hypothesis testing short notes

Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

• z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation and n is the size of the sample.
• t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\). s is the sample standard deviation.
• \(\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}\). \(O_{i}\) is the observed value and \(E_{i}\) is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

• One sample: z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
• Two samples: z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

• One sample: t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\).
• Two samples: t = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

\(H_{0}\): The population parameter is ≤ some value

\(H_{1}\): The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

\(H_{0}\): The population parameter is ≥ some value

\(H_{1}\): The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

\(H_{0}\): the population parameter = some value

\(H_{1}\): the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

• Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
• Step 2: Set up the alternative hypothesis.
• Step 3: Choose the correct significance level, \(\alpha\), and find the critical value.
• Step 4: Calculate the correct test statistic (z, t or \(\chi\)) and p-value.
• Step 5: Compare the test statistic with the critical value or compare the p-value with \(\alpha\) to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.

Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.

Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.

1 - \(\alpha\) = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).

\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15

z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

Related Articles:

• Probability and Statistics
• Data Handling

Important Notes on Hypothesis Testing

• Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
• It involves the setting up of a null hypothesis and an alternate hypothesis.
• There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
• Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

• Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) > 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 5, s = 18. \(\alpha\) = 0.05 Using the t-distribution table, the critical value is 2.132 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
• Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. \(H_{0}\): \(\mu\) = 80, \(H_{1}\): \(\mu\) ≠ 80 \(\overline{x}\) = 88, \(\mu\) = 80, n = 36, \(\sigma\) = 10. \(\alpha\) = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) z = \(\frac{88-80}{\frac{10}{\sqrt{36}}}\) = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
• Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) < 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = \(\frac{82-90}{\frac{18}{\sqrt{6}}}\) t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

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FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

What is the Alpha Level in Two Tail Hypothesis Testing?

To get the alpha level in a two tail hypothesis testing divide \(\alpha\) by 2. This is done as there are two rejection regions in the curve.

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• Knowledge Base

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

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 Hypothesis Testing with Examples

A comprehensible guide on hypothesis testing with examples and visualizations.

Neeraj Krishna

Towards Data Science

Most tutorials I’ve seen on hypothesis testing start with a prior assumption of the distribution, list down some definitions and formulae, and directly apply them to solve a problem.

However, in this tutorial, we will learn from the first principles. This will be an example-driven tutorial where we start with a basic example and build our way up to understand the foundations of hypothesis testing.

Let’s get started.

Which die did you pick?

Imagine there are two indistinguishable dice in front of you. One is fair, and the other is loaded. You randomly pick a die and toss it. After observing on which face it lands, can you determine which die you’ve picked?

The probability distribution of the dice is shown below:

In binary hypothesis testing problems, we’ll often be presented with two choices which we call hypotheses, and we’ll have to decide whether to pick one or the other.

The hypotheses are represented by H₀ and H₁ and are called null and alternate hypotheses respectively. In hypothesis testing, we either reject or accept the null hypothesis.

In our example, die 1 and die 2 are null and alternate hypotheses respectively.

If you think about it intuitively, if the die lands on 1 or 2, it’s more likely die 2 because it has more probability to land on 1 or 2. So the decision to accept or reject the null hypothesis depends on the distribution of the observations.

So we can say the goal of hypothesis testing is to draw a boundary and separate the observation space into two regions: the rejection region and the acceptance region.

If the observation falls in the rejection region, we reject the null hypothesis, else we accept it. Now, the decision boundary isn’t going to be perfect and we’re going to make errors. For example, it’s possible that die 1 lands on 1 or 2 and we mistake it for die 2; but there is less probability of this happening. We’ll learn how to calculate the probabilities of errors in the next section.

How do we determine the decision boundary? There’s a simple and effective method called the likelihood ratio test we’ll discuss next.

Likelihood ratio test

You’ve got to realize first the distribution of the observations depends on the hypotheses. Below I’ve plotted the distributions in our example under the two hypotheses:

Now, P(X=x;H₀) and P(X=x;H₁) represents the likelihood of observations under hypotheses H₀ and H₁ respectively. Their ratio tells us how likely one hypothesis is true over the other for different observations.

This ratio is called the likelihood ratio and is represented by L(X) . L(X) is a random variable that depends on the observation x .

In the likelihood ratio test, we reject the null hypothesis if the ratio is above a certain value i.e, reject the null hypothesis if L(X) > 𝜉 , else accept it. 𝜉 is called the critical ratio.

So this is how we can draw a decision boundary: we separate the observations for which the likelihood ratio is greater than the critical ratio from the observations for which it isn’t.

So the observations of the form {x | L(x) > 𝜉} fall into the rejection region while the rest of them fall into the acceptance region.

Let’s illustrate it with our dice example. The likelihood ratio can be calculated as:

The plot of the likelihood ratio looks like this:

Now the placement of the decision boundary comes down to choosing the critical ratio. Let’s assume the critical ratio is a value between 3/2 and 3/4 i.e., 3/4 < 𝜉 < 3/2 . Then our decision boundary looks like this:

Let’s discuss the errors associated with this decision. The first type of error occurs if observation x belongs to the rejection region but occurs under the null hypothesis. In our example, it means die 1 lands on 1 or 2.

This is called the false rejection error or the type 1 error. The probability of this error is represented by 𝛼 and can be computed as:

The second error occurs if observation x belongs to the acceptance region but occurs under the alternate hypothesis. This is called the false acceptance error or the type 2 error. The probability of this error is represented by 𝛽 and can be computed as:

In our example, the false rejection and the false acceptance error can be calculated as:

Let’s consider two other scenarios where the critical ratio takes the following values: 𝜉 > 3/2 and 𝜉 < 3/4 .

The type 1 and type 2 errors can be computed similarly.

Let’s plot both the errors for different values of 𝜉.

As the critical value 𝜉 increases, the rejection region becomes smaller. As a result, the false rejection probability 𝛼 decreases, while the false acceptance probability 𝛽 increases.

The likelihood ratio test offers the smallest errors

We could draw a boundary in the observation space anywhere. Why do we need to compute the likelihood ratio and go through all that? Let’s see why.

Below I’ve calculated the type I and type II errors for different boundaries.

The plot of Type I and Type II errors with their sum for different boundaries looks like this:

We can see for the optimum value of the critical ratio obtained from the likelihood ratio test, the sum of type I and type II errors is the least.

In other words, for a given false rejection probability, the likelihood ratio test offers the smallest possible false acceptance probability.

This is called the Neyman-Pearson Lemma. I’ve referenced the theoretical proof at the end of the article.

Likelihood ratio test for continuous distributions

In the above example, we didn’t discuss how to choose the value of the critical ratio 𝜉. The probability distributions were discrete, so a small change in the critical ratio 𝜉 will not affect the boundary.

When we are dealing with continuous distributions, we fix the value of the false rejection probability 𝛼 and calculate the critical ratio based on that.

But again, the process would be the same. Once we obtain the value of the critical ratio, we separate the observation space.

Typical choices for 𝛼 are 𝛼 = 0.01, 𝛼 = 0.05, or 𝛼 = 0.01 , depending on the degree of the undesirability of false rejection.

For example, if we’re dealing with a normal distribution, we could standardize it and look up the Z-table to find 𝜉 for a given 𝛼.

In this article, we’ve looked at the idea behind hypothesis testing and the intuition behind the process. The whole process can be summarized in the diagram below:

We start with two hypotheses H₀ and H₁ such that the distribution of the underlying data depends on the hypotheses. The goal is to prove or disprove the null hypothesis H₀ by finding a decision rule that maps the realized value of the observation x to one of the two hypotheses. Finally, we calculate the errors associated with the decision rule.

However, in the real world, the distinction between the two hypotheses wouldn’t be straightforward. So we’d have to do some workarounds to perform hypothesis testing. Let’s discuss this in the next article.

Hope you’ve enjoyed this article. Let’s connect.

Image and Diagram Credits

All the images, figures, and diagrams in this article are created by the author; unless explicitly mentioned in the caption.

Chapter 9 and section 3 of the book Introduction to Probability by Dimitri Bertsekas and John Tsitsiklis

Written by Neeraj Krishna

I write about effective learning, technology, and deep learning | 2x top writer | senior data scientist @MakeMyTrip

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The Ultimate Guide to Hypothesis Testing for beginners

sangeetha natarajan

Analytics Vidhya

We come across so many statements in our life whose truth cannot be verified that instant. Have you ever wondered which one to choose? or which statement will validate our truth? or is there a way to find statistically acceptable answers to our simple statements? Of course, yes!

How? Hypothesis Testing.

Hypothesis Testing is one of the most important concepts in Analytics yet many of us do not have a clear idea about what it is actually about.

Let’s consider the below claims made by some business organizations,

1. Average life of vegetarians is different than that of meat-eaters.

2. On a state driver’s test, about 40% pass the test on the first try.

3. Proportion of married people defaulting on loan repayment is less than the proportion of singles defaulting on loan repayment.

How do we statistically validate the above claims/verify its truth?

Well, this is where Hypothesis Testing comes into play. Let’s see more about this below.

