Hypothesis Test for Two Independent Means

When you want to find the difference of two means, you should use a hypothesis test for two independent means. You need two independent variables in this test. One must be categorical (i.e. men or women, employed or unemployed) and the other must be quantitative (i.e. temperature, salary). This test has five steps, which are outline and explained below.


You begin by making your null and alternate hypotheses. The null hypothesis is that the difference of the means of the two different variables is zero. This should be written in both a full sentence that is meaningful to the study, as well as in symbols, which appear as \mu_1-\mu_2=0. The alternate hypothesis will depend on what you are trying to prove. There are three options, either the difference of the means is less than zero, not equal to zero, or greater than zero. This hypothesis, too, should be written in a contextually meaningful sentence for the study, as well as in symbols. The three options for symbols are \mu_1-\mu_2 <0, \mu_1-\mu_2\neq 0, and \mu_1-\mu_2 >0 respectively.


The next step is to check three conditions to see if we can preform the study. You must make sure that the samples for both variables were randomly selected. You also need to check that both samples for the variables are less than ten percent of their individual population. Finally, you need to know if the data is nearly normal. There are two ways to check this final condition. If each variable has at least forty samples, then this final condition is met. Otherwise, look at the histograms and QQ plots for each variable. The histograms should look pretty symmetric, and the data points in the QQ plots should be close to the line. If all three conditions are met, you may proceed.

Test Statistic:

Now you need to calculate t. The formula for t is t=\frac{(\overline{y}_1-\overline{y}_2)-(\mu_1-\mu_2)}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}. Notice, however, that \mu_1-\mu_2=0 by the null hypothesis, so the formula can be rewritten as t=\frac{\overline{y}_1-\overline{y}_2}{\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}.

Null Distribution/p-value:

Using the value for t found above, use SPSS to find the p-value. With all this information, you should also sketch what the curve looks like and where the appropriate information lies within it.


Depending on your p-value above, you will either reject or fail to reject the null. If your p-value is less than \alpha, then you reject the null. If your p-value is greater than \alpha, then you fail to reject the null. Along with stating your conclusion, you must put it into a full sentence that is contextually meaningful for your study.

If you would like more information concerning the test or how to preform it, check out some of the following links:

  • This page has an outline of the process of this test, along with some examples. It is simple and easy to follow.
  • This site has a lengthy description. As an added bonus, it explains how to do certain steps in SPSS.
  • For the more visual learners, here is a video that goes through an example of this test. Instead of SPSS however, the author uses a calculator for his calculations.

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