Question

An athletic conditioning coach had a group of 9 randomly selected athletes run 200 meters (full effort), rest for two minutes, then run 200 meters again (full effort), trying their best not to lose any time in their second run. The athletes’ 200 meter times (in seconds) are shown below. Carry out a hypothesis test (at α = .01) to determine whether the mean difference in time between the first and second run is positive. (That is, that “on the average,” athletes run the second 200 meter trial significantly slower than the first.) [To receive full credit, your response should be sure to check the relevant requirement(s), state your hypotheses, find a test value, find a critical value or p-value (your choice), make your decision, and state your conclusion.]

First Trial	21	24	22	26	28	25	23	29	24
Second Trial	25	24	24	29	26	27	28	29	23

Answer 1

The following table is obtained:

	Sample 1	Sample 2	Difference = Sample 1 - Sample 2
	21	25	-4
	24	24	0
	22	24	-2
	26	29	-3
	28	26	2
	25	27	-2
	23	28	-5
	29	29	0
	24	23	1
Average	24.667	26.111	-1.444
St. Dev.	2.646	2.261	2.351
n	9	9	9

and the sample size is n = 9. For the score differences we have

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μD​ ≤ 0

Ha: μD​ > 0

This corresponds to a right-tailed test, for which a t-test for two paired samples be used.

(2) Rejection Region

Based on the information provided, the significance level is α=0.01, and the degrees of freedom are df = 8

Hence, it is found that the critical value for this right-tailed test is t_c = 2.896, for α=0.01 and df = 8

(3) Test Statistics

The t-statistic is computed as shown in the following formula:

(4) Decision about the null hypothesis

Since it is observed that t = -1.843 ≤tc​=2.896, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p = 0.9487 , and since p = 0.9487 ≥0.01, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is greater than μ2​, at the 0.01 significance level.