The t Test: Hypothesis Testing with Two Population Means or Proportions

The t test is a hypothesis test used to determine whether there is a significant difference between the means of two groups. 

When do you run a t test?

• When your independent variable (IV) is categorical,
• when your dependent variable (IV) is continuous, AND
• when your IV contains only two groups, populations, or proportions.

There are four types of t tests:

• Independent-samples t test with pooled variances
• Independent-samples t test with separate variances
• Dependent-samples t test
• Proportions t test

Each of these t tests relies on a different set of formulas, so it is important to pick the correct test.

Once you choose which t test to run, you must decide whether to run a one-tailed or two-tailed test. While this decision does not impact which formulas you use, the number of tails in your test affects your critical value. The critical value is the point you need to exceed to reject your null hypothesis. A two-tailed test will have two critical values (a negative and positive) while a one-tailed test will have one critical value (a negative or a positive).

How to Choose the Proper t Test

In the video below (originally recorded for my online applied statistics class), I discuss how to choose the correct t test. The book I mention in the video is Statistics for Criminology and Criminal Justice (3rd edition) by Jacinta M. Gau.

Conducting a t Test

To conduct a t test, one must first determine the null and alternative hypotheses. The null hypothesis states that there is no significant difference between the means of the two groups. The alternative hypothesis states there is a significant difference. The t test then calculates a t value, determining the probability of obtaining the observed difference in means if the null hypothesis is true.

While these general steps are the same for every t test, the formulas are unique to each test. Below are some some examples of how to work through each type of t test (originally recorded for my online applied statistics class).

Independent-Samples t Test with Pooled Variances

Independent-Samples t Test with Separate Variances

Dependent-Samples t Test

Two-Populations Test for Proportions and Percentages

The t test can provide valuable insights into the differences between two groups. However, it is important to carefully consider the assumptions and limitations of the test before drawing any conclusions.

By understanding the assumptions and limitations of the test, researchers can use it to gain valuable insights into their data and draw meaningful conclusions.