Question

You may need to use the appropriate technology to answer this 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	73
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	77
Golfer 15	68	70
Golfer 16	68	65
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?

(a)

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.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.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.

(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 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 1. If the interval contains 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 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 zero. If the interval contains 0, 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

a)

Ho :   µd=   0
Ha :   µd ╪   0

paired t test

Sample #1	Sample #2	difference , Di =sample1-sample2	(Di - Dbar)²
70	72	-2	1.21
71	72	-1	0.01
70	73	-3	4.41
72	71	1	3.61
70	69	1	3.61
67	67	0	0.81
71	68	3	15.21
68	73	-5	16.81
67	73	-6	26.01
70	69	1	3.61
72	72	0	0.81
72	70	2	8.41
70	73	-3	4.41
70	77	-7	37.21
68	70	-2	1.21
68	65	3	15.21
71	70	1	3.61
70	68	2	8.41
69	68	1	3.61
67	71	-4	9.61

	sample 1	sample 2	Di	(Di - Dbar)²
sum =	1393	1411	-18	167.800

mean of difference ,    D̅ =ΣDi / n =   -0.900
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    2.972

Level of Significance ,    α =    0.1
      
sample size ,    n =    20
      

std error , SE = Sd / √n =    2.9718   / √   20   =   0.6645      
                          
t-statistic = (D̅ - µd)/SE = (   -0.9000   -   0   ) /    0.6645   =   -1.354
                          
Degree of freedom, DF=   n - 1 =    19                  
  
p-value =        0.1915   [excel function: =t.dist.2t(t-stat,df) ]               
Conclusion:     p-value>α , Do not reject null hypothesis     

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)

point estimate of difference ,    D̅ =ΣDi / n =   -0.900

c)

sample size ,    n =    20          
Degree of freedom, DF=   n - 1 =    19   and α =    0.1  
t-critical value =    t α/2,df =    1.7291   [excel function: =t.inv.2t(α/2,df) ]      
                  
std dev of difference , Sd =    √ [ (Di-Dbar)²/(n-1) =    2.9718          
                  
std error , SE = Sd / √n =    2.9718   / √   20   =   0.6645
margin of error, E = t*SE =    1.7291   *   0.6645   =   1.1490
                  
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.