8. Testing for Differences in Means and Proportions: Paired Samples t-test
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 #12# 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 |
| #X:\,\text{Before}# | 270 | 262 | 261 | 223 | 259 | 251 | 252 | 220 | 269 | 233 | 259 | 256 |
| #Y:\,\text{After}# | 265 | 254 | 263 | 220 | 253 | 245 | 253 | 215 | 265 | 230 | 264 | 253 |
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\,\,#
#H_a : \mu_D\,\,#
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