Final answer:
The t-test compares groups based on their mean scores, using the Student's t-distribution when standard deviations are not known or sample sizes are small, and is appropriate for matched pairs or two independent means depending on the situation.
Step-by-step explanation:
The t-test compares groups based on their mean scores. When statistics students develop a technique to lower their anxiety levels, and they have paired anxiety level data at the start and end of the quarter, a test of matched pairs is appropriate. Given the IQ scores example and the known standard deviation, a Student's t-distribution is used for the hypothesis test if the sample size is less than 30, which is usually assumed for a t-test. For the diet example, since we have two groups with known population standard deviations, we would use a test of two independent means. In the final example comparing the mean of the final exam scores of two different classes, it's a test of two means, the standard deviations are assumed to be unknown, and we would use a Student's t-distribution for the test if the sample sizes are small or the population standard deviation is unknown.
When dealing with matched or paired samples for hypothesis testing, one must assume that the paired differences are normally distributed or the sample size is large enough for the Central Limit Theorem to apply if a t-test is being used.