What is the relationship between crime and punishment? We use data from 90 North Carolina counties to examine the question. County crime rates and other characteristics are observed over the period 1981-1987. The data are in the file crime. Use the 1985 data for this exercise.

a. Calculate the summary statistics for CRMRTE (crimes committed per person) and PRBARR (the probability of arrest \(=\) the ratio of arrests to offenses), including the maximums and minimums. Does there appear to be much variation from county to county in these variables?

b. Plot CRMRTE versus PRBARR. Do you observe a relationship between these variables?

c. Estimate the linear regression model \(C R M R T E=\beta_{1}+\beta_{2} P R B A R R+e\). If we increase the probability of arrest by \(10 \%\) what will be the effect on the crime rate? What is a \(95 \%\) interval estimate of this quantity?

d. Test the null hypothesis that there is no relationship between the county crime rate and the probability of arrest versus the alternative that there is an inverse relationship. State the null and alternative hypotheses in terms of the model parameters. Clearly, state the test statistic and its distribution if the null hypothesis is true and the test rejection region. Use the \(1 \%\) level of significance. What is your conclusion?