Question

A mid-distance running coach claims that his six-month training program significantly reduces the average time to complete a 1500-meter run. Five mid-distance runners were randomly selected before they were trained with the coach’s six-month training program and their completion time of 1500-meter run was recorded (in minutes). After six-months of training under the coach, the same five runner’s 1500 meter run time recorded again, the results are given below. Runner 1 2 3 4 5 Completion time before training 5.9 7.5 6.1 6.8 8.1 Completion time after training 5.4 7.1 6.2 6.5 7.8 18. Is the statement that the training program reduce the mean run time at 5% level of significance. (Hint: paired samples) 19. What conclusion you can draw about the claim of the coach?

Answer 1

The data provided is:

Runner	Completion time before training	Completion time after training
1	5.9	5.4
2	7.5	7.1
3	6.1	6.2
4	6.8	6.5
5	8.1	7.8

The following table is obtained:

	Sample 1	Sample 2	Difference = Sample 1 - Sample 2
	5.9	5.4	0.5
	7.5	7.1	0.4
	6.1	6.2	-0.1
	6.8	6.5	0.3
	8.1	7.8	0.3
Average	6.88	6.6	0.28
St. Dev.	0.928	0.908	0.228
n	5	5	5

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=0.05, and the degrees of freedom are df=4.

Hence, it is found that the critical value for this right-tailed test is tc​=2.132, for α=0.05 and df=4.

The rejection region for this right-tailed test is R=t:t>2.132.

(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=2.746>tc​=2.132, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0.0258, and since p=0.0258<0.05, it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population mean μ1​ is greater than μ2​, at the 0.05 significance level. Hence the coaches claim is true.

Graphically

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