Hypothesis and Hypothesis Testing: A hypothesis is nothing but an assumption/claim/proposition made about something which is later on tested statistically to verify its truth. Hypothesis Testing: Hypothesis testing is the process of checking the validity of the claim using evidence found in sample data.

A Hypothesis Testing consists of two contradictory statements namely,

1.Null Hypothesis

2.Alternative Hypothesis

N ull Hypothesis(H₀): In a null hypothesis, we claim that there is no relationship or difference between groups with respect to the value of the population parameter.

We begin the Hypothesis Test by assuming that the Null Hypothesis to be true and later on we retain/reject the Null Hypothesis based on the evidence found in sample data.

Note: The null statement must always contain some form of equality (=, ≤ or ≥)

A lternative Hypothesis (H ₐ or H ₁ ): In the alternative Hypothesis, we claim that there is some change /relationship between groups with respect to the value of the population parameter.

An alternative hypothesis always contradicts the Null Hypothesis and only any one of the above hypotheses could be true.

Note: An Alternative hypothesis is denoted using less than, greater than, or not equals symbols, i.e., (≠, >, or <).

Steps in Hypothesis Testing:

Now let’s see the steps involved in performing a Hypothesis Test.

1. Formulate the Null and Alternative Hypothesis

2. Decide the Significance Level (α)

3. Calculate the test statistic

4. Calculate the p-value

5. Decision to reject/retain the Null Hypothesis

1.Formulating the Null and Alternative Hypothesis: The first step in performing Hypothesis Testing is to describe the Null and Alternative Hypothesis in words. These Hypotheses are described using a population parameter such as mean, median, or proportion.

For Example, consider the below hypothesis,

H₀: The average life of vegetarians and meat-eaters are the same

H₁: The average life of vegetarians and meat-eaters are not the same

Where µᵥ and µₘ are the average life of vegetarians and meat-eaters respectively.

2. Proportion of married people defaulting on loan repayment is less than the proportion of singles defaulting on loan repayment.

H₀: The proportion of married people defaulting on loan repayment is not less than the proportion of singles defaulting.

H₁: The proportion of married people defaulting on loan repayment is less than the proportion of singles defaulting.

H₀: pₘ ≥ pₛ

H₁: pₘ < pₛ

where pₘ and pₛ are the proportion of married and single defaulters respectively.

After formulating the Null and Alternative hypothesis, based on the evidence from the sample data we retain/reject the Null Hypothesis.

2.Decide the Significance Level (α) : The Significance Level(α) is nothing but the maximum threshold set to reject the Null Hypothesis.

Usually, α is set as 0.05. This means that there is a 5% chance we will reject the Null Hypothesis even when it’s true. Here if p-value<0.05 we reject the null hypothesis even when there is a 5% chance for it to be true.

Critical Value and Critical region: The value of the statistic in the sampling distribution for which the probability is α is called the critical value.

The areas beyond the critical values are known as the critical region/rejection region and critical values are the values that indicate the edge of the critical region.

Note: Critical regions describe the entire area of values that rejects the null hypothesis. If the test statistic falls in the critical region, the null hypothesis will be rejected.

One-Tailed Test: When the critical/rejection region is on one side of the distribution it is known as a One-Tailed Test. In this case, the null hypothesis will be rejected if the test statistic is on one side of the distribution, either left or right.

Left Tailed Test: If the test is left-tailed, the critical/rejection region, with an area equal to α, will be on the left side of the distribution curve. In this case, the null hypothesis will be rejected if the test statistic is very small(as it will fall on the left end of the distribution).

Here, p-value = P[Test statistics <= observed value of the test statistic]

E.g.: college students take less than five years to graduate from college, on the average

H₁: μ < 5

This is a left-tailed test since the rejection region would consist of values less than 5.

Right Tailed Test: If the test is right-tailed, the critical /rejection region, with an area equal to α, will be on the right side of the distribution curve. In this case, the null hypothesis will be rejected if the test statistic is very large (or falls on the right end of the distribution).

Here, p-value = P [Test statistics >= observed value of the test statistic]

E.g.: A package of gum claims that the flavor lasts more than 39 minutes.

H₁: μ > 39

This is a right-tailed test since the rejection region would consist of values greater than 39.

Two-Tailed Test:

If the test is two-tailed, α must be divided by 2 and the critical /rejection regions will be at both ends of the distribution curve. Hence, in this case, the null hypothesis will be rejected when the test value is on either of two rejection regions on either side of the distribution.

For a two-tailed test,

p-value = 2 * P[Test statistics >= |observed value of the test statistic|]

3.Calculate the Test statistic:

A test statistic is nothing but the standardized difference between the value of the parameter estimated from the sample (such as sample mean) and the value of the null hypothesis (such as hypothesized population mean). It is a measure of how far the sample mean is from the hypothesized population mean.

test statistic = (x̄ — μ) / (σ / √n)

where, x̄ = sample mean

μ = population mean

σ = Standard Deviation of Population

n = Number of Observation

If the test statistic falls in the critical region, we reject the null hypothesis. Also, the larger the absolute value of the test statistic, the larger will be the standardized difference between hypothesized mean and sample mean, and hence the p-value, in this case, will be smaller with greater evidence against the null hypothesis.

There are various different ways to calculate the test statistics. Some of the commonly used tests are Z-test and T-Test. Although the test statistic calculations are mostly similar any of the above tests can be chosen based on the given population or distribution parameters.

T-Test: A T-test can be used when,

- The population distribution is normal or

- The sampling distribution is symmetric and the sample size is ≤ 15 or

- The sampling distribution is moderately skewed and the sample size is 16 ≤ n ≤ 30 or

-The sample size is greater than 30, without outliers.

Z-Test: A Z-test can be used when

-when the population is normally distributed and σ is known.

- The sample size n ≥ 30

Now consider the below example,

T he average height of a 7th grader five years ago was 145 cm with a standard deviation of 20 cm. From a random sample of 200 students, the average height of the sample students is found to be 147 cm. Are 7th graders now taller than they were before?

H₀: µ ≤ 145

Hₐ: µ > 145

Clearly, this is a right-tailed test since the rejection region (µ > 145) in this case lies on the right end of the sampling distribution.

sample mean, x̄ = 147

population mean, μ =145

Standard Deviation of Population, σ =20

Number of Observation, n =200

test statistic = (x̄ — μ) / (σ / √n) = (147–145) / (20/√200) = 1.414

The above test statistic means that the sample mean is 1.414 standard deviations away from the population mean or the standardized difference between the sample mean and hypothesized population mean is 1.414

For α = 0.05, the critical value for a one-tailed test is 1.64. Hence if the test statistic is greater than 1.64 it will be in the rejection region.

Clearly, our test statistic 1.414 is less than the critical value of 1.64 and hence does not fall in the rejection region. This means that the sample mean is not significantly different from the hypothesized population mean and hence we fail to reject the Null hypothesis

P-value: A p-value is nothing but the conditional probability of getting the test statistic given the null hypothesis is true.

P-value=P (Observing the test statistic| Null hypothesis is true)

If p-value <= significance level (α), we reject the null hypothesis in favor of the alternative hypothesis.

If p-value > significance level (α), we fail to reject the null hypothesis and hence accept the alternative hypothesis.

For the above example, the p-value can be stated as the probability of observing a test statistic of 1.414 when the null hypothesis (µ ≤ 145) is true.

i.e., P (z=1.414| µ ≤ 145) = P (z ≤ 1.414) =0 .078681.

clearly p-value, 0.0786 is greater than 0.05(α). Hence, we fail to reject the Null Hypothesis.

So, to summarize the process of Hypothesis Testing starts with the formulation of the Null and Alternative Hypothesis, setting the significance level(α) and then calculating the p-value based on which we accept or reject the Null Hypothesis.

Do provide your valuable feedbacks and comments below.

Written by sangeetha natarajan

Data Analyst|NLP|Statistics

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Understanding Hypothesis Testing

Hypothesis testing involves formulating assumptions about population parameters based on sample statistics and rigorously evaluating these assumptions against empirical evidence. This article sheds light on the significance of hypothesis testing and the critical steps involved in the process.

What is Hypothesis Testing?

A hypothesis is an assumption or idea, specifically a statistical claim about an unknown population parameter. For example, a judge assumes a person is innocent and verifies this by reviewing evidence and hearing testimony before reaching a verdict.

Hypothesis testing is a statistical method that is used to make a statistical decision using experimental data. Hypothesis testing is basically an assumption that we make about a population parameter. It evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. 

To test the validity of the claim or assumption about the population parameter:

• A sample is drawn from the population and analyzed.
• The results of the analysis are used to decide whether the claim is true or not.

Example: You say an average height in the class is 30 or a boy is taller than a girl. All of these is an assumption that we are assuming, and we need some statistical way to prove these. We need some mathematical conclusion whatever we are assuming is true.

