Question

In this problem, assume that the distribution of differences is approximately normal. Note: For degrees of freedom d.f. not in the Student's t table, use the closest d.f. that is smaller. In some situations, this choice of d.f. may increase the P-value by a small amount and therefore produce a slightly more "conservative" answer.

Do professional golfers play better in their last round? Let row B represent the score in the fourth (and final) round, and let row A represent the score in the first round of a professional golf tournament. A random sample of finalists in the British Open gave the following data for their first and last rounds in the tournament.

B: Last	72	66	74	71	71	72	68	68	74
A: First	67	69	63	71	65	71	71	71	71

Do the data indicate that the population mean score on the last round is higher than that on the first? Use a 5% level of significance. (Let d = B − A.)

(a) What is the level of significance?


State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test?

H₀: μ_d > 0; H₁: μ_d = 0; right-tailedH₀: μ_d = 0; H₁: μ_d < 0; left-tailed    H₀: μ_d = 0; H₁: μ_d ≠ 0; two-tailedH₀: μ_d = 0; H₁: μ_d > 0; right-tailed

(b) What sampling distribution will you use? What assumptions are you making?

The standard normal. We assume that d has an approximately normal distribution.The Student's t. We assume that d has an approximately normal distribution.    The Student's t. We assume that d has an approximately uniform distribution.The standard normal. We assume that d has an approximately uniform distribution.

What is the value of the sample test statistic? (Round your answer to three decimal places.)


(c) Find (or estimate) the P-value.

P-value > 0.250

0.125 < P-value < 0.250   

0.050 < P-value < 0.125

0.025 < P-value < 0.050

0.005 < P-value < 0.025

P-value < 0.005

Sketch the sampling distribution and show the area corresponding to the P-value.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.    At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

(e) State your conclusion in the context of the application.

Fail to reject the null hypothesis, there is sufficient evidence to claim that the population score on the last round is higher than that on the first.Reject the null hypothesis, there is sufficient evidence to claim that the population score on the last round is higher than that on the first.    Reject the null hypothesis, there is insufficient evidence to claim that the population score on the last round is higher than that on the first.Fail to reject the null hypothesis, there is insufficient evidence to claim that the population score on the last round is higher than that on the first.

Answer 1

We are given the following data and are asked to tets the hypothesis that population mean score is higher in the last round than that on the first round.

B:Last	72	66	74	71	71	72	68	68	74
A:First	67	69	63	71	65	71	71	71	71

a) The level of significance is 5% or 0.05.

Null hypothesis

i.e., population mean score is same for the last round and for the first round.

Alternate hypothesis

(right tailed)

i.e., population mean score for last round is higher than that on the first round.

b) We use the Student's t and assume that d has an approximately normal distribution since the sample size is small.

To find the the test statistic we find the mean and standard deviation for the last round and the first round.

Let the score of the last round be represented as x and and the score of the first round be represented as y.

xi	yi	di=xi-yi	di2
72	67	5	25
66	69	-3	9
74	63	11	121
71	71	0	0
71	65	6	36
72	71	1	1
68	71	-3	9
68	71	-3	9
74	71	3	9

Test statistic under H₀

Hence the test statistic value is 1.172.

c) The p-value is .137

Graph is not drawn to scale. The shaded region represents the area corresponding to the p-value of 0.137.

d) Since p-value is greater than significance level of 0.05, we fail to reject the null hypothesis.

Hence we conclude that at level, we fail to reject the null hypothesis and conclude that the data are not statistically significant.

e) Our conclusion in the context of the application is,

Fail to reject the null hypothesis, there is insufficient evidence to claim that the population score on the last round is higher than that on the first.