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.

ANOVA	df	SS	MS	F	Significance F
Regression	2	3,113.6	1,556.8	0.25	0.779
Residual	17	104,444.27	6,143.78
Total	19	107,557.8

	Coefficients	Standard Error	t Stat	p-value
Intercept	843.6066	128.6559	6.557	0.000
Poverty	−3.3654	5.1734	−0.6510	0.524
Income	-8.0888	23.0364	−0.3510	0.730

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 H₀ we cannot conclude that the poverty rate and the crime rate are linearly related.
• Reject H₀ we can conclude that the poverty rate and the crime rate are linearly related.
• Do not reject H₀ we can conclude that the poverty rate and the crime rate are linearly related.
• Reject H₀ we 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 significantly differs from zero.

• 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 does not significantly differ from zero.

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

(b-2) Do not reject H0 we cannot conclude that the poverty rate and the crime rate are linearly related.

(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.