Chapter 13.1, Problem 5PB

Does more education cause more crime? The FL Crime data file on the book’s website has data for the 67 counties in Florida on

y = crime rate: Annual number of crimes in county per 1000 population

x₁ = education: Percentage of adults in county with at least a high school education

x₂ = urbanization: Percentage in county living in an urban environment.

The figure shows a scatterplot matrix. MINITAB multiple regression results are also displayed.

Scatterplot matrix for crime rate, education, and urbanization.

Multiple regression for y = crime rate, x₁ = education, and x₂ = urbanization.

1. a. Find the predicted crime rate for a county that has 0% in an urban environment and (i) 70% high school graduation rate and (ii) 80% high school graduation rate.
2. b. Use results from part a to explain how education affects the crime rate, controlling for urbanization, interpreting the slope coefficient −0.58 of education.
3. c. Using the prediction equation, show that the equation relating crime rate and education when urbanization is fixed at (i) 0, (ii) 50, and (iii) 100, is as follows:

Sketch a plot with these lines and use it to interpret the effect of education on crime rate, controlling for urbanization.

1. d. The scatterplot matrix shows that education has a positive association with crime rate, but the multiple regression equation shows that the association is negative when we keep x₂ = urbanization fixed. Consider the hypothetical figure that follows. Sketch lines that represent (i) the prediction equation from a simple regression model using only education and ignoring the information on urbanization and (ii) the prediction equation from the multiple regression model for counties having urbanization = 50. Use these lines to explain the difference in the interpretation of the slope for education in simple and multiple regression models with regard to ignoring or controlling for urbanization. (Note: The reversal in the association between crime rate and education is an example of Simpson’s paradox; see Example 16 in Sec. 3.4 and Example 18 in Sec. 10.5).

Hypothetical scatterplot for crime rate and education, labeling by urbanization.