10, 2024 at 3:21 pm

Thank you so much, Jim, for the suggestions in terms of what I need to think about and consider! You are always so clear in your explanations!!!!

June 10, 2024 at 3:22 pm

You’re very welcome! Best of luck with your evaluations!

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Introduction to the Sign Test

The Sign Test stands as a fundamental non-parametric statistical method designed to compare two related samples, typically used in scenarios where more conventional tests such as the t-test cannot be applied due to the distributional characteristics of the data. It focuses on the direction (sign) of changes between paired observations rather than their numerical differences, offering a straightforward approach for assessing median differences.

Key Features of the Sign Test

• Non-parametric Nature : It does not assume a normal distribution of the data, making it suitable for a wide range of datasets, including ordinal data.
• Simplicity : The test relies solely on the signs (+ or -) of the differences between paired observations, disregarding their magnitudes.
• Application : Known as the binomial sign test, it operates under the hypothesis that the probability (p) of observing a positive difference is 0.5, reflecting no systematic bias between the two groups.

Practical Applications: Understanding Consumer Preferences

An exemplary application of the Sign Test can be demonstrated through a consumer preference study, such as comparing preferences between two popular soda brands, Pepsi and Coke, among a group of 10 consumers. By asking participants which brand they prefer and pairing their responses before and after a specific intervention (e.g., a blind taste test), researchers can apply the Sign Test to determine if there is a statistically significant preference for one product over the other.

Assumptions of the Sign Test

• Data Distribution : The test is distribution-free, meaning it does not require the data to follow a specific distribution pattern.
• Sample Origin : The data should originate from two related samples, which could represent the same group under different conditions or times.
• Dependence : Samples must be paired or matched, often reflecting a ‘before-and-after’ scenario, where the pairing is intrinsic to the research design.

Conducting the Sign Test in SPSS

To perform the Sign Test in SPSS, follow these steps:

• Navigate to the “menu” and select “Analysis.”
• Choose “Nonparametric” from the options.
• Click on “Two Related Samples” and select the “Sign Test.”

This process allows researchers to easily execute the test within SPSS, providing a user-friendly interface for analyzing paired data.

Conclusion: Evaluating the Sign Test’s Utility

While the Sign Test is considered less powerful than other statistical tests due to its focus on signs rather than magnitudes of change, its simplicity and applicability in situations where data do not meet the assumptions of parametric tests make it an invaluable tool in the researcher’s arsenal. By enabling the analysis of median differences between paired samples without stringent distributional requirements, the Sign Test facilitates the exploration of research questions across various domains, from consumer preferences to medical studies, where data may not adhere to normal distribution or when numerical data are not available.

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Types of sign test:

• One sample: We set up the hypothesis so that + and – signs are the values of random variables having equal size.
• Paired sample: This test is also called an alternative to the paired t-test .  This test uses the + and – signs in paired sample tests or in before-after study. In this test, null hypothesis is set up so that the sign of + and – are of equal size, or the population means are equal to the sample mean.
• Calculate the + and – sign for the given distribution.  Put a + sign for a value greater than the mean value, and put a – sign for a value less than the mean value.  Put 0 as the value is equal to the mean value; pairs with 0 as the mean value are considered ties.
• Denote the total number of signs by ‘n’ (ignore the zero sign) and the number of less frequent signs by ‘S.’
• Obtain the critical value (K) at .05 of the significance level by using the following formula in case of small samples:

Sign test in case of large sample:

• Compare the value of ‘S’ with the critical value (K). If the value of S is greater than the value of K, then the null hypothesis is accepted.  If the value of the S is less than the critical value of K, then the null hypothesis is accepted.  In the case of large samples, S is compared with the Z value.

Available in nonparametric tests, the following steps are involved in conducting a sign test in SPSS:

• Click on the “SPSS” icon from the start menu.  The following window will appear when we will click on the SPSS icon:

• Click on the “open data” icon and select the data.
• Select “nonparametric test” from the analysis menu and select “two related sample” from the nonparametric option.  As we click on the two related samples, the following window will appear:

Select the first paired variable and drag it to the right side in variable 1, and select the second paired variable and drag it to the right side in variable 2.  Select the “sign test” from the available test.  Click on “options” and select “descriptive” from there.  Now, click on the “ok” button. The result window for the sign test will appear.

In the result window, the first table will be of the descriptive statistics for sign test.  These will include the number of observations per sample, the mean, the SD, the minimum and the maximum value for sign tests in both samples.  The second table shows the frequency table.  This will show the number of negative sign, the number of positive sign for the number of ties, and the total number of observations.  In SPSS, the following table will appear for the descriptive table and frequency:

* Click here for assistance with conducting the sign test or other quantitative analyses.

