I will use , the Greek letter psi, for contrasts among population means. In each case the contrast is 0 when is true. It is convenient to work with whole numbers, and we can just as well work with is still equal to 0 when is equal to the average of and . To estimate a contrast, form the corresponding sample contrast by using the sample means in place of the population means. We can obtain the standard error of a contrast by using the rules for variances given early in the course.
Inference is based on t statistics.
The corresponding sample contrast is The standard error of is To test the null hypothesis use the t statistic with degrees of freedom that are associated with . Reading methods . In our example the coefficients are for and for . The P-value is large, so there is no evidence against . We have no reason to think that the two reading techniques differ in effectiveness.