Question

17. An exercise physiologist records the mile times in seconds for 10 college distance runners before and after a twelve-week training cycle. Is there evidence at the .05 level that the runners' mile times improved (were lower...) following the twelve weeks' training?

Before	After
249.9	250.2
278.8	266.2
254.2	244.7
261.8	252.7
238.8	237.6
267.2	269.7
267.8	256.8
259.8	263.1
254.5	252.1
255.4	243.5

What's the value of the test statistic (based on the After-Before differences)?

18. What's the p-value?

19. Which of the following is(are) the correct critical value(s)?

		-1.833
		-1.96, 1.96
		-2.33
		-2.821
		-2.262, 2.262

20.

What is the correct decision?

		Reject H0
		Don't Reject HA
		Reject HA
		Don't Reject H0

Answer 1

Solution:

(Part 17)

Here, we have to use the paired t test for the population mean difference. The null and alternative hypotheses for this test are given as below:

Null hypothesis: H₀: The runners’ average mile times are not improved.

Alternative hypothesis: H_a: The runners’ average mile times are improved.

H₀: µ_d = 0 versus H_a: µ_d < 0

This is a lower tailed test.

We are given level of significance = α = 0.05

The test statistic formula for this test is given as below:

t = (Dbar - µ_d)/[S_d/sqrt(n)]

From given data, we have

Dbar = -5.1600

S_d = 6.2593

n = 10

df = n – 1 = 9

α = 0.05

t = (Dbar - µ_d)/[S_d/sqrt(n)]

t = (-5.1600 - 0)/[ 6.2593/sqrt(10)]

t = -5.16/ 1.9793

t = -2.6069

(Part 18)

P-value = 0.0142

(by using t-table)

(Part 19)

Critical value = -1.833

(by using t-table)

(Part 20)

P-value < α = 0.05

So, we reject the null hypothesis

Reject H₀

There is sufficient evidence to conclude that the runners’ average mile times are improved.