Related Analysis:

• Conduct and Interpret a Wilcoxon Sign Test
• McNemar’s Test

Statistics Solutions can assist with your quantitative analysis by assisting you to develop your methodology and results chapters. The services that we offer include:

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This page offers all the basic information you need about the sign test. It is part of Statkat’s wiki module, containing similarly structured info pages for many different statistical methods. The info pages give information about null and alternative hypotheses, assumptions, test statistics and confidence intervals, how to find p values, SPSS how-to’s and more.

To compare the sign test with other statistical methods, go to Statkat's Comparison tool or practice with the sign test at Statkat's Practice question center

• 1. When to use
• 2. Null hypothesis
• 3. Alternative hypothesis
• 4. Assumptions
• 5. Test statistic
• 6. Sampling distribution
• 7. Significant?
• 8. Equivalent to
• 9. Example context

When to use?

Note that theoretically, it is always possible to 'downgrade' the measurement level of a variable. For instance, a test that can be performed on a variable of ordinal measurement level can also be performed on a variable of interval measurement level, in which case the interval variable is downgraded to an ordinal variable. However, downgrading the measurement level of variables is generally a bad idea since it means you are throwing away important information in your data (an exception is the downgrade from ratio to interval level, which is generally irrelevant in data analysis).

If you are not sure which method you should use, you might like the assistance of our method selection tool or our method selection table .

Null hypothesis

The sign test tests the following null hypothesis (H 0 ):

• H 0 : P(first score of a pair exceeds second score of a pair) = P(second score of a pair exceeds first score of a pair)
• H 0 : the population median of the difference scores is equal to zero

Alternative hypothesis

The sign test tests the above null hypothesis against the following alternative hypothesis (H 1 or H a ):

• H 1 two sided: P(first score of a pair exceeds second score of a pair) $\neq$ P(second score of a pair exceeds first score of a pair)
• H 1 right sided: P(first score of a pair exceeds second score of a pair) > P(second score of a pair exceeds first score of a pair)
• H 1 left sided: P(first score of a pair exceeds second score of a pair) < P(second score of a pair exceeds first score of a pair)
• H 1 two sided: the population median of the difference scores is different from zero
• H 1 right sided: the population median of the difference scores is larger than zero
• H 1 left sided: the population median of the difference scores is smaller than zero

Assumptions

Statistical tests always make assumptions about the sampling procedure that was used to obtain the sample data. So called parametric tests also make assumptions about how data are distributed in the population. Non-parametric tests are more 'robust' and make no or less strict assumptions about population distributions, but are generally less powerful. Violation of assumptions may render the outcome of statistical tests useless, although violation of some assumptions (e.g. independence assumptions) are generally more problematic than violation of other assumptions (e.g. normality assumptions in combination with large samples).

The sign test makes the following assumptions:

• Sample of pairs is a simple random sample from the population of pairs. That is, pairs are independent of one another

Test statistic

The sign test is based on the following test statistic:

Sampling distribution

Significant.

This is how you find out if your test result is significant:

• Check if $W$ observed in sample is in the rejection region or
• Find two sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
• Find right sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
• Find left sided $p$ value corresponding to observed $W$ and check if it is equal to or smaller than $\alpha$
• Check if $z$ observed in sample is at least as extreme as critical value $z^*$ or
• Find two sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
• Check if $z$ observed in sample is equal to or larger than critical value $z^*$ or
• Find right sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$
• Check if $z$ observed in sample is equal to or smaller than critical value $z^*$ or
• Find left sided $p$ value corresponding to observed $z$ and check if it is equal to or smaller than $\alpha$

Equivalent to

The sign test is equivalent to:

• McNemar's test : $W = b$, and $z^2 = X^2$
• Friedman test with two related groups

Example context

The sign test could for instance be used to answer the question:

How to perform the sign test in SPSS:

• Put the two paired variables in the boxes below Variable 1 and Variable 2
• Under Test Type, select the Sign test

How to perform the sign test in jamovi :

• Put the two paired variables in the box below Measures

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Hypothesis Testing

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A hypothesis test is a statistical inference method used to test the significance of a proposed (hypothesized) relation between population statistics (parameters) and their corresponding sample estimators . In other words, hypothesis tests are used to determine if there is enough evidence in a sample to prove a hypothesis true for the entire population.

