Compute z and arrive at a decision. . . ............. . . ... Exercises 1. Two machines fill bottles with a cough syrup. Workers check do the performance of each periodically by testing bottles re- moved at random from the production line. The sample statis- tics for machine 1 are as follows: The mean weight of the contents of 40 bottles was 202.6 milligrams, and the standard deviation of the sample was 3.3 milligrams. For machine 2, the mean weight of 50 bottles was 200.0 milligrams, and the stan- dard deviation was 2.0 milligrams. Using a two-tailed test and the 0.01 level of significance, test whether there is any differ- ence in the performance of the two machines. a. State both the null hypothesis and the alternate hypothesis. b. State the decision rule. c. Compute z and arrive at a decision. 12. You are conducting a study of the annual incomes of probation officers in metropolitan areas of fewer than 100,000 popula- tion and in metropolitan areas having greater than 500,000 population. Your sample data are: Population Population Less Greater Than 100,000 Than 500,000 Sample size 30 60 Sample mean $34,290 $34, 330 Sample standard deviation $135 $142 Test the hypothesis that the annual incomes of probation offi- cers in areas having greater than 500,000 population are sig- nificantly greater than those paid in areas of fewer than 100,000 population. Use the 0.05 level of significance. (Hint: Note that the problem includes the words "greater than.") Use a systematic approach by stating the null hypothesis, the alter- nate hypothesis, and so on. 13. A sociologist asserts that the mean length of courtship is longer before a second marriage than before a first. She bases this claim on the observation of (1) 80 first marriages in which the