3.4.1 Sampling distributions for the difference in two means

In the example of two exam versions, the teacher would like to evaluate whether there is convincing evidence that the difference in average scores between the two exams is not due to chance.

It will be useful to extend the $t$ distribution method from Section 3.3 to apply to a difference of means:

Our procedure for checking conditions mirrors what we did for large samples in Section 3.2. First, we verify the small sample conditions (independence and nearly normal data) for each sample separately, then we verify that the samples are also independent. For instance, if the teacher believes students in her class are independent, the exam scores are nearly normal, and the students taking each version of the exam were independent, then we can use the $t$ distribution for inference on the point estimate ${\overline{x}}_1-{\overline{x}}_2$.

The formula for the standard error of ${\overline{x}}_1-{\overline{x}}_2$, introduced in Section 3.2, also applies to small samples:

Because we will use the $t$ distribution, we will need to identify the appropriate degrees of freedom. This can be done using computer software. An alternative technique is to use the smaller of $n_1-1$ and $n_2-1$, which is the method we will apply.³⁵³⁵This technique for degrees of freedom is conservative with respect to a Type 1 Error; it is more difficult to reject the null hypothesis using this $df$ method.