The test considers two hypotheses: the null hypothesis , which is a statement meant to be tested, usually something like "there is no effect" with the intention of proving this false, and the alternate hypothesis , which is the statement meant to stand after the test is performed. The two hypotheses must be mutually exclusive ; moreover, in most applications, the two are complementary (one being the negation of the other). The test works by comparing the \(p\)-value to the level of significance (a chosen target). If the \(p\)-value is less than or equal to the level of significance, then the null hypothesis is rejected.

When analyzing data, only samples of a certain size might be manageable as efficient computations. In some situations the error terms follow a continuous or infinite distribution, hence the use of samples to suggest accuracy of the chosen test statistics. The method of hypothesis testing gives an advantage over guessing what distribution or which parameters the data follows.

Definitions and Methodology

Hypothesis test and confidence intervals.

In statistical inference, properties (parameters) of a population are analyzed by sampling data sets. Given assumptions on the distribution, i.e. a statistical model of the data, certain hypotheses can be deduced from the known behavior of the model. These hypotheses must be tested against sampled data from the population.

The null hypothesis \((\)denoted \(H_0)\) is a statement that is assumed to be true. If the null hypothesis is rejected, then there is enough evidence (statistical significance) to accept the alternate hypothesis \((\)denoted \(H_1).\) Before doing any test for significance, both hypotheses must be clearly stated and non-conflictive, i.e. mutually exclusive, statements. Rejecting the null hypothesis, given that it is true, is called a type I error and it is denoted \(\alpha\), which is also its probability of occurrence. Failing to reject the null hypothesis, given that it is false, is called a type II error and it is denoted \(\beta\), which is also its probability of occurrence. Also, \(\alpha\) is known as the significance level , and \(1-\beta\) is known as the power of the test. \(H_0\) \(\textbf{is true}\)\(\hspace{15mm}\) \(H_0\) \(\textbf{is false}\) \(\textbf{Reject}\) \(H_0\)\(\hspace{10mm}\) Type I error Correct Decision \(\textbf{Reject}\) \(H_1\) Correct Decision Type II error The test statistic is the standardized value following the sampled data under the assumption that the null hypothesis is true, and a chosen particular test. These tests depend on the statistic to be studied and the assumed distribution it follows, e.g. the population mean following a normal distribution. The \(p\)-value is the probability of observing an extreme test statistic in the direction of the alternate hypothesis, given that the null hypothesis is true. The critical value is the value of the assumed distribution of the test statistic such that the probability of making a type I error is small.

Methodologies: Given an estimator \(\hat \theta\) of a population statistic \(\theta\), following a probability distribution \(P(T)\), computed from a sample \(\mathcal{S},\) and given a significance level \(\alpha\) and test statistic \(t^*,\) define \(H_0\) and \(H_1;\) compute the test statistic \(t^*.\) \(p\)-value Approach (most prevalent): Find the \(p\)-value using \(t^*\) (right-tailed). If the \(p\)-value is at most \(\alpha,\) reject \(H_0\). Otherwise, reject \(H_1\). Critical Value Approach: Find the critical value solving the equation \(P(T\geq t_\alpha)=\alpha\) (right-tailed). If \(t^*>t_\alpha\), reject \(H_0\). Otherwise, reject \(H_1\). Note: Failing to reject \(H_0\) only means inability to accept \(H_1\), and it does not mean to accept \(H_0\).

Assume a normally distributed population has recorded cholesterol levels with various statistics computed. From a sample of 100 subjects in the population, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is larger than 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the \(p\)-value approach with significance level \(\alpha=0.05:\) Define \(H_0\): \(\mu=200\). Define \(H_1\): \(\mu>200\). Since our values are normally distributed, the test statistic is \(z^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{100}}}\approx 3.09\). Using a standard normal distribution, we find that our \(p\)-value is approximately \(0.001\). Since the \(p\)-value is at most \(\alpha=0.05,\) we reject \(H_0\). Therefore, we can conclude that the test shows sufficient evidence to support the claim that \(\mu\) is larger than \(200\) mg/dL.

If the sample size was smaller, the normal and \(t\)-distributions behave differently. Also, the question itself must be managed by a double-tail test instead.

