Question

Excel is recommended for this problem.

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: One or more of the parameters is not equal to zero.
H_a: β₁ = β₂ = 0H₀: β₀ = 0
H_a: β₀ ≠ 0    H₀: β₁ = β₂ = 0
H_a: One or more of the parameters is not equal to zero.H₀: β₀ ≠ 0
H_a: β₀ = 0H₀: β₁ = β₂ = 0
H_a: 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.

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.

Do not reject H₀. There is sufficient 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
H_a: β₁ = 0H₀: β₁ = 0
H_a: β₁ ≠ 0    H₀: β₁ ≥ 0
H_a: β₁ < 0H₀: β₁ ≤ 0
H_a: β₁ > 0H₀: β₁ = 0
H_a: 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?

Reject H₀. There is insufficient 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.

  Do not reject H₀. There is sufficient 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
H_a: β₂ > 0H₀: β₂ ≤ 0
H_a: β₂ > 0    H₀: β₂ ≥ 0
H_a: β₂ < 0H₀: β₂ ≠ 0
H_a: β₂ = 0H₀: β₂ = 0
H_a: 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?

Do not 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.    

Reject H₀. There is insufficient 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.

Answer 1

Regression Analysis
	R²	0.563
	Adjusted R²	0.512	n	20
	R	0.751	k	2
	Std. Error	0.054	Dep. Var.	R/IP
ANOVA table
Source	SS	df	MS	F	p-value
Regression	0.0635	2	0.0317	10.97	.0009
Residual	0.0492	17	0.0029
Total	0.1127	19
Regression output					confidence interval
variables	coefficients	std. error	t (df=17)	p-value	95% lower	95% upper	std. coeff.
Intercept	0.5365	0.0814	6.590	4.59E-06	0.3648	0.7083	0.000
SO/IP	-0.2483	0.0718	-3.459	.0030	-0.3998	-0.0968	-0.567
HR/IP	1.0319	0.4359	2.367	.0300	0.1123	1.9516	0.388

ANSWERS:

(a)

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

F- score = 10.97

p- value = 0.001

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

(b)

H₀: β₁ = 0
H_a: β₁ ≠ 0

t- score =-3.46

p- value = 0.003

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

(c)

H₀: β₂ = 0
H_a: β₂ ≠ 0

t- score = 2.37

p- value = 0.030

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