T Test for Two Population Means

The Case Study and the Data

It is not difficult to find any school system that relies on testing for a variety of different purposes, such as adhering to different federal and state guidelines, for example, the No Child Left Behind Act of 2001. Another purpose might be to chart school progress or determine differences between classrooms at the same grade level. To examine whether such differences are significant, a t test for independent means can be applied. Table 1 shows the sample data set for 25 children, one set from Susan Graves's classroom and one set from Jack Longer's classroom.

The Assumptions Underlying the t Test

Three important assumptions underlie the use of the t test for independent means:

• 1.
  The test variable (which in this example is the math test score for each student) is normally distributed. If the sample is large enough (15 or more per group), this assumption is fairly resistant to being violated, but if the scores are not normal, then a larger sample size might be needed.
• 2.
  The variances for each of the test variables in both groups are equal to one another.
• 3.
  The cases in each of the samples are random in nature, and the scores on the test variable are independent of one another. If this assumption is violated, then the resulting t value should not be trusted.

The Research Hypothesis

The null hypothesis associated with this analysis is that there is a difference between the population [Page 991]means of the two samples. This is a nondirectional test and can be stated as follows:

where μ equals the population mean.

Computing the t Value

One formula for computing the t value is

where

X¯1 is the mean for Group 1, which in this example is 79.48;

X¯2 is the mean for Group 2, which in this example is 73.91;

n1 is the number of participants in Group 1, which in this example is 25;

n2 is the number of participants in Group 2, which in this example is 22;

Figure 1 Group Statistics, Independent Samples Test

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