Assume a population's cholesterol levels are recorded and various statistics are computed. From a sample of 25 subjects, the sample mean was 214.12 mg/dL (milligrams per deciliter), with a sample standard deviation of 45.71 mg/dL. Perform a hypothesis test, with significance level 0.05, to test if there is enough evidence to conclude that the population mean is not equal to 200 mg/dL. Hypothesis Test We will perform a hypothesis test using the \(p\)-value approach with significance level \(\alpha=0.05\) and the \(t\)-distribution with 24 degrees of freedom: Define \(H_0\): \(\mu=200\). Define \(H_1\): \(\mu\neq 200\). Using the \(t\)-distribution, the test statistic is \(t^*=\frac{\bar X - \mu_0}{\frac{s}{\sqrt{n}}}=\frac{214.12 - 200}{\frac{45.71}{\sqrt{25}}}\approx 1.54\). Using a \(t\)-distribution with 24 degrees of freedom, we find that our \(p\)-value is approximately \(2(0.068)=0.136\). We have multiplied by two since this is a two-tailed argument, i.e. the mean can be smaller than or larger than. Since the \(p\)-value is larger than \(\alpha=0.05,\) we fail to reject \(H_0\). Therefore, the test does not show sufficient evidence to support the claim that \(\mu\) is not equal to \(200\) mg/dL.

The complement of the rejection on a two-tailed hypothesis test (with significance level \(\alpha\)) for a population parameter \(\theta\) is equivalent to finding a confidence interval \((\)with confidence level \(1-\alpha)\) for the population parameter \(\theta\). If the assumption on the parameter \(\theta\) falls inside the confidence interval, then the test has failed to reject the null hypothesis \((\)with \(p\)-value greater than \(\alpha).\) Otherwise, if \(\theta\) does not fall in the confidence interval, then the null hypothesis is rejected in favor of the alternate \((\)with \(p\)-value at most \(\alpha).\)

• Statistics (Estimation)
• Normal Distribution
• Correlation
• Confidence Intervals

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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 & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

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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.

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 that 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. This data should be representative of the population to infer conclusions accurately.

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.

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.

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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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.

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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.

If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning!

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.

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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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Hypothesis testing.

Key Topics:

• Basic approach
• Null and alternative hypothesis
• Decision making and the p -value
• Z-test & Nonparametric alternative

Basic approach to hypothesis testing

• State a model describing the relationship between the explanatory variables and the outcome variable(s) in the population and the nature of the variability. State all of your assumptions .
• Specify the null and alternative hypotheses in terms of the parameters of the model.
• Invent a test statistic that will tend to be different under the null and alternative hypotheses.
• Using the assumptions of step 1, find the theoretical sampling distribution of the statistic under the null hypothesis of step 2. Ideally the form of the sampling distribution should be one of the “standard distributions”(e.g. normal, t , binomial..)
• Calculate a p -value , as the area under the sampling distribution more extreme than your statistic. Depends on the form of the alternative hypothesis.
• Choose your acceptable type 1 error rate (alpha) and apply the decision rule : reject the null hypothesis if the p-value is less than alpha, otherwise do not reject.

• \(\frac{\bar{X}-\mu_0}{\sigma / \sqrt{n}}\)
• general form is: (estimate - value we are testing)/(st.dev of the estimate)
• z-statistic follows N(0,1) distribution
• 2 × the area above |z|, area above z,or area below z, or
• compare the statistic to a critical value, |z| ≥ z α/2 , z ≥ z α , or z ≤ - z α
• Choose the acceptable level of Alpha = 0.05, we conclude …. ?

Making the Decision

It is either likely or unlikely that we would collect the evidence we did given the initial assumption. (Note: “likely” or “unlikely” is measured by calculating a probability!)

If it is likely , then we “ do not reject ” our initial assumption. There is not enough evidence to do otherwise.

If it is unlikely , then:

• either our initial assumption is correct and we experienced an unusual event or,
• our initial assumption is incorrect

In statistics, if it is unlikely, we decide to “ reject ” our initial assumption.

Example: Criminal Trial Analogy

First, state 2 hypotheses, the null hypothesis (“H 0 ”) and the alternative hypothesis (“H A ”)

• H 0 : Defendant is not guilty.
• H A : Defendant is guilty.

Usually the H 0 is a statement of “no effect”, or “no change”, or “chance only” about a population parameter.

While the H A , depending on the situation, is that there is a difference, trend, effect, or a relationship with respect to a population parameter.

• It can one-sided and two-sided.
• In two-sided we only care there is a difference, but not the direction of it. In one-sided we care about a particular direction of the relationship. We want to know if the value is strictly larger or smaller.

Then, collect evidence, such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, handwriting samples, etc. (In statistics, the data are the evidence.)

Next, you make your initial assumption.

• Defendant is innocent until proven guilty.

In statistics, we always assume the null hypothesis is true .

Then, make a decision based on the available evidence.

• If there is sufficient evidence (“beyond a reasonable doubt”), reject the null hypothesis . (Behave as if defendant is guilty.)
• If there is not enough evidence, do not reject the null hypothesis . (Behave as if defendant is not guilty.)

