9.4 REVIEW OF CONCEPTS
The t distributions are similar to the z distribution, except that in the former, we must estimate the standard deviation from the sample. When estimating the standard deviation, we make a mathematical correction to adjust for the increased likelihood of error. After estimating the standard deviation, the t statistic is calculated like the z statistic for a distribution of means. The t distributions can be used in three ways: (1) to compare the mean of a sample to a population mean when we don’t know the population standard deviation (single-sample t test), (2) to compare two samples with a within-groups design (paired-samples t test), and (3) to compare two samples with a between-groups design (independent-samples t test—introduced in Chapter 10).
Like z tests, single-sample t tests are conducted in the rare cases in which we have one sample that we’re comparing to a known population. The difference is that we only have to know the mean of the population to conduct a single-sample t test. There are many t distributions, one for every possible sample size. We look up the appropriate critical values on the t table based on degrees of freedom, a number calculated from the sample size. We can calculate a confidence interval and an effect size (Cohen’s d), for a single-sample t test.
The Paired-Samples t Test
We use a paired-samples t test when we have two samples, and the same participants are in both samples; to conduct the test, we calculate a difference score for every individual in the study. The comparison distribution is a distribution of mean difference scores instead of the distribution of means that we used with a single-sample t test. Aside from the comparison distribution, the steps of hypothesis testing are similar to those for a single-sample t test. As we can with a z test and a single-sample t test, we can calculate a confidence interval and an effect size (Cohen’s d) for a paired-samples t test.