Question

In​ baseball, League A allows a designated hitter​ (DH) to bat for the​ pitcher, who is typically a weak hitter. In League​ B, the pitcher must bat. The common belief is that this results in League A teams scoring more runs. In interleague​ play, when League A teams visit League B​ teams, the League A pitcher must bat.​ So, if the DH does result in more​ runs, it would be expected that league A teams will score more runs in League A park than when visiting League B parks. To test this​ claim, a random sample of runs scored by league A teams with and without their DH is given in the accompanying table. Complete parts​ a) through​ d) below.

legue a park (with DH)	Legue b park (without DH)
7	0
2	1
4	6
6	3
2	5
3	6
12	8
9	3
3	5
14	5
3	5
7	2
5	2
5	4
2	1
14	2
6	4
6	9
6	10
6	1
5	3
7	7
8	7
4	2
13	4
7	9
5	3
0	2

a) Draw​ side-by-side boxplots of the number of runs scored by League A teams with and without their DH. Choose the correct graph below.

A.

051015AB

Two boxplots, one above the other, share a horizontal axis labeled from 0 to 15 in increments of 1. The bottom boxplot is labeled A and has vertical line segments drawn at 4, 6, and 7. A box encloses the vertical line segments, and horizontal line segments extend from both sides of the box to 0 and 13. An x is plotted at 14. The top boxplot is labeled B and has vertical line segments at 3, 4.5, and 7. A box encloses the vertical line segments, and horizontal line segments extend from both sides of the box to 0 and 11.

B.

051015AB

Two boxplots, one above the other, share a horizontal axis labeled from 0 to 15 in increments of 1. The bottom boxplot is labeled A and has vertical line segments drawn at 4, 6, and 7. A box encloses the vertical line segments, and horizontal line segments extend from both sides of the box to 0 and 9. Three x's are plotted at 12, 13, and 14. The top boxplot is labeled B and has vertical line segments at 2, 3.5, and 6. A box encloses the vertical line segments, and horizontal line segments extend from both sides of the box to 0 and 10.

C.

051015AB

Two boxplots, one above the other, share a horizontal axis labeled from 0 to 15 in increments of 1. The bottom boxplot is labeled A and has vertical line segments drawn at 3, 5, and 6. A box encloses the vertical line segments, and horizontal line segments extend from both sides of the box to 0 and 12. Two x's are plotted at 13 and 14. The top boxplot is labeled B and has vertical line segments at 2, 3.5, and 6. A box encloses the vertical line segments, and horizontal line segments extend from both sides of the box to 0 and 12.

D.

051015AB

Two boxplots, one above the other, share a horizontal axis labeled from 0 to 15 in increments of 1. The bottom boxplot is labeled A and has vertical line segments drawn at 4, 6, and 7. A box encloses the vertical line segments, and horizontal line segments extend from both sides of the box to 1 and 14. The top boxplot is labeled B and has vertical line segments at 2, 3.5, and 6. A box encloses the vertical line segments, and horizontal line segments extend from both sides of the box to 0 and 12.

Does there appear to be a difference in the number of runs between these​ situations?

A. No but the number of runs scored in a League A park appear to be slightly higher than the number of runs scored in a League B park.

B. Yes because the number of runs scored in a League B park appear to have a higher median than the number of runs scored in a League A park.

C.Yes because the number of runs scored in a League A park appear to have a higher median than the number of runs scored in a League B park.

D.No because the number of runs scored in a League A park is about the same as the number of runs scored in a League B park.

​b) Explain why a hypothesis test may be used to test whether the mean number of runs scored for the two types of ballparks differ.

Select all that apply.

A.Each sample has the same sample size.

B.Each sample is obtained independently of the other.

C.Each sample size is small relative to the size of its population.

D.Each sample is a simple random sample.

E.Each sample size is large.

​c) Test whether the mean number of runs scored in a League A park is greater than the mean number of runs scored in a League B park at the

alphaα=0.05 level of significance.

Determine the null and alternative hypotheses for this test. Let mu Subscript Upper AμA

represent the mean number of runs scored by a League A team in a League A park and let

mu Subscript Upper BμB represent the mean number of runs scored by a League A team in a League B park.

Upper H 0H0​:

▼

sigma Subscript Upper AσA

pp mu Subscript Upper AμA

▼

greater than>

equals=

less than<

not equals≠

▼

sigma Subscript Upper BσB

mu Subscript Upper BμB

p 0p0

versus

Upper H 1H1​:

▼

mu Subscript Upper AμA

pp

sigma Subscript Upper AσA

▼

greater than>

equals=

less than<

not equals≠

▼

p0 mu Subscript Upper BμB sigma Subscript Upper BσB Find t0​,the test statistic for this hypothesis test. t0=nothing

​(Round to two decimal places as​ needed.)

Determine the​ P-value for this test.

​P-value=

​(Round to three decimal places as​ needed.)

State the appropriate conclusion. Choose the correct answer below.

A.Do not reject Upper H0. There is not sufficient evidenceThere is not sufficient evidence at the level of significance to conclude that games played with a designated hitter result in more runs.

B.Reject Upper H 0H0.There is not sufficient evidence at the level of significance to conclude that games played with a designated hitter result in more runs.

C.Do not reject Upper H0.There is sufficient evidenceat the level of significance to conclude that games played with a designated hitter result in more runs.

D.Reject Upper H0. There is sufficient evidenceThere is sufficient evidence at the level of significance to conclude that games played with a designated hitter result in more runs.

​d) Construct a 95​% confidence interval for the mean difference in the number of runs scored by League A teams in a League A park and the number of runs scored by League A teams in a League B park. Interpret the interval.

Lower​ bound:

Upper​ bound:

​(Round to three decimal places as​ needed.)

Interpret the confidence interval. Select the correct choice below and fill in the answer boxes to complete your choice.

​(Round to three decimal places as needed. Use ascending​ order)

A. We are 95​%confident the difference between the mean number of runs scored in a League A park and the mean number of runs scored in a League B park is between nothing and nothing.The confidence interval does not containdoes not contain ​zero, so there is sufficient evidence to conclude there is a difference in the mean number of runs scored with or without the DH.

B. We are 95​% confident the difference between the mean number of runs scored in a League A park and the mean number of runs scored in a League B park is between nothing and nothing.The confidence interval contains ​zero, so there is notis not sufficient evidence to conclude there is a difference in the mean number of runs scored with or without the DH.

Answer 1

Following are the box plot of the data:

14(a)

C.Yes because the number of runs scored in a League A park appear to have a higher median than the number of runs scored in a League B park.

(b)

E.Each sample size is large.

(c)

Hypotheses are:

(d)

Following is the output of independent sample t-test:

The p-value is: 0.017

Since p-value is less than 0.05 so we reject the null hypothesis.

D.Reject Upper H0. There is sufficient evidenceThere is sufficient evidence at the level of significance to conclude that games played with a designated hitter result in more runs.

(d)

Lower bound: 0.147

Upper bound: 3.567

Correct option is:

A. We are 95​%confident the difference between the mean number of runs scored in a League A park and the mean number of runs scored in a League B park is between nothing and nothing.The confidence interval does not containdoes not contain ​zero, so there is sufficient evidence to conclude there is a difference in the mean number of runs scored with or without the DH.