In Chapter 5, Exercise 9, we studied the variability of crime rates and police expenditures in the eastern and Midwestern United States. We’ve now been asked to investigate the hypothesis that the number of crimes is related to police expenditures per capita because states with higher crime rates are likely to increase their police force, thereby spending more on the number of officers on the street.
A. Construct a scatter diagram of the number of crimes and police expenditures per capita, with number of crimes as the predictor variable. What can you say about the relationship between these two variables based on the scatterplot?
B. Find the least-squares regression equation that predicts police expenditures per capita from the number of crimes. What is the slope? What is the intercept?
C. Calculate the coefficient of determination and provide an interpretation.
D. If the number of crimes increased by 100 for a state, by how much would you predict police expenditures capita to increase?
E. Does it make sense to predict police expenditures per capita when the number of crimes is equal to zero? Why or not?
We do not have the data on either the crime rates or police expenditures.
If a set of data has mean=12 and standard deviation=3, then the coefficient of variation is [x]%
A. To construct a scatter diagram of the number of crimes and police expenditures per capita with number of crimes as the predictor variable, you will need the data for these two variables. Once you have this data, you can plot a point on the scatter diagram for each state, where the x-coordinate represents the number of crimes and the y-coordinate represents the police expenditures per capita. Once you plot all the points, you can examine the scatterplot to assess the relationship between the two variables.
B. To find the least-squares regression equation that predicts police expenditures per capita from the number of crimes, you can use statistical software or a calculator with regression capabilities. By performing a linear regression analysis, you can obtain the equation of the fitted line. In this case, the equation will be in the form y = mx + b, where y represents police expenditures per capita, x represents the number of crimes, m represents the slope, and b represents the intercept.
C. The coefficient of determination, denoted as r^2, measures the proportion of the variability in the dependent variable (police expenditures per capita) that can be explained by the independent variable (number of crimes). To calculate r^2, you can square the correlation coefficient. Once you have obtained the value of r^2, you can interpret it as the percentage of the variability in police expenditures per capita that can be accounted for by the number of crimes.
D. To predict how much police expenditures per capita would increase if the number of crimes increased by 100 for a state, you can use the regression equation obtained in part B. Substitute the new value of the number of crimes (x) into the equation and calculate the corresponding predicted value of police expenditures per capita (y).
E. It may not make sense to predict police expenditures per capita when the number of crimes is equal to zero. This is because the regression equation is based on the relationship between the number of crimes and police expenditures per capita. If there are no crimes in a state, it would not be appropriate to use this equation to make predictions about police expenditures. In this case, the intercept of the regression equation would represent the base level of police expenditures per capita in the absence of any crimes, which may not be meaningful in practice.