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 2 294.3 147.2 9.73E-01 Residual 17 91,413.94 5,377.29 Total 19 91,708.30 Coefficients Standard Error t Stat p-value Lower 95% Upper 95% Intercept 760.09 93.0941 8.1648 0.0000 563.68 956.50 Poverty −0.0245 5.8042 −0.0042 0.9967 −12.27 12.22 Income 3.2598 14.1819 0.2299 0.8209 −26.66 33.18 a. Specify the sample regression equation. (Negative values should be indicated by a minus sign. Report your answers to 4 decimal places.) Crimeˆ Crime ^ = + 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 H0Picture we cannot conclude the poverty rate and the crime rate are linearly related. Reject H0Picture the poverty rate and the crime rate are linearly related. Do not reject H0Picture we can conclude the poverty rate and the crime rate are linearly related. Reject H0Picture 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 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. 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

Solution:-

(a) crime =823.03 ?1.5527*Poverty ?7.7945*Income

(b-1) H₀: ?₁ = 0; H_A: ?₁ ? 0

(b-2) Since the p-value of Poveryt is 0.7997 which is larger than 0.05, we do not reject the null hypothesis.

So answer is Do not reject H0 the poverty rate and the crime rate are not linearly related.

(c-1) (?46.70, 31.11)

(c-2) Since the interval includes 0, we do not reject the null hypothesis.

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

(d-1)H₀: ?₁ = ?₂ = 0; H_A: At least one ? _j ? 0

(d-2) Since the p-value of F tset is 0.8877 which is larger than 0.05, we do not reject the null hypothesis.

Answer: No, since the null hypothesis is not rejected.

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