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 2,517.3 1,258.7 0.29 0.749 Residual 17 72,837.53 4,284.56 Total 19 75,354.8 Coefficients Standard Error t Stat p-value Intercept 716.6835 86.0322 8.330 0.000 Poverty 3.3717 4.7573 0.7090 0.488 Income 3.6612 14.3119 0.2560 0.801 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? 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 "tα/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; 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? 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

ANOVA df SS MS F Significance F

Regression 2 2,517.3 1,258.7 0.29 0.749

Residual 17 72,837.53 4,284.56

Total 19 75,354.8

Coefficients Standard Error t Stat p-value

Intercept 716.6835 86.0322 8.330 0.000

Poverty 3.3717 4.7573 0.7090 0.488

Income 3.6612 14.3119 0.2560 0.801

b-1.

H0: β1 = 0; HA: β1 ≠ 0

b-2.

p-value for Poverty rate is 0.488.

Since, p-value is greater than 0.05 significance level, we fail to reject null hypothesis H0 and conclude that there is no strong evidence that poverty rate and the crime rate are linearly related.

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

c-1.

df for residual is 17.

For 95% confidence interval, = 0.05

Critical value of t at  /2 = 0.025 and df = 17 is 2.110

95% confidence interval for the slope coefficient of income

(3.6612 - 2.11 * 14.3119 , 3.6612 + 2.11 * 14.3119)

(-26.54, 33.86)

c-2.

Since the 95% confidence interval for the slope coefficient of income contains the value 0,

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.

P-value for F statistic is 0.749

Since, p-value is greater than 0.05 significance level, we fail to reject null hypothesis H0 and conclude that there is no strong evidence that poverty rate and income jointly significant in explaining the crime rate.

No, since the null hypothesis is not rejected.