CHAPTER 11 Comparing Average Values between Groups 147

Surveying Student t tests

In this section, we present the general approach to conducting a Student t test.

We walk through the computational steps common to the different kinds of t

tests, including one-group, paired, and independent. As we do that, we explain

the computational differences between the different test types. Finally, we

demonstrate how to run the t tests using open source software R, and explain

how to interpret the output (see Chapter 4 for more information about getting

started with R).

Understanding the general approach to a t test

As reviewed earlier, t tests are designed to compare two means only. If you mea-

sure the means of two groups, you see that they almost always come out to be

different numbers. The Student t tests are intended to answer the question, Is the

observed difference in means larger than what you would expect from random fluctua-

tions alone? The different t tests take the same general approach to answer this

question, using the following steps:

1.

Calculate the difference (D) between the mean values you are comparing.

2.

Calculate the precision of the difference, which is the magnitude of the

random fluctuations in that difference.

For the t test, calculate the standard error (SE) of that difference (see

Chapter 10 for a refresher on SE).

3.

Calculate the test statistic, which in this case is t.

The test statistic expresses the size of the D relative to the size of its SE. That

is: t

D

SE

/

.

4.

Calculate the degrees of freedom (df) of the t statistic.

df is a tricky concept, but is easy to calculate. For t, the df is the total number of

observations minus the number of means you calculated from those

observations.

5.

Use the t and df to calculate the p value.

The p value is the probability that random fluctuations alone could produce a t

value at least as large as the value you just calculated based upon the Student t

distribution.

The Student t statistic is always calculated using the general equation D/SE. Each

specific type of t test we discussed earlier  — including one-group, paired,

unpaired, and Welch — calculates D, SE, and df slightly differently. These different

calculations are summarized in Table 11-1.