If the observed outcome, e.g., a sample statistic, is surprising under the assumption that the null hypothesis is true, but more probable if the alternative is true, then this outcome is evidence against H 0 and in favor of H A .

An observed effect so large that it would rarely occur by chance is called statistically significant (i.e., not likely to happen by chance).

Using the p -value to make the decision

The p -value represents how likely we would be to observe such an extreme sample if the null hypothesis were true. The p -value is a probability computed assuming the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. Since it's a probability, it is a number between 0 and 1. The closer the number is to 0 means the event is “unlikely.” So if p -value is “small,” (typically, less than 0.05), we can then reject the null hypothesis.

Significance level and p -value

Significance level, α, is a decisive value for p -value. In this context, significant does not mean “important”, but it means “not likely to happened just by chance”.

α is the maximum probability of rejecting the null hypothesis when the null hypothesis is true. If α = 1 we always reject the null, if α = 0 we never reject the null hypothesis. In articles, journals, etc… you may read: “The results were significant ( p <0.05).” So if p =0.03, it's significant at the level of α = 0.05 but not at the level of α = 0.01. If we reject the H 0 at the level of α = 0.05 (which corresponds to 95% CI), we are saying that if H 0 is true, the observed phenomenon would happen no more than 5% of the time (that is 1 in 20). If we choose to compare the p -value to α = 0.01, we are insisting on a stronger evidence!

So, what kind of error could we make? No matter what decision we make, there is always a chance we made an error.

Errors in Criminal Trial:

Errors in Hypothesis Testing

Type I error (False positive): The null hypothesis is rejected when it is true.

• α is the maximum probability of making a Type I error.

Type II error (False negative): The null hypothesis is not rejected when it is false.

• β is the probability of making a Type II error

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

The power of a statistical test is its probability of rejecting the null hypothesis if the null hypothesis is false. That is, power is the ability to correctly reject H 0 and detect a significant effect. In other words, power is one minus the type II error risk.

\(\text{Power }=1-\beta = P\left(\text{reject} H_0 | H_0 \text{is false } \right)\)

Which error is worse?

Type I = you are innocent, yet accused of cheating on the test. Type II = you cheated on the test, but you are found innocent.

This depends on the context of the problem too. But in most cases scientists are trying to be “conservative”; it's worse to make a spurious discovery than to fail to make a good one. Our goal it to increase the power of the test that is to minimize the length of the CI.

We need to keep in mind:

• the effect of the sample size,
• the correctness of the underlying assumptions about the population,
• statistical vs. practical significance, etc…

(see the handout). To study the tradeoffs between the sample size, α, and Type II error we can use power and operating characteristic curves.

What type of error might we have made?

Type I error is claiming that average student height is not 65 inches, when it really is. Type II error is failing to claim that the average student height is not 65in when it is.

We rejected the null hypothesis, i.e., claimed that the height is not 65, thus making potentially a Type I error. But sometimes the p -value is too low because of the large sample size, and we may have statistical significance but not really practical significance! That's why most statisticians are much more comfortable with using CI than tests.

There is a need for a further generalization. What if we can't assume that σ is known? In this case we would use s (the sample standard deviation) to estimate σ.

If the sample is very large, we can treat σ as known by assuming that σ = s . According to the law of large numbers, this is not too bad a thing to do. But if the sample is small, the fact that we have to estimate both the standard deviation and the mean adds extra uncertainty to our inference. In practice this means that we need a larger multiplier for the standard error.

We need one-sample t -test.

One sample t -test

• Assume data are independently sampled from a normal distribution with unknown mean μ and variance σ 2 . Make an initial assumption, μ 0 .

• t-statistic: \(\frac{\bar{X}-\mu_0}{s / \sqrt{n}}\) where s is a sample st.dev.
• t-statistic follows t -distribution with df = n - 1
• Alpha = 0.05, we conclude ….

Testing for the population proportion

Let's go back to our CNN poll. Assume we have a SRS of 1,017 adults.

We are interested in testing the following hypothesis: H 0 : p = 0.50 vs. p > 0.50

What is the test statistic?

If alpha = 0.05, what do we conclude?

We will see more details in the next lesson on proportions, then distributions, and possible tests.

Hypothesis Testing: Sign Test

What is a sign test.