Defining Hypotheses

• Null hypothesis (H 0 ): In statistics, the null hypothesis is a general statement or default position that there is no relationship between two measured cases or no relationship among groups. In other words, it is a basic assumption or made based on the problem knowledge. Example : A company’s mean production is 50 units/per da H 0 : [Tex]\mu [/Tex] = 50.
• Alternative hypothesis (H 1 ): The alternative hypothesis is the hypothesis used in hypothesis testing that is contrary to the null hypothesis.  Example: A company’s production is not equal to 50 units/per day i.e. H 1 : [Tex]\mu [/Tex] [Tex]\ne [/Tex] 50.

Key Terms of Hypothesis Testing

• Level of significance : It refers to the degree of significance in which we accept or reject the null hypothesis. 100% accuracy is not possible for accepting a hypothesis, so we, therefore, select a level of significance that is usually 5%. This is normally denoted with  [Tex]\alpha[/Tex] and generally, it is 0.05 or 5%, which means your output should be 95% confident to give a similar kind of result in each sample.
• P-value: The P value , or calculated probability, is the probability of finding the observed/extreme results when the null hypothesis(H0) of a study-given problem is true. If your P-value is less than the chosen significance level then you reject the null hypothesis i.e. accept that your sample claims to support the alternative hypothesis.
• Test Statistic: The test statistic is a numerical value calculated from sample data during a hypothesis test, used to determine whether to reject the null hypothesis. It is compared to a critical value or p-value to make decisions about the statistical significance of the observed results.
• Critical value : The critical value in statistics is a threshold or cutoff point used to determine whether to reject the null hypothesis in a hypothesis test.
• Degrees of freedom: Degrees of freedom are associated with the variability or freedom one has in estimating a parameter. The degrees of freedom are related to the sample size and determine the shape.

Why do we use Hypothesis Testing?

Hypothesis testing is an important procedure in statistics. Hypothesis testing evaluates two mutually exclusive population statements to determine which statement is most supported by sample data. When we say that the findings are statistically significant, thanks to hypothesis testing. 

One-Tailed and Two-Tailed Test

One tailed test focuses on one direction, either greater than or less than a specified value. We use a one-tailed test when there is a clear directional expectation based on prior knowledge or theory. The critical region is located on only one side of the distribution curve. If the sample falls into this critical region, the null hypothesis is rejected in favor of the alternative hypothesis.

One-Tailed Test

There are two types of one-tailed test:

• Left-Tailed (Left-Sided) Test: The alternative hypothesis asserts that the true parameter value is less than the null hypothesis. Example: H 0 ​: [Tex]\mu \geq 50 [/Tex] and H 1 : [Tex]\mu < 50 [/Tex]
• Right-Tailed (Right-Sided) Test : The alternative hypothesis asserts that the true parameter value is greater than the null hypothesis. Example: H 0 : [Tex]\mu \leq50 [/Tex] and H 1 : [Tex]\mu > 50 [/Tex]

Two-Tailed Test

A two-tailed test considers both directions, greater than and less than a specified value.We use a two-tailed test when there is no specific directional expectation, and want to detect any significant difference.

Example: H 0 : [Tex]\mu = [/Tex] 50 and H 1 : [Tex]\mu \neq 50 [/Tex]

To delve deeper into differences into both types of test: Refer to link

What are Type 1 and Type 2 errors in Hypothesis Testing?

In hypothesis testing, Type I and Type II errors are two possible errors that researchers can make when drawing conclusions about a population based on a sample of data. These errors are associated with the decisions made regarding the null hypothesis and the alternative hypothesis.

• Type I error: When we reject the null hypothesis, although that hypothesis was true. Type I error is denoted by alpha( [Tex]\alpha [/Tex] ).
• Type II errors : When we accept the null hypothesis, but it is false. Type II errors are denoted by beta( [Tex]\beta [/Tex] ).

How does Hypothesis Testing work?

Step 1: define null and alternative hypothesis.

State the null hypothesis ( [Tex]H_0 [/Tex] ), representing no effect, and the alternative hypothesis ( [Tex]H_1 [/Tex] ​), suggesting an effect or difference.

We first identify the problem about which we want to make an assumption keeping in mind that our assumption should be contradictory to one another, assuming Normally distributed data.

Step 2 – Choose significance level

Select a significance level ( [Tex]\alpha [/Tex] ), typically 0.05, to determine the threshold for rejecting the null hypothesis. It provides validity to our hypothesis test, ensuring that we have sufficient data to back up our claims. Usually, we determine our significance level beforehand of the test. The p-value is the criterion used to calculate our significance value.

Step 3 – Collect and Analyze data.

Gather relevant data through observation or experimentation. Analyze the data using appropriate statistical methods to obtain a test statistic.

Step 4-Calculate Test Statistic

The data for the tests are evaluated in this step we look for various scores based on the characteristics of data. The choice of the test statistic depends on the type of hypothesis test being conducted.

There are various hypothesis tests, each appropriate for various goal to calculate our test. This could be a Z-test , Chi-square , T-test , and so on.

• Z-test : If population means and standard deviations are known. Z-statistic is commonly used.
• t-test : If population standard deviations are unknown. and sample size is small than t-test statistic is more appropriate.
• Chi-square test : Chi-square test is used for categorical data or for testing independence in contingency tables
• F-test : F-test is often used in analysis of variance (ANOVA) to compare variances or test the equality of means across multiple groups.

We have a smaller dataset, So, T-test is more appropriate to test our hypothesis.

T-statistic is a measure of the difference between the means of two groups relative to the variability within each group. It is calculated as the difference between the sample means divided by the standard error of the difference. It is also known as the t-value or t-score.

Step 5 – Comparing Test Statistic:

In this stage, we decide where we should accept the null hypothesis or reject the null hypothesis. There are two ways to decide where we should accept or reject the null hypothesis.

Method A: Using Crtical values

Comparing the test statistic and tabulated critical value we have,

• If Test Statistic>Critical Value: Reject the null hypothesis.
• If Test Statistic≤Critical Value: Fail to reject the null hypothesis.

Note: Critical values are predetermined threshold values that are used to make a decision in hypothesis testing. To determine critical values for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Method B: Using P-values

We can also come to an conclusion using the p-value,

• If the p-value is less than or equal to the significance level i.e. ( [Tex]p\leq\alpha [/Tex] ), you reject the null hypothesis. This indicates that the observed results are unlikely to have occurred by chance alone, providing evidence in favor of the alternative hypothesis.
• If the p-value is greater than the significance level i.e. ( [Tex]p\geq \alpha[/Tex] ), you fail to reject the null hypothesis. This suggests that the observed results are consistent with what would be expected under the null hypothesis.

Note : The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one observed in the sample, assuming the null hypothesis is true. To determine p-value for hypothesis testing, we typically refer to a statistical distribution table , such as the normal distribution or t-distribution tables based on.

Step 7- Interpret the Results

At last, we can conclude our experiment using method A or B.

Calculating test statistic

To validate our hypothesis about a population parameter we use statistical functions . We use the z-score, p-value, and level of significance(alpha) to make evidence for our hypothesis for normally distributed data .

1. Z-statistics:

When population means and standard deviations are known.

[Tex]z = \frac{\bar{x} – \mu}{\frac{\sigma}{\sqrt{n}}}[/Tex]

• [Tex]\bar{x} [/Tex] is the sample mean,
• μ represents the population mean, 
• σ is the standard deviation
• and n is the size of the sample.

2. T-Statistics

T test is used when n<30,

t-statistic calculation is given by:

[Tex]t=\frac{x̄-μ}{s/\sqrt{n}} [/Tex]

• t = t-score,
• x̄ = sample mean
• μ = population mean,
• s = standard deviation of the sample,
• n = sample size

3. Chi-Square Test

Chi-Square Test for Independence categorical Data (Non-normally distributed) using:

[Tex]\chi^2 = \sum \frac{(O_{ij} – E_{ij})^2}{E_{ij}}[/Tex]

• [Tex]O_{ij}[/Tex] is the observed frequency in cell [Tex]{ij} [/Tex]
• i,j are the rows and columns index respectively.
• [Tex]E_{ij}[/Tex] is the expected frequency in cell [Tex]{ij}[/Tex] , calculated as : [Tex]\frac{{\text{{Row total}} \times \text{{Column total}}}}{{\text{{Total observations}}}}[/Tex]

Real life Examples of Hypothesis Testing

Let’s examine hypothesis testing using two real life situations,

Case A: D oes a New Drug Affect Blood Pressure?

Imagine a pharmaceutical company has developed a new drug that they believe can effectively lower blood pressure in patients with hypertension. Before bringing the drug to market, they need to conduct a study to assess its impact on blood pressure.

• Before Treatment: 120, 122, 118, 130, 125, 128, 115, 121, 123, 119
• After Treatment: 115, 120, 112, 128, 122, 125, 110, 117, 119, 114

Step 1 : Define the Hypothesis

• Null Hypothesis : (H 0 )The new drug has no effect on blood pressure.
• Alternate Hypothesis : (H 1 )The new drug has an effect on blood pressure.

Step 2: Define the Significance level

Let’s consider the Significance level at 0.05, indicating rejection of the null hypothesis.

