1) Although crime rates in the U.S. are generally falling, in the past much has been written about the relative effectiveness of raising the probability of arrest and increasing the average sentence length.

One study using state-level data from 1996 regressed the number of arrests (measured in thousands) on the number of police officers (measured in thousands), the average sentence length (measured in years), the unemployment rate and the average income (measured in dollars). The study concludes that average sentence length is a more effective deterrent than the probability of arrest.

a) It's possible that average sentence length has a nonlinear effect on the crime rate. For example, there might be a very pronounced effect for adding an additional year to a sentence of less than five years while there is a much smaller effect from adding an additional year to a sentence of more than 15 years. Suggest a method for modeling this specific type of nonlinearity.

b) How would you test my conjecture in part a?

c) I also suspect that the effect of average sentence lengths is more pronounced when potential criminals realize that there is a high probability of being caught. Please describe a method for modeling this specific type of nonlinearity.

d) How would you test my conjecture in part c?