Mathematics

Q1: As a parole supervisor for your state, you were asked to run a trial anti-recidivism program for a group of 200 randomly-selected parolees. The program didn’t cost any more than standard parole, and you weren’t looking for a miracle cure for recidivism. You just wanted to see if offenders in this program did better than the state overall average. You know that of the 24,036 state parolees who left parole in the past 3 years, a total of 13,461 were returned to incarceration.

Of the 200 people in your program, 97 were returned to incarceration. Using the chi-square statistic, can you say at the 95% confidence level that your program did better than the state average for returns to incarceration?

Q2: In your city the population is 45% white, 23% black, 21% Hispanic, and 11% none of the above. You have just gone through the yearly applicants for the police department. Your applicant pool consisted of 240 white applicants, 88 black applicants, 112 Hispanic applicants, and 41 applicants not identifying as any of those groups. In terms of demographic breakdown, was your applicant pool statistically different than your city’s general demographic profile (at the 95% confidence level)?

IF NOT: A city council member looks at the numbers and percentages and says, “wait, but these percentages are different than the city’s general demographic percentages. Why are you saying that there is no difference between the applicant pool and the city demographic profile?” How do you respond?

IF SO: What is the group with the largest disproportionality (meaning the biggest relative difference between observed and expected values)? Try to come up with a possible explanation for that group’s under-representation or over-representation in the applicant pool.

Q3: A man is convicted of murder and is to be sentenced soon. The death penalty is a possible sentence, You want to know if there is a difference between judges and police officers as to whether they believe the offender should get the death penalty. You’ve sampled judges and police officers and here are your results:

Life without Parole         Death penalty
Judges                     152                      35
Police officers          823                    302

Write out your hypotheses, calculate the chi-square test of association, and indicate whether there is a relationship between profession (i.e., being a judge vs being a police officer) and support for the death penalty for the offender.

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