If the evidence suggests less than a 5% chance of observing the results due to random variation.

Step 3 : Compute the test statistic

Using paired T-test analyze the data to obtain a test statistic and a p-value.

The test statistic (e.g., T-statistic) is calculated based on the differences between blood pressure measurements before and after treatment.

t = m/(s/√n)

• m  = mean of the difference i.e X after, X before
• s  = standard deviation of the difference (d) i.e d i ​= X after, i ​− X before,
• n  = sample size,

then, m= -3.9, s= 1.8 and n= 10

we, calculate the , T-statistic = -9 based on the formula for paired t test

Step 4: Find the p-value

The calculated t-statistic is -9 and degrees of freedom df = 9, you can find the p-value using statistical software or a t-distribution table.

thus, p-value = 8.538051223166285e-06

Step 5: Result

• If the p-value is less than or equal to 0.05, the researchers reject the null hypothesis.
• If the p-value is greater than 0.05, they fail to reject the null hypothesis.

Conclusion: Since the p-value (8.538051223166285e-06) is less than the significance level (0.05), the researchers reject the null hypothesis. There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

Python Implementation of Case A

Let’s create hypothesis testing with python, where we are testing whether a new drug affects blood pressure. For this example, we will use a paired T-test. We’ll use the scipy.stats library for the T-test.

Scipy is a mathematical library in Python that is mostly used for mathematical equations and computations.

We will implement our first real life problem via python,

import numpy as np from scipy import stats # Data before_treatment = np . array ([ 120 , 122 , 118 , 130 , 125 , 128 , 115 , 121 , 123 , 119 ]) after_treatment = np . array ([ 115 , 120 , 112 , 128 , 122 , 125 , 110 , 117 , 119 , 114 ]) # Step 1: Null and Alternate Hypotheses # Null Hypothesis: The new drug has no effect on blood pressure. # Alternate Hypothesis: The new drug has an effect on blood pressure. null_hypothesis = "The new drug has no effect on blood pressure." alternate_hypothesis = "The new drug has an effect on blood pressure." # Step 2: Significance Level alpha = 0.05 # Step 3: Paired T-test t_statistic , p_value = stats . ttest_rel ( after_treatment , before_treatment ) # Step 4: Calculate T-statistic manually m = np . mean ( after_treatment - before_treatment ) s = np . std ( after_treatment - before_treatment , ddof = 1 ) # using ddof=1 for sample standard deviation n = len ( before_treatment ) t_statistic_manual = m / ( s / np . sqrt ( n )) # Step 5: Decision if p_value <= alpha : decision = "Reject" else : decision = "Fail to reject" # Conclusion if decision == "Reject" : conclusion = "There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different." else : conclusion = "There is insufficient evidence to claim a significant difference in average blood pressure before and after treatment with the new drug." # Display results print ( "T-statistic (from scipy):" , t_statistic ) print ( "P-value (from scipy):" , p_value ) print ( "T-statistic (calculated manually):" , t_statistic_manual ) print ( f "Decision: { decision } the null hypothesis at alpha= { alpha } ." ) print ( "Conclusion:" , conclusion )

T-statistic (from scipy): -9.0 P-value (from scipy): 8.538051223166285e-06 T-statistic (calculated manually): -9.0 Decision: Reject the null hypothesis at alpha=0.05. Conclusion: There is statistically significant evidence that the average blood pressure before and after treatment with the new drug is different.

In the above example, given the T-statistic of approximately -9 and an extremely small p-value, the results indicate a strong case to reject the null hypothesis at a significance level of 0.05. 

• The results suggest that the new drug, treatment, or intervention has a significant effect on lowering blood pressure.
• The negative T-statistic indicates that the mean blood pressure after treatment is significantly lower than the assumed population mean before treatment.

Case B : Cholesterol level in a population

Data: A sample of 25 individuals is taken, and their cholesterol levels are measured.

Cholesterol Levels (mg/dL): 205, 198, 210, 190, 215, 205, 200, 192, 198, 205, 198, 202, 208, 200, 205, 198, 205, 210, 192, 205, 198, 205, 210, 192, 205.

Populations Mean = 200

Population Standard Deviation (σ): 5 mg/dL(given for this problem)

Step 1: Define the Hypothesis

• Null Hypothesis (H 0 ): The average cholesterol level in a population is 200 mg/dL.
• Alternate Hypothesis (H 1 ): The average cholesterol level in a population is different from 200 mg/dL.

As the direction of deviation is not given , we assume a two-tailed test, and based on a normal distribution table, the critical values for a significance level of 0.05 (two-tailed) can be calculated through the z-table and are approximately -1.96 and 1.96.

The test statistic is calculated by using the z formula Z = [Tex](203.8 – 200) / (5 \div \sqrt{25}) [/Tex] ​ and we get accordingly , Z =2.039999999999992.

Step 4: Result

Since the absolute value of the test statistic (2.04) is greater than the critical value (1.96), we reject the null hypothesis. And conclude that, there is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL

Python Implementation of Case B

import scipy.stats as stats import math import numpy as np # Given data sample_data = np . array ( [ 205 , 198 , 210 , 190 , 215 , 205 , 200 , 192 , 198 , 205 , 198 , 202 , 208 , 200 , 205 , 198 , 205 , 210 , 192 , 205 , 198 , 205 , 210 , 192 , 205 ]) population_std_dev = 5 population_mean = 200 sample_size = len ( sample_data ) # Step 1: Define the Hypotheses # Null Hypothesis (H0): The average cholesterol level in a population is 200 mg/dL. # Alternate Hypothesis (H1): The average cholesterol level in a population is different from 200 mg/dL. # Step 2: Define the Significance Level alpha = 0.05 # Two-tailed test # Critical values for a significance level of 0.05 (two-tailed) critical_value_left = stats . norm . ppf ( alpha / 2 ) critical_value_right = - critical_value_left # Step 3: Compute the test statistic sample_mean = sample_data . mean () z_score = ( sample_mean - population_mean ) / \ ( population_std_dev / math . sqrt ( sample_size )) # Step 4: Result # Check if the absolute value of the test statistic is greater than the critical values if abs ( z_score ) > max ( abs ( critical_value_left ), abs ( critical_value_right )): print ( "Reject the null hypothesis." ) print ( "There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL." ) else : print ( "Fail to reject the null hypothesis." ) print ( "There is not enough evidence to conclude that the average cholesterol level in the population is different from 200 mg/dL." )

Reject the null hypothesis. There is statistically significant evidence that the average cholesterol level in the population is different from 200 mg/dL.

Limitations of Hypothesis Testing

• Although a useful technique, hypothesis testing does not offer a comprehensive grasp of the topic being studied. Without fully reflecting the intricacy or whole context of the phenomena, it concentrates on certain hypotheses and statistical significance.
• The accuracy of hypothesis testing results is contingent on the quality of available data and the appropriateness of statistical methods used. Inaccurate data or poorly formulated hypotheses can lead to incorrect conclusions.
• Relying solely on hypothesis testing may cause analysts to overlook significant patterns or relationships in the data that are not captured by the specific hypotheses being tested. This limitation underscores the importance of complimenting hypothesis testing with other analytical approaches.

Hypothesis testing stands as a cornerstone in statistical analysis, enabling data scientists to navigate uncertainties and draw credible inferences from sample data. By systematically defining null and alternative hypotheses, choosing significance levels, and leveraging statistical tests, researchers can assess the validity of their assumptions. The article also elucidates the critical distinction between Type I and Type II errors, providing a comprehensive understanding of the nuanced decision-making process inherent in hypothesis testing. The real-life example of testing a new drug’s effect on blood pressure using a paired T-test showcases the practical application of these principles, underscoring the importance of statistical rigor in data-driven decision-making.

Frequently Asked Questions (FAQs)

1. what are the 3 types of hypothesis test.

There are three types of hypothesis tests: right-tailed, left-tailed, and two-tailed. Right-tailed tests assess if a parameter is greater, left-tailed if lesser. Two-tailed tests check for non-directional differences, greater or lesser.

2.What are the 4 components of hypothesis testing?

Null Hypothesis ( [Tex]H_o [/Tex] ): No effect or difference exists. Alternative Hypothesis ( [Tex]H_1 [/Tex] ): An effect or difference exists. Significance Level ( [Tex]\alpha [/Tex] ): Risk of rejecting null hypothesis when it’s true (Type I error). Test Statistic: Numerical value representing observed evidence against null hypothesis.

3.What is hypothesis testing in ML?

Statistical method to evaluate the performance and validity of machine learning models. Tests specific hypotheses about model behavior, like whether features influence predictions or if a model generalizes well to unseen data.

4.What is the difference between Pytest and hypothesis in Python?

Pytest purposes general testing framework for Python code while Hypothesis is a Property-based testing framework for Python, focusing on generating test cases based on specified properties of the code.

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• Hypothesis Testing

What is Hypothesis Testing?

Hypothesis testing in statistics refers to analyzing an assumption about a population parameter. It is used to make an educated guess about an assumption using statistics. With the use of sample data, hypothesis testing makes an assumption about how true the assumption is for the entire population from where the sample is being taken.  

