Paired Samples t-test: Test Statistic and p-value

The government of Canada wants to know whether the legalization of marihuana has had any effect on the rate of drug-related offenses. To investigate this matter, a researcher selects a simple random sample of #13# cities and compares the rates of drug-related offenses before #(X)# and after #(Y)# the legalization was implemented.

The values in the table below are the number of drug-related offenses per #100#,#000# residents:

City	1	2	3	4	5	6	7	8	9	10	11	12	13
#X:\,\text{Before}#	263	255	225	242	238	264	232	258	269	224	248	258	253
#Y:\,\text{After}#	255	243	231	236	242	249	227	250	273	222	256	254	265

You may assume that the population distributions of drug-related offenses both before and after the legalization are normal.

The researcher plans on using a paired samples #t#-test to determine whether the legalization of marihuana has had a significant effect on the number of drug-related offenses.

Define #D=Y-X#.

State the null and alternative hypotheses of the proposed test.

#H_0 : \mu_D\,\,#
#H_a : \mu_D\,\,#

8. Testing for Differences in Means and Proportions: Paired Samples t-test

Paired Samples t-test: Test Statistic and p-value