Question

Scores in the first and fourth (final) rounds for a sample of 20 golfers who competed in golf tournaments are shown in the following table.

Player	First Round	Final Round
Golfer 1	70	72
Golfer 2	71	72
Golfer 3	70	74
Golfer 4	72	71
Golfer 5	70	69
Golfer 6	67	67
Golfer 7	71	68
Golfer 8	68	73
Golfer 9	67	73
Golfer 10	70	69

Player	First Round	Final Round
Golfer 11	72	72
Golfer 12	72	70
Golfer 13	70	73
Golfer 14	70	75
Golfer 15	68	70
Golfer 16	68	66
Golfer 17	71	70
Golfer 18	70	68
Golfer 19	69	68
Golfer 20	67	71

Suppose you would like to determine if the mean score for the first round of a golf tournament event is significantly different than the mean score for the fourth and final round. Does the pressure of playing in the final round cause scores to go up? Or does the increased player concentration cause scores to come down?

Use

α = 0.10

to test for a statistically significantly difference between the population means for first- and fourth-round scores.

State the null and alternative hypotheses. (Use μ_d = mean score first round − mean score fourth round.)

H₀:

μ_d > 0

H_a:

μ_d ≤ 0

H₀:

μ_d = 0

H_a:

μ_d ≤ 0

    

H₀:

μ_d ≠ 0

H_a:

μ_d = 0

H₀:

μ_d = 0

H_a:

μ_d ≠ 0

H₀:

μ_d ≤ 0

H_a:

μ_d > 0

Calculate the value of the test statistic. (Round your answer to three decimal places.)

Calculate the p-value. (Round your answer to four decimal places.)

p-value =

State your conclusion.

Reject H₀. There is insufficient evidence to conclude that the mean score for the first round of a golf tournament is significantly different than the mean score for the fourth and final round.Reject H₀. There is sufficient evidence to conclude that the mean score for the first round of a golf tournament is significantly different than the mean score for the fourth and final round.    Do not Reject H₀. There is sufficient evidence to conclude that the mean score for the first round of a golf tournament is significantly different than the mean score for the fourth and final round.Do not reject H₀. There is insufficient evidence to conclude that the mean score for the first round of a golf tournament is significantly different than the mean score for the fourth and final round.

(b)

What is the point estimate of the difference between the two population means? (Use mean score first round − mean score fourth round.)

For which round is the population mean score lower?

The mean of the fourth round scores was lower than the mean of the first round scores.The mean of the first round scores was lower than the mean of the fourth round scores.    

(c)

What is the margin of error for a 90% confidence interval estimate for the difference between the population means? (Round your answer to two decimal places.)

Could this confidence interval have been used to test the hypothesis in part (a)? Explain.

Yes. One could check to see if the 90% confidence interval includes a difference of zero. If the interval does not contain 0, the difference is not statistically significant.

Yes. One could check to see if the 90% confidence interval includes a difference of 1. If the interval does not contain 1, the difference is not statistically significant.    

Yes. One could check to see if the 90% confidence interval includes a difference of zero. If the interval contains 0, the difference is not statistically significant.

Yes. One could check to see if the 90% confidence interval includes a difference of 1. If the interval contains 1, the difference is not statistically significant.

No. One can not use a confidence interval to test hypothesis in part (a) because hypothesis tests and confidence intervals are two different things.

Answer 1

Excel output:

Player	First Round	Final Round		t-Test: Paired Two Sample for Means
Golfer 1	70	72
Golfer 2	71	72			First Round	Final Round
Golfer 3	70	74		Mean	69.65	70.55
Golfer 4	72	71		Variance	2.765789474	6.05
Golfer 5	70	69		Observations	20	20
Golfer 6	67	67		Pearson Correlation	0.139601116
Golfer 7	71	68		Hypothesized Mean Difference	0
Golfer 8	68	73		df	19
Golfer 9	67	73		t Stat	-1.452966315
Golfer 10	70	69		P(T<=t) one-tail	0.081275
Golfer 11	72	72		t Critical one-tail	1.729132812
Golfer 12	72	70		P(T<=t) two-tail	0.162549999
Golfer 13	70	73		t Critical two-tail	2.093024054
Golfer 14	70	75
Golfer 15	68	70
Golfer 16	68	66
Golfer 17	71	70
Golfer 18	70	68
Golfer 19	69	68
Golfer 20	67	71

Answer:

Part a.

  Value of the test statistic= -1.453

p-value=0.1625

Since p-value>0.10, so

Do not reject H0. There is insufficient evidence to conclude that the mean score for the first round of a golf tournament is significantly different than the mean score for the fourth and final round.

Part b. The point estimate of the difference between the two population means=69.65-70.55=-0.9

Part c.

90% CI for (-0.9-1.07, -0.9+1.07)=(-1.97, 0.17).

Yes. One could check to see if the 90% confidence interval includes a difference of zero. If the interval contains 0, the difference is not statistically significant.