Any hypothetical statement we make may or may not be valid, and it is then our responsibility to provide evidence for its possibility. To approach any hypothesis, we follow these four simple steps that test its validity.

First, we formulate two hypothetical statements such that only one of them is true. By doing so, we can check the validity of our own hypothesis.

The next step is to formulate the statistical analysis to be followed based upon the data points.

Then we analyze the given data using our methodology.

The final step is to analyze the result and judge whether the null hypothesis will be rejected or is true.

Let’s look at several hypothesis testing examples:

It is observed that the average recovery time for a knee-surgery patient is 8 weeks. A physician believes that after successful knee surgery if the patient goes for physical therapy twice a week rather than thrice a week, the recovery period will be longer. Conduct hypothesis for this statement. 

David is a ten-year-old who finishes a 25-yard freestyle in the meantime of 16.43 seconds. David’s father bought goggles for his son, believing that it would help him to reduce his time. He then recorded a total of fifteen 25-yard freestyle for David, and the average time came out to be 16 seconds. Conduct a hypothesis.

A tire company claims their A-segment of tires have a running life of 50,000 miles before they need to be replaced, and previous studies show a standard deviation of 8,000 miles. After surveying a total of 28 tires, the mean run time came to be 46,500 miles with a standard deviation of 9800 miles. Is the claim made by the tire company consistent with the given data? Conduct hypothesis testing. 

All of the hypothesis testing examples are from real-life situations, which leads us to believe that hypothesis testing is a very practical topic indeed. It is an integral part of a researcher's study and is used in every research methodology in one way or another. 

Inferential statistics majorly deals with hypothesis testing. The research hypothesis states there is a relationship between the independent variable and dependent variable. Whereas the null hypothesis rejects this claim of any relationship between the two, our job as researchers or students is to check whether there is any relation between the two.  

Hypothesis Testing in Research Methodology

Now that we are clear about what hypothesis testing is? Let's look at the use of hypothesis testing in research methodology. Hypothesis testing is at the centre of research projects. 

What is Hypothesis Testing and Why is it Important in Research Methodology?

Often after formulating research statements, the validity of those statements need to be verified. Hypothesis testing offers a statistical approach to the researcher about the theoretical assumptions he/she made. It can be understood as quantitative results for a qualitative problem. 

(Image will be uploaded soon)

Hypothesis testing provides various techniques to test the hypothesis statement depending upon the variable and the data points. It finds its use in almost every field of research while answering statements such as whether this new medicine will work, a new testing method is appropriate, or if the outcomes of a random experiment are probable or not.

Procedure of Hypothesis Testing

To find the validity of any statement, we have to strictly follow the stepwise procedure of hypothesis testing. After stating the initial hypothesis, we have to re-write them in the form of a null and alternate hypothesis. The alternate hypothesis predicts a relationship between the variables, whereas the null hypothesis predicts no relationship between the variables.

After writing them as H 0 (null hypothesis) and H a (Alternate hypothesis), only one of the statements can be true. For example, taking the hypothesis that, on average, men are taller than women, we write the statements as:

H 0 : On average, men are not taller than women.

H a : On average, men are taller than women. 

Our next aim is to collect sample data, what we call sampling, in a way so that we can test our hypothesis. Your data should come from the concerned population for which you want to make a hypothesis. 

What is the p value in hypothesis testing? P-value gives us information about the probability of occurrence of results as extreme as observed results.

You will obtain your p-value after choosing the hypothesis testing method, which will be the guiding factor in rejecting the hypothesis. Usually, the p-value cutoff for rejecting the null hypothesis is 0.05. So anything below that, you will reject the null hypothesis. 

A low p-value means that the between-group variance is large enough that there is almost no overlapping, and it is unlikely that these came about by chance. A high p-value suggests there is a high within-group variance and low between-group variance, and any difference in the measure is due to chance only.

What is statistical hypothesis testing?

When forming conclusions through research, two sorts of errors are common: A hypothesis must be set and defined in statistics during a statistical survey or research. A statistical hypothesis is what it is called. It is, in fact, a population parameter assumption. However, it is unmistakable that this idea is always proven correct. Hypothesis testing refers to the predetermined formal procedures used by statisticians to determine whether hypotheses should be accepted or rejected. The process of selecting hypotheses for a given probability distribution based on observable data is known as hypothesis testing. Hypothesis testing is a fundamental and crucial issue in statistics. 

Why do I Need to Test it? Why not just prove an alternate one?

The quick answer is that you must as a scientist; it is part of the scientific process. Science employs a variety of methods to test or reject theories, ensuring that any new hypothesis is free of errors. One protection to ensure your research is not incorrect is to include both a null and an alternate hypothesis. The scientific community considers not incorporating the null hypothesis in your research to be poor practice. You are almost certainly setting yourself up for failure if you set out to prove another theory without first examining it. At the very least, your experiment will not be considered seriously.

Types of Hypothesis Testing

There are several types of hypothesis testing, and they are used based on the data provided. Depending on the sample size and the data given, we choose among different hypothesis testing methodologies. Here starts the use of hypothesis testing tools in research methodology.

Normality- This type of testing is used for normal distribution in a population sample. If the data points are grouped around the mean, the probability of them being above or below the mean is equally likely. Its shape resembles a bell curve that is equally distributed on either side of the mean.

T-test- This test is used when the sample size in a normally distributed population is comparatively small, and the standard deviation is unknown. Usually, if the sample size drops below 30, we use a T-test to find the confidence intervals of the population. 

Chi-Square Test- The Chi-Square test is used to test the population variance against the known or assumed value of the population variance. It is also a better choice to test the goodness of fit of a distribution of data. The two most common Chi-Square tests are the Chi-Square test of independence and the chi-square test of variance.

ANOVA- Analysis of Variance or ANOVA compares the data sets of two different populations or samples. It is similar in its use to the t-test or the Z-test, but it allows us to compare more than two sample means. ANOVA allows us to test the significance between an independent variable and a dependent variable, namely X and Y, respectively.

Z-test- It is a statistical measure to test that the means of two population samples are different when their variance is known. For a Z-test, the population is assumed to be normally distributed. A z-test is better suited in the case of large sample sizes greater than 30. This is due to the central limit theorem that as the sample size increases, the samples are considered to be distributed normally. 

FAQs on Hypothesis Testing

1. Mention the types of hypothesis Tests.

There are two types of a hypothesis tests:

Null Hypothesis: It is denoted as H₀.

Alternative Hypothesis: IT is denoted as H₁ or Hₐ.

2. What are the two errors that can be found while performing the null Hypothesis test?

While performing the null hypothesis test there is a possibility of occurring two types of errors,

Type-1: The type-1 error is denoted by (α), it is also known as the significance level. It is the rejection of the true null hypothesis. It is the error of commission.

Type-2: The type-2 error is denoted by (β). (1 - β) is known as the power test. The false null hypothesis is not rejected. It is the error of the omission. 

3. What is the p-value in hypothesis testing?

During hypothetical testing in statistics, the p-value indicates the probability of obtaining the result as extreme as observed results. A smaller p-value provides evidence to accept the alternate hypothesis. The p-value is used as a rejection point that provides the smallest level of significance at which the null hypothesis is rejected. Often p-value is calculated using the p-value tables by calculating the deviation between the observed value and the chosen reference value. 

It may also be calculated mathematically by performing integrals on all the values that fall under the curve and areas far from the reference value as the observed value relative to the total area of the curve. The p-value determines the evidence to reject the null hypothesis in hypothesis testing.

4. What is a null hypothesis?

The null hypothesis in statistics says that there is no certain difference between the population. It serves as a conjecture proposing no difference, whereas the alternate hypothesis says there is a difference. When we perform hypothesis testing, we have to state the null hypothesis and alternative hypotheses such that only one of them is ever true. 

By determining the p-value, we calculate whether the null hypothesis is to be rejected or not. If the difference between groups is low, it is merely by chance, and the null hypothesis, which states that there is no difference among groups, is true. Therefore, we have no evidence to reject the null hypothesis.

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S.3.3 hypothesis testing examples.

• Example: Right-Tailed Test
• Example: Left-Tailed Test
• Example: Two-Tailed Test

Brinell Hardness Scores

An engineer measured the Brinell hardness of 25 pieces of ductile iron that were subcritically annealed. The resulting data were:

The engineer hypothesized that the mean Brinell hardness of all such ductile iron pieces is greater than 170. Therefore, he was interested in testing the hypotheses:

H 0 : μ = 170 H A : μ > 170

The engineer entered his data into Minitab and requested that the "one-sample t -test" be conducted for the above hypotheses. He obtained the following output:

Descriptive Statistics

$\mu$: mean of Brinelli

Null hypothesis    H₀: $\mu$ = 170 Alternative hypothesis    H₁: $\mu$ > 170

The output tells us that the average Brinell hardness of the n = 25 pieces of ductile iron was 172.52 with a standard deviation of 10.31. (The standard error of the mean "SE Mean", calculated by dividing the standard deviation 10.31 by the square root of n = 25, is 2.06). The test statistic t * is 1.22, and the P -value is 0.117.

