3. The Death Penalty Some US states have the death penalty, and others don't, and laws have changed over time. In addition to changes in murder rates, we will also consider whether the death penalty was in force in each state and each year. Using this information, we would like to investigate how the presence of the death penalty affects the murder rate of a state. Question 3.1. We want to know whether the death penalty causes a change in the murder rate. Why is it not sufficient to compare murder rates in places and times when the death penalty was in force with places and times when it wasn't? Write your answer here, replacing this text. A Natural Experiment1 In order to attempt to investigate the causal relationship between the death penalty and murder rates, we're going to take advantage of a natural experiment. A natural experiment happens when something other than experimental design applies a treatment to one group and not to another control group, and we have some hope that the treatment and control groups don't have any other systematic differences. Our natural experiment is this: in 1972, a Supreme Court decision called Furman v. Georgia banned the death penalty throughout the US. Suddenly, many states went from having the death penalty to not having the death penalty As a first step, let's see how murder rates changed before and after the court decision. We'll define the test as follows: Population: All the states that had the death penalty before the 1972 abolition. (There is no control group for the states that already lacked the death penalty in 1972, so we must omit them.) This includes all US states except Alaska, Hawaii, Maine, Michigan, Wisconsin, and Minnesota. Treatment group: The states in that population, in 1973 (the year after 1972). Control group: The states in that population, in 1971 (the year before 1972). Null hypothesis: Murder rates in 1971 and 1973 come from the same distribution Alternative hypothesis: Murder rates were higher in 1973 than they were in 1971. Our alternative hypothesis is related to our suspicion that murder rates increase when the death penalty is eliminated. Question 3.2: Should we use an A/B test to test these hypotheses? If yes, what is our 'A' group and what is our 'B' group? Write your answer here, replacing this text. The death penalty table below describes whether each state allowed the death penalty in 1971. 7): non_death_penalty_states = make_array( 'Alaska', 'Hawaii', 'Maine', 'Michigan', 'Wisconsin', 'Minnesota') def had_death penalty_in_1971(state): "" "Returns True if the argument is the name of a state that had the death penalty in 1971."" # The implementation of this function uses a bit of syntax # we haven't seen before. Just trust that it behaves as its #documentation claims. return state not in non_death penalty_states states = murder_rates.group('State').select('State') death_penalty - states.with_column('Death penalty', states.apply(had_death penalty_in_1971, 0)) death_penalty Question 3.3: Use the death penalty and murder_rates tables to find murder rates in 1971 for states with the death penalty before the abolition. Create a new table preban rates that contains the same information as murder_rates, along with a column Death Penalty that contains booleans (True or False) describing if states had the death penalty in 1971. In [138]: States that had death penalty in 1972 preban rates ... preban rates In (): ok.grade("q3_3"); Question 3.4: Create a table postban_rates that contains the same information as preban_rates, but for 1973 instead of 1971. postban_rates should only contain the states found in praban_rates. In [144): postban_rates -... postban_rates - postban_rates.sort("State") postban rates In (): ok.grade('q3_4"); Question 3.5: Use preban_rates copy and postban_rates to create a table change_in_death_rates that contains each state's population, murder rate, and whether or not that state had the death penalty for both 1971 and 1973. Hint: tbl_1.append(tbl_2) with create a new table that includes rows from both tbl_1 and tbl_2 . Both tables must have the exactly the same columns, in the same order. In [74]: preban_rates_copy - preban_rates.copy change in death rates -... change_in_death_rates Run the cell below to view the distribution of death rates during the pre-ban and post-ban time periods. In (75): change_in_death_rates.hist('Murder Rate', group - Death Penalty') Question 3.6: Create a table rate_means that contains the average murder rates for the states that had the death penalty and the states that didn't have the death penalty. It should have two columns: one indicating if the penalty was in place, and one that contains the average murder rate for each group. In (86): rate_means = ... rate means In [ ]: ok.grade("3_6"); Question 3.7: We want to figure out if there is a difference between the distribution of death rates in 1971 and 1973. Specifically, we want to test if murder rates were higher in 1973 than they were in 1971. What should the test statistic be? How does it help us differentiate whether the data supports the null and alternative? If you are in lab, confirm your answer with a lab TA/LA before moving on. Write your answer here, replacing this text. Question 3.8: Set observed_difference to the observed test statistic using the rate_means tablo In [107]: observed difference - ... observed difference In [ ]: ok.grade("q3_8"); Question 3.9: Given a table like change in death rates, a value column label, and a group column group_label, write a function that calculates the appropriate test statistic. In [110]: def find_test_stat (table, labels_col, values_col). find_test_stat(change_in_death_rates, "Death Penalty", "Murder Rate") In ok.grade"39")