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. Use Table 2 and Table 4.

ANOVA	df	SS	MS	F	Significance F
Regression	2	188,246.8	94,123.4		9.04E-07
Residual	17	45,457.32	2,673.96
Total	19	233,704.1

	Coefficients	Standard Error	t Stat	p-value	Lower 95%	Upper 95%
Intercept	−301.62	549.7135	−0.5487	0.5903	−1,461.52	858.28
Poverty	53.1597	14.2198	3.7384	0.0016	23.16	83.16
Income	4.9472	8.2566	0.5992	0.5569	−12.47	22.37

a.	Specify the sample regression equation. (Negative values should be indicated by a minus sign. Round your answers to 4 decimal places.)

  CrimeˆCrime^ = +  Poverty +  Income

b-1.

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

	H0: β1 = 0; HA: β1 ≠ 0
	H0: β1 ≤ 0; HA: β1 > 0
	H0: β1 ≥ 0; HA: β1 < 0

        

b-2.

At the 5% significance level, what is the conclusion to the test?

	Reject H0; the poverty rate and the crime rate are linearly related.
	Reject H0; the poverty rate and the crime rate are not linearly related.
	Do not reject H0; we can conclude the poverty rate and the crime rate are linearly related.
	Do not reject H0; we cannot conclude 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 your answers to 2 decimal places.)

Confidence interval

to

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; HA: At least one β j ≠ 0
	H0: β1 = β2 = 0; HA: At least one β j > 0
	H0: β1 = β2 = 0; HA: 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?

	Yes, since the null hypothesis is rejected.
	Yes, since the null hypothesis is not rejected.
	No, since the null hypothesis is rejected.
	No, since the null hypothesis is not rejected.

Answer 1

a)

The model is:

b-1)

H0: β1 = 0; HA: β1 ≠ 0

b-2)

The p-value is 0.0016

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

Reject H0; the poverty rate and the crime rate are linearly related.

c-1

The 95% confidence interval for income is

(−12.47 , 22.37)

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)

The p-value of F test is 0.0000

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

Yes, since the null hypothesis is rejected.