It seems to make common sense that a dependent (repeated measures) design would yield a lower critical value for t than an independent sample design since in the former variance should be lower. Yet when I run the two different t-tests in Excel using the same data I get a lower critical value for t using the independent t-test.
Am I missing something or do I just need more coffee?
Best Answer
You are misinterpreting the meaning of critical value of t which is, excepting your sample size, unrelated to your data.
Different samples will surely produce different t statistics. And of course, if you have the same sample size in group 1 as in group 2 (or in measurement 1 as in measurement 2), then the paired and unpaired t test statistics be different numbers for the same sample. However, the critical value of t is not a statistic drawn from your sample data. Rather it is a value of t—given your sample size/degrees of freedom, and preferred rate of making a type I error—that marks the threshold between "reject" and "fail to reject" the null hypothesis. We can know what this critical value of t is (given the sample size), because we have a theory describing the behavior of Student's t distribution, specifically with values of t that are unlikely to occur if the null hypothesis is true.