A criminologist studied the relationship between level of education and crime ra

Question

A criminologist studied the relationship between level of education and crime rate. He collected data from 84 medium-sized US counties. Two variables were measured:X-the percentage of individuals having at least a high-school diploma: and Y-the crime rate (crimes reported per 100,000 residents) in the previous year. A snapshot of the data and scatter plot are shown here: County Crines/100,000 Percent-of-High-school-graduates 179 9100 Figure 1: Scatter plot of Crime rate vs. Percentage of high school graduates Some sumary statistics are also given: Perform analysis under the simple linear regression model (a) Based on the scatter plot, comment on the relatlonship between peroentage of high (b) Caleulate the least squares estimatos:. Write down the fitted regression (e) Caleulnte error sum of squares (SSE) and mean squnced error (MSE). What is the line. Interpret and (d) Calculate the standard errors for the LS estinators , respectively. , (e) Test whether or not there is a linear assoriation between crime rate and peroentage of high selool graduates at significance level 0.01. State the ull and alternative hy- potheues, the test statistie, its null distribution, the decision rule and the conclasion (f) What is an unbiased estimator for ,? Coustruct a 99% confidence interval for . Interpret your confidence interval. (g) Construct a 95% confidence interval for the mean crime rate for counties with per- centage of high school graduates being 85. Interpret your confidence interval. (h) County A has a high-school graduates percentage being 83. What is the predicted crime rate of county A? Construct a 95% prediction interval for the crime rate. Compare this interval with the one from part (g), what do you find? (i) Would additional assumption be needed in order to condact parts (e)(b)? If so. please state what it is

Explanation / Answer

Using Minitab:

Source DF Adj SS Adj MS F-Value P-Value
Regression 1 3026162 3026162 7.34 0.054
% of high school graduates 1 3026162 3026162 7.34 0.054
Error 4 1650202 412550
Lack-of-Fit 3 1640120 546707 54.23 0.099
Pure Error 1 10082 10082
Total 5 4676364

Model Summary

S R-sq R-sq(adj) R-sq(pred)
642.301 64.71% 55.89% 0.00%

Coefficients

Term Coef SE Coef T-Value P-Value VIF
Constant 16356 3060 5.35 0.006
% of high school graduates -105.2 38.8 -2.71 0.054 1.00

a. R suared=64.71% there is positive relationship between two variables

b.

Regression Equation

Crimes = 16356 - 105.2 % of high school graduates

here b0=16356 and b1=105.2%

c.

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