If the engineer set his significance level α at 0.05 and used the critical value approach to conduct his hypothesis test, he would reject the null hypothesis if his test statistic t * were greater than 1.7109 (determined using statistical software or a t -table):

Since the engineer's test statistic, t * = 1.22, is not greater than 1.7109, the engineer fails to reject the null hypothesis. That is, the test statistic does not fall in the "critical region." There is insufficient evidence, at the \(\alpha\) = 0.05 level, to conclude that the mean Brinell hardness of all such ductile iron pieces is greater than 170.

If the engineer used the P -value approach to conduct his hypothesis test, he would determine the area under a t n - 1 = t 24 curve and to the right of the test statistic t * = 1.22:

In the output above, Minitab reports that the P -value is 0.117. Since the P -value, 0.117, is greater than \(\alpha\) = 0.05, the engineer fails to reject the null hypothesis. There is insufficient evidence, at the \(\alpha\) = 0.05 level, to conclude that the mean Brinell hardness of all such ductile iron pieces is greater than 170.

Note that the engineer obtains the same scientific conclusion regardless of the approach used. This will always be the case.

Height of Sunflowers

A biologist was interested in determining whether sunflower seedlings treated with an extract from Vinca minor roots resulted in a lower average height of sunflower seedlings than the standard height of 15.7 cm. The biologist treated a random sample of n = 33 seedlings with the extract and subsequently obtained the following heights:

The biologist's hypotheses are:

H 0 : μ = 15.7 H A : μ < 15.7

The biologist entered her data into Minitab and requested that the "one-sample t -test" be conducted for the above hypotheses. She obtained the following output:

$\mu$: mean of Height

Null hypothesis    H₀: $\mu$ = 15.7 Alternative hypothesis    H₁: $\mu$ < 15.7

The output tells us that the average height of the n = 33 sunflower seedlings was 13.664 with a standard deviation of 2.544. (The standard error of the mean "SE Mean", calculated by dividing the standard deviation 13.664 by the square root of n = 33, is 0.443). The test statistic t * is -4.60, and the P -value, 0.000, is to three decimal places.

Minitab Note. Minitab will always report P -values to only 3 decimal places. If Minitab reports the P -value as 0.000, it really means that the P -value is 0.000....something. Throughout this course (and your future research!), when you see that Minitab reports the P -value as 0.000, you should report the P -value as being "< 0.001."

If the biologist set her significance level \(\alpha\) at 0.05 and used the critical value approach to conduct her hypothesis test, she would reject the null hypothesis if her test statistic t * were less than -1.6939 (determined using statistical software or a t -table):s-3-3

Since the biologist's test statistic, t * = -4.60, is less than -1.6939, the biologist rejects the null hypothesis. That is, the test statistic falls in the "critical region." There is sufficient evidence, at the α = 0.05 level, to conclude that the mean height of all such sunflower seedlings is less than 15.7 cm.

If the biologist used the P -value approach to conduct her hypothesis test, she would determine the area under a t n - 1 = t 32 curve and to the left of the test statistic t * = -4.60:

In the output above, Minitab reports that the P -value is 0.000, which we take to mean < 0.001. Since the P -value is less than 0.001, it is clearly less than \(\alpha\) = 0.05, and the biologist rejects the null hypothesis. There is sufficient evidence, at the \(\alpha\) = 0.05 level, to conclude that the mean height of all such sunflower seedlings is less than 15.7 cm.

Note again that the biologist obtains the same scientific conclusion regardless of the approach used. This will always be the case.

Gum Thickness

A manufacturer claims that the thickness of the spearmint gum it produces is 7.5 one-hundredths of an inch. A quality control specialist regularly checks this claim. On one production run, he took a random sample of n = 10 pieces of gum and measured their thickness. He obtained:

The quality control specialist's hypotheses are:

H 0 : μ = 7.5 H A : μ ≠ 7.5

The quality control specialist entered his data into Minitab and requested that the "one-sample t -test" be conducted for the above hypotheses. He obtained the following output:

$\mu$: mean of Thickness

Null hypothesis    H₀: $\mu$ = 7.5 Alternative hypothesis    H₁: $\mu \ne$ 7.5

The output tells us that the average thickness of the n = 10 pieces of gums was 7.55 one-hundredths of an inch with a standard deviation of 0.1027. (The standard error of the mean "SE Mean", calculated by dividing the standard deviation 0.1027 by the square root of n = 10, is 0.0325). The test statistic t * is 1.54, and the P -value is 0.158.

If the quality control specialist sets his significance level \(\alpha\) at 0.05 and used the critical value approach to conduct his hypothesis test, he would reject the null hypothesis if his test statistic t * were less than -2.2616 or greater than 2.2616 (determined using statistical software or a t -table):

Since the quality control specialist's test statistic, t * = 1.54, is not less than -2.2616 nor greater than 2.2616, the quality control specialist fails to reject the null hypothesis. That is, the test statistic does not fall in the "critical region." There is insufficient evidence, at the \(\alpha\) = 0.05 level, to conclude that the mean thickness of all of the manufacturer's spearmint gum differs from 7.5 one-hundredths of an inch.

If the quality control specialist used the P -value approach to conduct his hypothesis test, he would determine the area under a t n - 1 = t 9 curve, to the right of 1.54 and to the left of -1.54:

In the output above, Minitab reports that the P -value is 0.158. Since the P -value, 0.158, is greater than \(\alpha\) = 0.05, the quality control specialist fails to reject the null hypothesis. There is insufficient evidence, at the \(\alpha\) = 0.05 level, to conclude that the mean thickness of all pieces of spearmint gum differs from 7.5 one-hundredths of an inch.

Note that the quality control specialist obtains the same scientific conclusion regardless of the approach used. This will always be the case.

In our review of hypothesis tests, we have focused on just one particular hypothesis test, namely that concerning the population mean \(\mu\). The important thing to recognize is that the topics discussed here — the general idea of hypothesis tests, errors in hypothesis testing, the critical value approach, and the P -value approach — generally extend to all of the hypothesis tests you will encounter.

Hypothesis testing

When interpreting research findings, researchers need to assess whether these findings may have occurred by chance. Hypothesis testing is a systematic procedure for deciding whether the results of a research study support a particular theory which applies to a population.

Hypothesis testing uses sample data to evaluate a hypothesis about a population . A hypothesis test assesses how unusual the result is, whether it is reasonable chance variation or whether the result is too extreme to be considered chance variation.

Basic concepts

• Null and research hypothesis

Probability value and types of errors

Effect size and statistical significance.

• Directional and non-directional hypotheses

Null and research hypotheses

To carry out statistical hypothesis testing, research and null hypothesis are employed:

• Research hypothesis : this is the hypothesis that you propose, also known as the alternative hypothesis HA. For example:

H A: There is a relationship between intelligence and academic results.

H A: First year university students obtain higher grades after an intensive Statistics course.

H A; Males and females differ in their levels of stress.

• The null hypothesis (H o ) is the opposite of the research hypothesis and expresses that there is no relationship between variables, or no differences between groups; for example:

H o : There is no relationship between intelligence and academic results.

H o:  First year university students do not obtain higher grades after an intensive Statistics course.

H o : Males and females will not differ in their levels of stress.

The purpose of hypothesis testing is to test whether the null hypothesis (there is no difference, no effect) can be rejected or approved. If the null hypothesis is rejected, then the research hypothesis can be accepted. If the null hypothesis is accepted, then the research hypothesis is rejected.

In hypothesis testing, a value is set to assess whether the null hypothesis is accepted or rejected and whether the result is statistically significant:

• A critical value is the score the sample would need to decide against the null hypothesis.
• A probability value is used to assess the significance of the statistical test. If the null hypothesis is rejected, then the alternative to the null hypothesis is accepted.

The probability value, or p value , is the probability of an outcome or research result given the hypothesis. Usually, the probability value is set at 0.05: the null hypothesis will be rejected if the probability value of the statistical test is less than 0.05. There are two types of errors associated to hypothesis testing:

• What if we observe a difference – but none exists in the population?
• What if we do not find a difference – but it does exist in the population?

These situations are known as Type I and Type II errors:

• Type I Error: is the type of error that involves the rejection of a null hypothesis that is actually true (i.e. a false positive).
• Type II Error:  is the type of error that occurs when we do not reject a null hypothesis that is false (i.e. a false negative).

These errors cannot be eliminated; they can be minimised, but minimising one type of error will increase the probability of committing the other type.

The probability of making a Type I error depends on the criterion that is used to accept or reject the null hypothesis: the p value or alpha level . The alpha is set by the researcher, usually at .05, and is the chance the researcher is willing to take and still claim the significance of the statistical test.). Choosing a smaller alpha level will decrease the likelihood of committing Type I error.

For example, p<0.05  indicates that there are 5 chances in 100 that the difference observed was really due to sampling error – that 5% of the time a Type I error will occur or that there is a 5% chance that the opposite of the null hypothesis is actually true.

