Question

For a sample of 20 New England cities, a sociologist studies the crime rate in each city (crimes per 100,000 residents) as a function of its poverty rate (in %) and its median income (in $1,000s). A portion of the regression results is as follows. [You may find it useful to reference the t table.]

ANOVA	df	SS	MS	F	Significance F
Regression	2	14,616.0	7,308.0	1.50	0.252
Residual	17	83,082.77	4,887.22
Total	19	97,698.8

	Coefficients	Standard Error	t Stat	p-value
Intercept	895.5077	87.3533	10.252	0.000
Poverty	−8.9628	5.6537	−1.5850	0.131
Income	1.0354	14.6046	0.0710	0.944

b-1. Choose the appropriate hypotheses to test whether the poverty rate and the crime rate are linearly related.

• H₀: β₁ = 0; H_A: β₁ ≠ 0

• H₀: β₁ ≤ 0; H_A: β₁ > 0

• H₀: β₁ ≥ 0; H_A: β₁ < 0

b-2. At the 5% significance level, what is the conclusion to the test?

• Do not reject H₀we cannot conclude that the poverty rate and the crime rate are linearly related.
• Reject H₀we can conclude that the poverty rate and the crime rate are linearly related.
• Do not reject H₀we can conclude that the poverty rate and the crime rate are linearly related.
• Reject H₀we cannot conclude that the poverty rate and the crime rate are linearly related.

c-1. Construct a 95% confidence interval for the slope coefficient of income. (Negative values should be indicated by a minus sign. Round "t_α/2,df" value to 3 decimal places, and final answers to 2 decimal places.)

c-2. Using the confidence interval, determine whether income influences the crime rate at the 5% significance level.

• Income is significant in explaining the crime rate, since its slope coefficient does not significantly differ from zero.

• Income is not significant in explaining the crime rate, since its slope coefficient significantly differs from zero.

• Income is significant in explaining the crime rate, since its slope coefficient significantly differs from zero.

• Income is not significant in explaining the crime rate, since its slope coefficient does not significantly differ from zero.

d-1. Choose the appropriate hypotheses to determine whether the poverty rate and income are jointly significant in explaining the crime rate.

• H₀: β₁ = β₂ = 0; H_A: At least one β _j > 0

• H₀: β₁ = β₂ = 0; H_A: At least one β _j < 0

• H₀: β₁ = β₂ = 0; H_A: At least one β _j ≠ 0

d-2. At the 5% significance level, are the poverty rate and income jointly significant in explaining the crime rate?

• No, since the null hypothesis is not rejected.

• Yes, since the null hypothesis is rejected.

• No, since the null hypothesis is rejected.

• Yes, since the null hypothesis is not rejected.

Answer 1

b-1)

H0: β1 = 0; HA: β1 ≠ 0

since p value >0.05

Do not reject H0 we cannot conclude that the poverty rate and the crime rate are linearly related.

c-1)

estimated slope b=			1.0354
standard error of slope=sb==		14.6046
for 95 % confidence and 17degree of freedom critical t=	2.1100
95% confidence interval =b1 -/+ t*standard error=		(-29.78 ,31.85)

c-2)

Income is not significant in explaining the crime rate, since its slope coefficient does not significantly differ from zero.

d-1)H0: β1 = β2 = 0; HA: At least one β j ≠ 0

d-2)

No, since the null hypothesis is not rejected.