Question

In: Statistics and Probability

A statistical program is recommended. Data showing the values of several pitching statistics for a random...

A statistical program is recommended.

Data showing the values of several pitching statistics for a random sample of 20 pitchers from the American League of Major League Baseball is provided.

Player Team W L ERA SO/IP HR/IP R/IP
Verlander, J DET 24 5 2.40 1.00 0.10 0.29
Beckett, J BOS 13 7 2.89 0.91 0.11 0.34
Wilson, C TEX 16 7 2.94 0.92 0.07 0.40
Sabathia, C NYY 19 8 3.00 0.97 0.07 0.37
Haren, D LAA 16 10 3.17 0.81 0.08 0.38
McCarthy, B OAK 9 9 3.32 0.72 0.06 0.43
Santana, E LAA 11 12 3.38 0.78 0.11 0.42
Lester, J BOS 15 9 3.47 0.95 0.10 0.40
Hernandez, F SEA 14 14 3.47 0.95 0.08 0.42
Buehrle, M CWS 13 9 3.59 0.53 0.10 0.45
Pineda, M SEA 9 10 3.74 1.01 0.11 0.44
Colon, B NYY 8 10 4.00 0.82 0.13 0.52
Tomlin, J CLE 12 7 4.25 0.54 0.15 0.48
Pavano, C MIN 9 13 4.30 0.46 0.10 0.55
Danks, J CWS 8 12 4.33 0.79 0.11 0.52
Guthrie, J BAL 9 17 4.33 0.63 0.13 0.54
Lewis, C TEX 14 10 4.40 0.84 0.17 0.51
Scherzer, M DET 15 9 4.43 0.89 0.15 0.52
Davis, W TB 11 10 4.45 0.57 0.13 0.52
Porcello, R DET 14 9 4.75 0.57 0.10 0.57

An estimated regression equation was developed to predict the average number of runs given up per inning pitched (R/IP) given the average number of strikeouts per inning pitched (SO/IP) and the average number of home runs per inning pitched (HR/IP).

R/IP = 0.5365 - 0.2483 SO/IP + 1.032 HR/IP

(a)

Use the F test to determine the overall significance of the relationship.

State the null and alternative hypotheses.

H0: β0 = 0
Ha: β0 ≠ 0

H0: β0 ≠ 0
Ha: β0 = 0    

H0: β1 = β2 = 0
Ha: One or more of the parameters is not equal to zero.

H0: One or more of the parameters is not equal to zero.
Ha: β1 = β2 = 0

H0: β1 = β2 = 0
Ha: All the parameters are not equal to zero.

Calculate the test statistic. (Round your answer to two decimal places.) ____

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

p-value = ____

What is your conclusion at the 0.05 level of significance?

Reject H0. There is insufficient evidence to conclude that there is a significant overall relationship.

Do not reject H0. There is sufficient evidence to conclude that there is a significant overall relationship.    

Reject H0. There is sufficient evidence to conclude that there is a significant overall relationship.

Do not reject H0. There is insufficient evidence to conclude that there is a significant overall relationship.

(b)

Use the t test to determine the significance of SO/IP.

State the null and alternative hypotheses.

H0: β1 ≥ 0
Ha: β1 < 0

H0: β1 ≠ 0
Ha: β1 = 0    

H0: β1 ≤ 0
Ha: β1 > 0

H0: β1 = 0
Ha: β1 > 0

H0: β1 = 0
Ha: β1 ≠ 0

Find the value of the test statistic for β1. (Round your answer to two decimal places.) ____

Find the p-value for β1. (Round your answer to three decimal places.)

p-value = ____

What is your conclusion at the 0.05 level of significance?

Do not reject H0. There is sufficient evidence to conclude that SO/IP is a significant factor.

Reject H0. There is sufficient evidence to conclude that SO/IP is a significant factor.    

Reject H0. There is insufficient evidence to conclude that SO/IP is a significant factor.

Do not reject H0. There is insufficient evidence to conclude that SO/IP is a significant factor.

Use the t test to determine the significance of HR/IP.

State the null and alternative hypotheses.

H0: β2 ≠ 0
Ha: β2 = 0

H0: β2 ≤ 0
Ha: β2 > 0    

H0: β2 = 0
Ha: β2 ≠ 0

H0: β2 = 0
Ha: β2 > 0

H0: β2 ≥ 0
Ha: β2 < 0

Find the value of the test statistic for β2. (Round your answer to two decimal places.) ____

Find the p-value for β2. (Round your answer to three decimal places.)

p-value = ____

What is your conclusion at the 0.05 level of significance?

Reject H0. There is insufficient evidence to conclude that HR/IP is a significant factor.

Reject H0. There is sufficient evidence to conclude that HR/IP is a significant factor.    

Do not reject H0. There is sufficient evidence to conclude that HR/IP is a significant factor.

Do not reject H0. There is insufficient evidence to conclude that HR/IP is a significant factor.

Solutions

Expert Solution

using excel>data>data analysis>Regression

we have

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.750642
R Square 0.563463
Adjusted R Square 0.512106
Standard Error 0.053785
Observations 20
ANOVA
df SS MS F Significance F
Regression 2 0.063477 0.031738 10.97144 0.000871
Residual 17 0.049178 0.002893
Total 19 0.112655
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept 0.536508 0.081409 6.59031 4.59E-06 0.364751 0.708265
SO/IP -0.24835 0.071807 -3.45856 0.003002 -0.39985 -0.09685
HR/IP 1.031931 0.435884 2.367444 0.030037 0.112296 1.951567

(a)

State the null and alternative hypotheses.

H0: β1 = β2 = 0
Ha: All the parameters are not equal to zero.

the test statistic. F =10.97

p-value = 0.001

conclusion :

Reject H0. There is sufficient evidence to conclude that there is a significant overall relationship.

(b) the null and alternative hypotheses.

H0: β1 = 0
Ha: β1 ≠ 0

the value of the test statistic for β1 =-3.459

p-value =0.003

conclusion

Reject H0. There is sufficient evidence to conclude that SO/IP is a significant factor.    

the null and alternative hypotheses.

H0: β2 ≠ 0
Ha: β2 = 0

the value of the test statistic for β2 is 2.367

p-value = 0.030

conclusion

Reject H0. There is sufficient evidence to conclude that HR/IP is a significant factor.    


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