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 $11$ 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
$X:\,\text{Before}$	227	238	251	222	235	222	232	265	233	220	227
$Y:\,\text{After}$	228	226	242	209	238	210	228	260	236	214	222

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 $D=X-Y$ .

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