A sign test is an inferential technique to assess two competing hypotheses about the population medians across one or two samples. We can use the sign test for three specific purposes:

One-Sample: This tests whether the population median is less than, greater than, or not equal to a prespecified value. The b statistic counts the number of observations greater than the median assumed in the null hypothesis, which states that the population median equals the testing value. The resulting p-value tells us how likely observing the evidence we have for the alternative hypothesis or more when the null hypothesis is true. If the p-value is less than the specified significance level (e.g., less than 0.05), we reject the null hypothesis in favor of the alternate hypothesis, which states that the population median is less than, greater than, or not equal to the testing value. Otherwise, we fail to reject the null hypothesis, indicating we do not have significant evidence for the alternative hypothesis.

Two Independent Samples: This tests whether the difference of two population medians is lesser, greater than, or not equal to a prespecified value, which we frequently take to be zero. The 𝛘² statistic compares the observed data with what we expect under the null hypothesis, which states that the difference in population medians equals the testing value; usually, we take the testing value to be zero (e.g., the medians are the same). The resulting p-value tells us how likely observing the evidence we have for the alternative hypothesis or more when the null hypothesis is true. If the p-value is less than the specified significance level (e.g., less than 0.05), we reject the null hypothesis in favor of the alternate hypothesis, which states that the difference of population medians is less than, greater than, or not equal to the testing value. Otherwise, we fail to reject the null hypothesis, indicating we do not have significant evidence for the alternative hypothesis. This test is called Mood's median test, an extension of the sign test.

Paired Samples: This tests whether the population median of differences is lesser, greater than, or not equal to a prespecified value, which we frequently take to be zero. This procedure is used for a pre-post design. The b statistic compares the observed data with what we expect under the null hypothesis, which states that the population median of differences equals the testing value; usually, we take the testing value to be zero (e.g., the differences have a median of zero). The resulting p-value tells us how likely observing the evidence we have for the alternative hypothesis or more when the null hypothesis is true. If the p-value is less than the specified significance level (e.g., less than 0.05), we reject the null hypothesis in favor of the alternate hypothesis, which states that the population median difference is less than, greater than, or not equal to the testing value. Otherwise, we fail to reject the null hypothesis, indicating we do not have significant evidence for the alternative hypothesis.

The sign test can be used under the following conditions.

1. The observations are representative of the population of interest and independent.

2. The observations are continuous. The sign test is sensitive to data where there can be several observations at the assumed median, which can cause unreliable results.

How to use this app?

Step 1: To use this app, go to the 'Dataset & Hypothesis' tab and upload your .csv type dataset, or select a sample dataset.

Step 2: Next, you must select the type of sign-test (One-Sample, Two Independent Samples, or Paired Sample).

Step 3: You can check the assumptions in the 'Summary & Assumptions' tab.

Step 4: You can check the result of the selected hypothesis test procedure (test statistics, decision making, and test visualization) in the 'Hypothesis Test' and 'Confidence Interval' tabs.

Step 5 (Optional): We also provide the results of a bootstrap approach for computing a confidence interval and a randomization test. These are alternatives to the sign test that can be used to evaluate hypotheses about the data using resampling.

Please contact us if you have any questions at [email protected].

Within the sign test app, we provide the penguin data that includes measurements for penguin species inhabiting islands in Palmer Archipelago and made available through the palmerpenguins library for R (Gorman et al., 2014). Suppose researchers aimed to evaluate whether Adelie and Chinstrap penguins have differing bill depth (mm). This is a classic example of a scenario requiring the sign test framework.

Here, we have three samples of observations (the species) and a continuous attribute (bill depth). We will use the sign test procedure to evaluate whether the data support the claim that Adelie and Chinstrap penguins have different population median bill depths.

First, we load the sign test app. Second, we click 'Sample Data' to load the penguin data. Once the data are loaded, we select the quantitative variable (bill_depth_mm) and the categorical variable (species). Ensure that we have chosen the two-sample Mood's median test (independent), a two-sample extension of the sign test. Then we specify the independent samples as the Adelie and Chinstrap.

The first step of conducting the sign test procedure requires us to evaluate the assumptions. When we click 'Assumptions', a data summary provides our first look at the data.

The bill depth (mm) is a continuous variable, making it suitable for Mood's median test. These data were collected from many penguin nests across three different islands in Palmer Archipelago, meaning the data are likely representative. We trust that the researchers collected data in a way that made the observations near independent.

The 'Hypothesis Test' tab shows the result of Mood's median test. As we might expect after observing the plotted data, there is not significant evidence that the population median bill depths (mm) differ across Adelie and Chinstrap penguins (𝛘²=0.007, p=0.9357).