With a p<0.01, there will be 1 chance in 100 that the difference observed was really due to sampling error – 1% of the time a Type I error will occur.

The p level is specified before analysing the data. If the data analysis results in a probability value below the α (alpha) level, then the null hypothesis is rejected; if it is not, then the null hypothesis is not rejected.

When the null hypothesis is rejected, the effect is said to be statistically significant. However, statistical significance does not mean that the effect is important.

A result can be statistically significant, but the effect size may be small. Finding that an effect is significant does not provide information about how large or important the effect is. In fact, a small effect can be statistically significant if the sample size is large enough.

Information about the effect size, or magnitude of the result, is given by the statistical test. For example, the strength of the correlation between two variables is given by the coefficient of correlation, which varies from 0 to 1.

• A hypothesis that states that students who attend an intensive Statistics course will obtain higher grades than students who do not attend would be directional.
• A non-directional hypothesis states that there will be differences between students who attend do or don’t attend an intensive Statistics course, but we don’t know what group will get higher grades than the other. The hypothesis only states that they will obtain different grades.

The hypothesis testing process

The hypothesis testing process can be divided into five steps:

• Restate the research question as research hypothesis and a null hypothesis about the populations.
• Determine the characteristics of the comparison distribution.
• Determine the cut off sample score on the comparison distribution at which the null hypothesis should be rejected.
• Determine your sample’s score on the comparison distribution.
• Decide whether to reject the null hypothesis.

This example illustrates how these five steps can be applied to text a hypothesis:

• Let’s say that you conduct an experiment to investigate whether students’ ability to memorise words improves after they have consumed caffeine.
• The experiment involves two groups of students: the first group consumes caffeine; the second group drinks water.
• Both groups complete a memory test.
• A randomly selected individual in the experimental condition (i.e. the group that consumes caffeine) has a score of 27 on the memory test. The scores of people in general on this memory measure are normally distributed with a mean of 19 and a standard deviation of 4.
• The researcher predicts an effect (differences in memory for these groups) but does not predict a particular direction of effect (i.e. which group will have higher scores on the memory test). Using the 5% significance level, what should you conclude?

Step 1 : There are two populations of interest.

Population 1: People who go through the experimental procedure (drink coffee).

Population 2: People who do not go through the experimental procedure (drink water).

• Research hypothesis: Population 1 will score differently from Population 2.
• Null hypothesis: There will be no difference between the two populations.

Step 2 : We know that the characteristics of the comparison distribution (student population) are:

Population M = 19, Population SD= 4, normally distributed. These are the mean and standard deviation of the distribution of scores on the memory test for the general student population.

Step 3 : For a two-tailed test (the direction of the effect is not specified) at the 5% level (25% at each tail), the cut off sample scores are +1.96 and -1.99.

Step 4 : Your sample score of 27 needs to be converted into a Z value. To calculate Z = (27-19)/4= 2 ( check the Converting into Z scores section if you need to review how to do this process)

Step 5 : A ‘Z’ score of 2 is more extreme than the cut off Z of +1.96 (see figure above). The result is significant and, thus, the null hypothesis is rejected.

You can find more examples here:

• Statistics (RMIT Learning Lab)

Some commonly used statistical techniques

Correlation analysis, multiple regression.

• Analysis of variance

Chi-square test for independence

Correlation analysis explores the association between variables . The purpose of correlational analysis is to discover whether there is a relationship between variables, which is unlikely to occur by sampling error. The null hypothesis is that there is no relationship between the two variables. Correlation analysis provides information about:

• The direction of the relationship: positive or negative- given by the sign of the correlation coefficient.
• The strength or magnitude of the relationship between the two variables- given by the correlation coefficient, which varies from 0 (no relationship between the variables) to 1 (perfect relationship between the variables).
• Direction of the relationship.

A positive correlation indicates that high scores on one variable are associated with high scores on the other variable; low scores on one variable are associated with low scores on the second variable . For instance, in the figure below, higher scores on negative affect are associated with higher scores on perceived stress

A negative correlation indicates that high scores on one variable are associated with low scores on the other variable. The graph shows that a person who scores high on perceived stress will probably score low on mastery. The slope of the graph is downwards- as it moves to the right. In the figure below, higher scores on mastery are associated with lower scores on perceived stress.

Fig 2. Negative correlation between two variables. Adapted from Pallant, J. (2013). SPSS survival manual: A step by step guide to data analysis using IBM SPSS (5th ed.). Sydney, Melbourne, Auckland, London: Allen & Unwin

2. The strength or magnitude of the relationship

The strength of a linear relationship between two variables is measured by a statistic known as the correlation coefficient , which varies from 0 to -1, and from 0 to +1. There are several correlation coefficients; the most widely used are Pearson’s r and Spearman’s rho. The strength of the relationship is interpreted as follows:

• Small/weak: r= .10 to .29
• Medium/moderate: r= .30 to .49
• Large/strong: r= .50 to 1

It is important to note that correlation analysis does not imply causality. Correlation is used to explore the association between variables, however, it does not indicate that one variable causes the other. The correlation between two variables could be due to the fact that a third variable is affecting the two variables.

Multiple regression is an extension of correlation analysis. Multiple regression is used to explore the relationship between one dependent variable and a number of independent variables or predictors . The purpose of a multiple regression model is to predict values of a dependent variable based on the values of the independent variables or predictors. For example, a researcher may be interested in predicting students’ academic success (e.g. grades) based on a number of predictors, for example, hours spent studying, satisfaction with studies, relationships with peers and lecturers.

A multiple regression model can be conducted using statistical software (e.g. SPSS). The software will test the significance of the model (i.e. does the model significantly predicts scores on the dependent variable using the independent variables introduced in the model?), how much of the variance in the dependent variable is explained by the model, and the individual contribution of each independent variable.

Example of multiple regression model

From Dunn et al. (2014). Influence of academic self-regulation, critical thinking, and age on online graduate students' academic help-seeking.

In this model, help-seeking is the dependent variable; there are three independent variables or predictors. The coefficients show the direction (positive or negative) and magnitude of the relationship between each predictor and the dependent variable. The model was statistically significant and predicted 13.5% of the variance in help-seeking.

t-Tests are employed to compare the mean score on some continuous variable for two groups . The null hypothesis to be tested is there are no differences between the two groups (e.g. anxiety scores for males and females are not different).

If the significance value of the t-test is equal or less than .05, there is a significant difference in the mean scores on the variable of interest for each of the two groups. If the value is above .05, there is no significant difference between the groups.

t-Tests can be employed to compare the mean scores of two different groups (independent-samples t-test ) or to compare the same group of people on two different occasions ( paired-samples t-test) .

In addition to assessing whether the difference between the two groups is statistically significant, it is important to consider the effect size or magnitude of the difference between the groups. The effect size is given by partial eta squared (proportion of variance of the dependent variable that is explained by the independent variable) and Cohen’s d (difference between groups in terms of standard deviation units).

In this example, an independent samples t-test was conducted to assess whether males and females differ in their perceived anxiety levels. The significance of the test is .004. Since this value is less than .05, we can conclude that there is a statistically significant difference between males and females in their perceived anxiety levels.

Whilst t-tests compare the mean score on one variable for two groups, analysis of variance is used to test more than two groups . Following the previous example, analysis of variance would be employed to test whether there are differences in anxiety scores for students from different disciplines.

Analysis of variance compare the variance (variability in scores) between the different groups (believed to be due to the independent variable) with the variability within each group (believed to be due to chance). An F ratio is calculated; a large F ratio indicates that there is more variability between the groups (caused by the independent variable) than there is within each group (error term). A significant F test indicates that we can reject the null hypothesis; i.e. that there is no difference between the groups.

Again, effect size statistics such as Cohen’s d and eta squared are employed to assess the magnitude of the differences between groups.

In this example, we examined differences in perceived anxiety between students from different disciplines. The results of the Anova Test show that the significance level is .005. Since this value is below .05, we can conclude that there are statistically significant differences between students from different disciplines in their perceived anxiety levels.

Chi-square test for independence is used to explore the relationship between two categorical variables. Each variable can have two or more categories.

For example, a researcher can use a Chi-square test for independence to assess the relationship between study disciplines (e.g. Psychology, Business, Education,…) and help-seeking behaviour (Yes/No). The test compares the observed frequencies of cases with the values that would be expected if there was no association between the two variables of interest. A statistically significant Chi-square test indicates that the two variables are associated (e.g. Psychology students are more likely to seek help than Business students). The effect size is assessed using effect size statistics: Phi and Cramer’s V .

In this example, a Chi-square test was conducted to assess whether males and females differ in their help-seeking behaviour (Yes/No). The crosstabulation table shows the percentage of males of females who sought/didn't seek help. The table 'Chi square tests' shows the significance of the test (Pearson Chi square asymp sig: .482). Since this value is above .05, we conclude that there is no statistically significant difference between males and females in their help-seeking behaviour.

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Hypothesis Testing in Statistics – Short Notes + PPT

“Truth can be stated in a thousand different ways, yet each one can be true…” Swami Vivekananda

What is ‘Test of Hypothesis’?

