Question

Does listening to music while studying hinder students’ learning? Two AP Statistics students designed an experiment to find out. They selected a random sample of 30 students from their medium-sized high school to participate. Each subject was given 10 minutes to memorize two different lists of 20 words, once while listening to music and once in silence. The order of the two-word lists was determined at random; so was the order of the treatments. The difference in the number of words recalled (music − silence) was recorded for each subject. A paired t-test on the differences yielded t = −3.01 and P-value = 0.0027.

(a) State appropriate hypotheses for the paired t-test. Be sure to define your parameter.

(b) What are the degrees of freedom for the paired t-test?

(c) Interpret the P-value in context. What conclusion should the students draw?

(d) Describe a Type I error and a Type II error in this setting. Which mistake could students have made based on your answer to part (c)?

Answer 1

(a) The appropriate hypotheses for the paired t-test in this experiment are:

Null hypothesis (H0): The mean difference in the number of words recalled between listening to music and silence is zero.

Alternative hypothesis (Ha): The mean difference in the number of words recalled between listening to music and silence is not zero.

Parameter: The parameter of interest is the mean difference in the number of words recalled (music - silence) for the population of students.

(b) The degrees of freedom for the paired t-test are calculated as (n - 1), where n is the number of paired observations. In this case, since there are 30 students in the sample, the degrees of freedom would be (30 - 1) = 29.

(c) The P-value of 0.0027 indicates the probability of observing a t-value as extreme as -3.01 (or more extreme) under the null hypothesis. In other words, if the null hypothesis were true (the mean difference is zero), there is only a 0.27% chance of obtaining a t-value as extreme as -3.01.

Since the P-value is less than the conventional significance level of 0.05 (or 5%), we reject the null hypothesis. Therefore, the students can conclude that listening to music while studying has a significant effect on the number of words recalled.

(d) In this setting:

- Type I error: This refers to rejecting the null hypothesis when it is actually true. It means concluding that there is a significant difference in the number of words recalled due to listening to music while studying when, in reality, there is no such difference.

- Type II error: This refers to failing to reject the null hypothesis when it is actually false. It means failing to conclude that there is a significant difference in the number of words recalled due to listening to music while studying when, in reality, there is a difference.

Based on the answer to part (c), the students rejected the null hypothesis and concluded that listening to music while studying has a significant effect on the number of words recalled. Therefore, if the null hypothesis were true (no actual difference), the students could have made a Type I error by incorrectly concluding that there is a difference.