To provide context, we can interpret the corresponding boostrapping confidence interval for the population median difference. We are 95% confident that the true population median difference of bill depths (Adelie - Chinstrap) is between -0.60 mm and 0.50 mm. Note that zero is on this interval, indicating that it is plausible that the population medians are the same, which agrees with the interpretation of the test.

The randomization test produces results close to the sign test. Specifically, the randomization test produces a p-value of 0.855, which leads us to conclude that there is not significant evidence that the population median bill depths (mm) differ across Adelie and Chinstrap penguins. Note that this is the result of random resampling, and if you run the inference yourself, the result may vary slightly.

Gorman KB, Williams TD, Fraser WR (2014) Ecological Sexual Dimorphism and Environmental Variability within a Community of Antarctic Penguins (Genus Pygoscelis). PLoS ONE 9(3): e90081. doi:10.1371/journal.pone.0090081

Within the sign test app, we provide the MFAP4 data, including measurements for Hepatitis C patients collected by the German network of Excellence for Viral Hepatitis and studied by Bracht et al. (2016). Suppose these researchers wanted to show that the human microfibrillar-associated protein 4 (MFAP4, U/ml) is increased for hepatitis C patients. The researchers can use the sign test to evaluate whether the population median log-2 transformed MFAP4 is greater than that of healthy patients, which we take to be 1.71 U/ml (Zhang et al., 2019) on the log-2 scale.

Here, we have one sample of observations and a continuous attribute (MFAP4 log-2 U/ml). We will use the sign test procedure to evaluate whether the data support the claim that the population median log-2 MFAP4 level in hepatitis C patients is larger than 1.71.

First, we load the sign test app. Second, we click 'Sample Data' to load the MFAP4 data. Once the data are loaded, we select the variable (log2.MFAP4).

The log-2 MFAP4 U/ml is a continuous variable, making it suitable for the sign test. In their paper, Bracht et al. (2016) tell us these data were collected at different sites using a protocol meant to reduce bias, meaning the data are likely to be representative. We trust that the researchers collected data in a way that made the observations near independent.

The 'Hypothesis Test' tab shows the result of the sign test procedure. As we might expect after viewing graphs of the data, there is significant evidence that the population median MFAP4 U/ml level in hepatitis C patients is larger than 1.71 (b=530, p<0.0001).

We can interpret the corresponding confidence interval for the population median to provide context. We are 95% confident that the true population median log-2 MFAP4 level of hepatitis C patients is between 3.2016 and 3.406 U/ml on the log-2 scale. Note that the values the interval covers are larger than 1.71, indicating that the population median log-2 MFAP4 level for hepatitis C patients is larger than 1.71.

The randomization test produces results similar to the parametric result. Specifically, the randomization test produces a p-value < 0.0001, which leads us to conclude that there is significant evidence that the population median log-2 MFAP4 level is larger than 1.71. Note that this is the result of random sampling, and if you run the inference yourself, the result may vary slightly.

The same is true for the confidence interval. Using the bootstrap confidence interval, we are 95% confident that the true population median log-2 MFAP4 level among hepatitis C patients is between 3.2016 and 3.406. Note that this too is the result of random sampling, and if you run the inference yourself, the result may vary slightly.

Bracht, T., Molleken, C., Ahrens, M., Poschmann, G., Schlosser, A., Eisenacher, M., ... & Sitek, B. (2016). Evaluation of the biomarker candidate MFAP4 for non-invasive assessment of hepatic fibrosis in hepatitis C patients. Journal of Translational Medicine, 14(1), 1-9.

Zhang, X., Li, H., Kou, W., Tang, K., Zhao, D., Zhang, J., ... & Xu, Y. (2019). Increased plasma microfibrillar-associated protein 4 is associated with atrial fibrillation and more advanced left atrial remodelling. Archives of Medical Science, 15(3), 632-640.

Within the sign test app, we provide U.S. News and World Report's College Data that includes measurements for many U.S. Colleges from the 1995 issue of U.S. News and World Report and made available through the ISLR library in R (James et al., 2017). Suppose we aimed to evaluate whether private schools have a higher percentage of new students coming from the top 10% of their high school class than public schools.

Here, we have two samples of observations (private/public) and a discrete attribute (percent of new students in the top 10% of their high school class). We will use Mood's median test to evaluate whether the data support the claim that there is a difference in the median percent of new students coming from the top 10% of their high school class.

First, we load the sign test app. Second, we click 'Sample Data' to load the U.S. News College data.

The first step of conducting the sign test procedure requires us to evaluate the assumptions. When we click 'Summary & Assumptions', we get our first look at the data.