Ø   Test of Hypothesis (Hypothesis Testing) is a process of testing of the significance regarding the parameters of the population on the basis of sample drawn from it. Ø   Test of hypothesis is also called as ‘Test of Significance’. Ø   J. Neyman and E.S. Pearson initiated the practice of testing of hypothesis in statistics.

What is the purpose of Hypothesis Testing?

Ø   The main purpose of hypothesis testing is to help the researcher in reaching a conclusion regarding the population by examining a sample taken from that population. Ø   The hypothesis testing does not provide proof for the hypothesis. Ø   The test only indicates whether the hypothesis is supported or not supported by the available data.

What is Hypothesis?

Ø   Hypothesis is a statement about one or more populations.

Ø   It is a statement about the parameters of the population about which the statement is made.

Ø   Example:

$   A doctor hypothesized: “The drug ‘X’ is ineffective in 99% of cases of which it is used”.

$   “The average pass percentage of central university degree programme is 98”.

Ø   Through the hypothesis testing the researcher or investigator can determine whether or not such statements are compatible with the available data.

Types of Hypothesis

Ø   There are TWO types of hypothesis.

                        (A).   Research Hypothesis

                        (B).   Statistical Hypothesis

(A). Research Hypothesis

Ø   Research Hypothesis is “a tentative solution for the problem being investigated”.

Ø   It is the supposition (guess) that motivates the research.

Ø   In research, the researcher determines whether or not their supposition can be supported through scientific investigation.

Ø   The research hypothesis directly leads to statistical hypothesis.

(B). Statistical Hypothesis

Details of the Statistical hypothesis are discussed in the “Steps or Components in Testing of Statistical Hypothesis”.

Steps / Components in Testing of Statistical Hypothesis:

Ø   The statistical hypothesis testing consists of following Steps / Components

(1).      Data (variable)

(2).      Statistical Hypothesis

(3).      Test Statistic

(4).      Decision Rule

(5).      Significance Level

(6).      Statistical Decision

(7).      p – Value

(1). Data (variable)

Ø   Data is the information collected from the population.

Ø   It may be the observation of a natural phenomenon, Result of an experiment, Data from a survey or a secondary data.

Ø   The nature of data determines the type of statistical test to be selected.

Ø   All the features of the data such as continuous, discontinuous, quantitative or qualitative etc. matters in the process of hypothesis testing.

(2). Statistical Hypothesis

Ø   Statistical hypothesis is a statement about the population which we want to verify on the basis of information available from the sample.

Ø   A statistical hypothesis is stated in such a way that they may be evaluated by appropriate statistical techniques.

Ø   There are TWO types of statistical hypothesizes:

(a).  Null hypothesis

(b).  Alternative hypothesis

(a). Null Hypothesis

Ø   The Null hypothesis is the hypothesis to be tested by test statistic.

Ø  Null hypothesis is denoted as H 0 .

Ø   Usually the null hypothesis stated as the ‘Hypothesis of No Difference’.

Ø   The statement is created complementary to the conclusion that the researcher is seeking to reach through his research.

Ø   Usually stated in the negative terms of the original research hypothesis.

Ø   Example: The drug ‘X’ DO NOT induces apoptosis in cancerous cells.

Ø   In the statistical testing process, the null hypothesis is either:

$   Rejected

$   Not rejected (Fail to be rejected / accepted)

Ø   If the null hypothesis is not rejected, we say that the data on which the test is based do not provide sufficient evidence to cause the rejection of null hypothesis.

Ø   If the null hypothesis is rejected in the testing process, we say that the data at hand are not compatible with the null hypothesis but are supportive for some other hypothesis (commonly called as alternative hypothesis).

(b). Alternative Hypothesis

Ø   Alternate hypothesis is created in a negative meaning of the null hypothesis.

Ø   It is denoted as H 1 or H A .

Ø   Usually the alternative hypothesis and research hypothesis are the same.

Ø   Example: The drug ‘X’ induces apoptosis in cancerous cells.

How to state the statistical hypothesis?

Ø   The null hypothesis should contain an equality sign (=, ≤ or ≥).

Ø   Example: The population mean (μ) is not 100.

$   H0:       μ = 100

$   H1:       μ ≠ 100

Ø   Example: The population means is greater than 100.

$   H0:       μ ≤ 100

$   H1:       μ > 100

Ø   Example: The population mean is less than 100.

$   H0:       μ ≥ 100

$   H1:       μ < 100

Things to remember when constructing the Null Hypothesis:

$   What you expected to conclude with the study should be placed in the alternative hypothesis.

$   The null hypothesis should contain a statement of equality (=, ≤, ≥).

$   The null hypothesis is the hypothesis to be tested.

$   The null hypothesis and alternative hypothesis should be complementary.

(3). Test Statistic

Ø   Test statistic is the statistic computed from the data sample.

Ø   There are many possible values that the test statistic can adopt.

Ø   Test value of the statistic depends on the nature of the sample.

Ø   The test statistic is the decision maker in hypothesis testing.

Ø   Decision is to reject or not reject the null hypothesis.

Ø   General formula for test statistic: (applicable to most of the test statistic but not to all)

x̄            : mean

μ0         : hypothesized value of population mean

σ/√n     : Standard error

(4). Decision Rule

Ø   All the possible values that the test statistic can assume are points on the horizontal axis of a graph of the distribution of the test statistic.

Ø   The values are divided into two groups:

1.      Values of the rejection region

2.      Values of the non-rejection region

Ø   The decision rule tell us to reject the null hypothesis if the values of the test statistic that we compute from our sample is one of the values in the rejection region and not to reject the null hypothesis if the computed values of the test statistic is on the values in the non-rejection region.

(5) Significance Level

Ø   Level of significance is the probability of rejecting a true null hypothesis in the statistical testing procedure.

Ø   The level of significance is a probability value and it is denoted as ‘α’.

Ø   The significance level decide the decision value to go the rejection region or to the non-rejection region.

Ø  Due to the ‘Level of significance’ the test statistic is often called as ‘Significance Test’.

Ø   If we reject a true null hypothesis we are committed an error.

Ø   Thus, you have to ensure that the probability of rejecting a true null hypothesis is very small.

Ø   Thus, we select a small value of α to ensure the probability of rejecting a true null hypothesis is very less.

Ø   The frequently used α values are 0.01 (99%), 0.05 (95%).

Ø   Explanation : if we select 0.01 (99%) as the significance level, it means that we are 99% confident in our decision but still there is 1% change for our decision being false.

(6). Statistical Decision

Ø   It is the decision of rejecting or not rejecting the null hypothesis.

Ø   We reject the null hypothesis if the computed value of the test statistic is fall in the rejection region.

Ø   We will NOT reject the null hypothesis if the computed value falls in the non-rejection region.

Ø   Conclusion:

Ø   If we reject H 0 , we conclude that H A is true.

Ø   If we fail to reject H 0 , we conclude that the H 0 may be true.

Ø   When a null hypothesis is not rejected one should not say that the null hypothesis accepted but we say that null hypothesis is not rejected.

Ø   We usually avoid the usage ‘accept’, because we may have committed a type II error.

Learn more: Statistical Errors (Type I and Type II Errors)

(7). p-Value

Ø   p-value is the smallest value of α for which we can reject a null hypothesis.

Ø   A p-value is the probability that the computed value for a test statistic is at least as extreme as specified value of the test statistic when the null hypothesis is true.

Daniel, W.W., 1999. Biostatistics: A foundation for analysis in the health sciences 9th edition. John Wiley & Sons inc.: USA.

Khan, I.A. and Khanum, A., 2012, Fundamentals of Biostatistics, 3rd edition (revised), Ukaaz Publications, Hyderabad, India.

Kothari, C.R., 2004. Research methodology: Methods and techniques. New Age International, India.

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What Is Hypothesis Testing?

• How It Works

4 Step Process

The bottom line.

• Fundamental Analysis

Hypothesis Testing: 4 Steps and Example

Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population or a data-generating process. The word "population" will be used for both of these cases in the following descriptions.

Key Takeaways

• Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data.
• The test provides evidence concerning the plausibility of the hypothesis, given the data.
• Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed.
• The four steps of hypothesis testing include stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

How Hypothesis Testing Works

In hypothesis testing, an  analyst  tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis. Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.

The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.

• State the hypotheses.
• Formulate an analysis plan, which outlines how the data will be evaluated.
• Carry out the plan and analyze the sample data.
• Analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

Example of Hypothesis Testing

If an individual wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct. Mathematically, the null hypothesis is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.

A random sample of 100 coin flips is taken, and the null hypothesis is tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.

If there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."

When Did Hypothesis Testing Begin?

Some statisticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”

What are the Benefits of Hypothesis Testing?

Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.

What are the Limitations of Hypothesis Testing?

Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.

Hypothesis testing refers to a statistical process that helps researchers determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. All hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

Sage. " Introduction to Hypothesis Testing ," Page 4.

Elder Research. " Who Invented the Null Hypothesis? "

Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ."

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