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 shown in the accompanying table. Use Table 2 and Table 4.

ANOVA	df	SS	MS	F	Significance F
Regression	2	2,517.3	1,258.6		7.49E-01
Residual	17	72,837.53	4,284.56
Total	19	75,354.80

	Coefficients	Standard Error	t Stat	p-value	Lower 95%	Upper 95%
Intercept	716.68	86.0322	8.3304	0.0000	535.17	898.20
Poverty	3.3717	4.7573	0.7088	0.4881	−6.67	13.41
Income	3.6612	14.3119	0.2558	0.8012	−26.53	33.86

a.	Specify the sample regression equation. (Negative values should be indicated by a minus sign. Report 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 hypothesis test?
	Do not reject H0 we cannot conclude the poverty rate and the crime rate are linearly related. Reject H0 the poverty rate and the crime rate are linearly related. Do not reject H0 we can conclude the poverty rate and the crime rate are linearly related. Reject H0 the poverty rate and the crime rate are not 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 is significant in explaining the crime rate at the 5% significance level.
	Income is not 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. Income is significant in explaining the crime rate, since its slope coefficient does not significantly differ from zero. Income is 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 βj > 0 H0: β1 = β2 = 0; HA: At least one βj < 0 H0: β1 = β2 = 0; HA: At least one βj ≠ 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

Given :

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 shown in the accompanying table.

ANOVA	df	SS	MS	F	Significance F
Regression	2	2,517.3	1,258.6		7.49E-01
Residual	17	72,837.53	4,284.56
Total	19	75,354.80

	Coefficients	Standard Error	t Stat	p-value	Lower 95%	Upper 95%
Intercept	716.68	86.0322	8.3304	0.0000	535.17	898.20
Poverty	3.3717	4.7573	0.7088	0.4881	−6.67	13.41
Income	3.6612	14.3119	0.2558	0.8012	−26.53	33.86

a) The sample regression equation is

${\hat{y}}$ = $\beta$ 0 + $\beta$ 1X1 + $\beta$ 2X2

= 716.6800 + 3.3717 X1 + 3.6612 X2

b ) Hypothesis test :

1) The null and alternative hypothesis is

H₀: β₁ = 0; H_A: β₁ ≠ 0

2) P-value = 0.4881

Since P-value is greater than significance level 0.05, we do not reject the null hypothesis.

Do not reject H_0. We can not conclude that poverty rate and the crime rate are linearly related.

C) 95% confidence interval for the slope coefficient of income.

CI = (-26.53, 33.86)

Since the interval includes 0, we do not reject the null hypothesis.

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

d) 1) Hypothesis test :

The null and alternative hypothesis is

H₀: β₁ = β₂ = 0; H_A: At least one β_j ≠ 0

2) Since the p-value of F tset is 0.7490 is greater than 0.05, we do not reject the null hypothesis.

No, since the null hypothesis is not rejected