Question

A researcher wanted to see if live reggae music improved students’ math test scores. He selected a sample of 61 students from his university and gave them a math test. The students then studied for two and a half hours while an acoustic reggae band played quietly. The students then took another math test of the same difficulty. The researcher’s variable was the change in test score for each student — a positive change meant the student did better, while a negative change meant a student did worse. He found that the changes had an approximately normal distribution with sample mean 6.5 and sample standard deviation 12. Let μ be the population average change — that is, the average change in scores if all students in the university participated in the experiment. Suppose we wish to test the hypothesis that μ is positive. (For this question, assume the sample is random.)

(a) Write down mathematical null and alternative hypotheses in terms of μ. (Make sure you identify which is the null and which is the alternative.)

(b) A good test statistic to use here is T = (X-bar - u0)/(s/sqrt(n)) ,where X_bar is the sample mean, μ0 is the null hypothesis mean (in this case, zero), s is the sample standard deviation, and n is the sample size. Using the data, calculate the observed value of T.

(c) Using R, calculate a one-tailed P-value. (Note: Here, you may either use a normal distribution or a t-distribution; it doesn’t make much substantive difference.)

(d) Does the study provide convincing evidence that the population mean change is positive? Explain why or why not.

(e) Does the study provide convincing evidence that live reggae music improves students’ math test scores? Explain why or why not.

Answer 1

a)

b)

T = (6.5 - 0 )/(12/sqrt(61))

= 4.23055

c)

p-value =
1-pnorm(4.23055)

= 0.00001165

d)

since p-value < alpha

we reject the null hypothesis

the study provides convincing evidence that the population mean change is positive