The percentage of new students in the top 10% of their high school class is a discrete variable. For any school, this percentage can only increase or decrease by 100/n, where n is the number of students enrolled. However, there are enough unique observations that there aren't many ties, so we proceed with caution. We won't get into how U.S. News conducts its ratings, but it has been heavily scrutinized in the media. For demonstration purposes, we will proceed assuming that the data are representative. The data may be representative, but we'd have to do more digging.

The 'Hypothesis Test' tab shows the result of the Mood's median test. As we might expect after viewing the data, there is significant evidence that the population median percentage of new students coming from the top 10% of their high school class differs across institution types (𝛘²=18.285, p < 0.0001).

To provide context, we can interpret the corresponding bootstrap confidence interval for the population median difference. We are 95% confident that the true population median percentage of new students coming from the top 10% of their high school class (private-public) is between 2 and 8 percentage points. Note that the values the interval covers are larger than 0, indicating that the population median is larger for private schools than public schools.

The randomization test produces a p-value < 0.0001, which leads us to the same conclusion as Mood's median test. Note that this is the result of random sampling, and if you run the inference yourself, the result may vary slightly.

Gareth James, Daniela Witten, Trevor Hastie and Rob Tibshirani (2017). ISLR: Data for an Introduction to Statistical Learning with Applications in R. R package version 1.2. https://CRAN.R-project.org/package=ISLR

Within the sign test app, we provide the well-being data of undergraduate college students collected by Binfet et al. (2021). Suppose the researchers wanted to show that reported loneliness is decreased among undergraduate college students after contact with canines. The researchers can use the sign test to evaluate whether the population median loneliness is greater before canine contact than after. Loneliness is measured using the UCLA Loneliness Scale (Russell, 1996), the average of twenty questions answered on a one to four scale.

Here, we have two samples of observations (before/after) and a discrete attribute (self-reported loneliness). We will use the paired samples sign test to evaluate whether the population median loneliness is greater before canine contact than after.

First, we load the sign test app. Second, we click 'Sample Data' to load Binfet's Canine Data: Contact Group. Once the data are loaded, we select the `after' variable (lonely2) and the 'before' variable (lonely1). Ensure to choose Paired two-sample sign test (dependent).

The first step of conducting the sign test procedure requires us to evaluate the assumptions. When we click 'Assumptions', we get our first look at the data.

The loneliness score of participants is the average of twenty questions answered on a one to four scale, making it a discrete variable. For any participant, this score can only increase or decrease by 1/80, the difference in the average when adjusting one answer by 1. However, there are enough unique observations that there aren't many ties, so we proceed with caution. Binfet et al. (2021) recruited undergraduate students from one mid-sized Canadian University who were enrolled in a psychology course offering bonus credit for participating in research studies. While this sample may be representative of undergraduate students at midsized Canadian universities who take psychology courses, it may not represent all undergraduate students (e.g., non-Canadian institutions, students who don't take psychology courses, etc.).

The 'Hypothesis Test' tab shows the result of the sign test procedure. There is significant evidence that the population median loneliness is greater before canine contact than after (b=49, p < 0.0001).

To provide context, we can interpret the corresponding confidence interval for the population median difference. We are 95% confident that the true population median loneliness (after-before) is between -0.1 and -0.05. Note that this difference is based on the average of responses on a 1 to 4 scale. That is, the difference is significant but not large. Note that the interval only covers values less than 0, indicating that the population median loneliness is larger before compared to after.

The randomization test produces a p-value < 0.0001, which leads us to the same conclusion. Note that this is the result of random sampling, and if you run the inference yourself, the result may vary slightly.

The same is true for the confidence interval. Using the bootstrap confidence interval, we are 95% confident that the true population median loneliness (after-before) is between -0.1 and -0.05. Note that this too is the result of random sampling, and if you run the inference yourself, the result may vary slightly.

Binfet, J. T., Green, F. L., & Draper, Z. A. (2022). The Importance of Client–Canine Contact in Canine-Assisted Interventions: A Randomized Controlled Trial. Anthrozoös, 35(1), 1-22.

Russell, D. W. (1996). UCLA Loneliness Scale (Version 3): Reliability, validity, and factor structure. Journal of personality assessment, 66(1), 20-40.

• Data Preview
• Summary & Assumptions
• Hypothesis Test
• Confidence Interval
• Randomization Test
• Bootstrapping

Graphical Summary

Assumptions, make sure that you satisfy all the assumptions for the sign test, numerical summary, interpretation, hypothesis test details, confidence interval details, hypothesis test graphical summary, hypothesis test interpretation, confidence interval graphical summary, confidence interval interpretation.