The weights of the apples grown on an orchard are normally distributed. the mean weight has been m0 = 9.500 ounces. because of the inclement weather this year, the farmer would like to know if the mean weight has decreased. a random sample of n = 16 apples is selected. the sample mean is x = 9.320 ounces and the sample standard deviation is s = 0.18 ounces. a test needs to be conducted to test if the population mean has decreased at a significance level a = 0.05 .

23. the null and alternative hypotheses should be (3)

24. using the action limits to set up the decision rule, the decision rule should be (3)

25. the test statistic of this test is (3)

26. the conclusion of this test should be (3)

Aprisoner is trapped in a cell containing three doors. the first door leads to a tunnel that returns him to his cell after two days of travel. the second leads to a tunnel that returns him to his cell after three days of travel. the third door leads immediately to freedom. (a) assuming that the prisoner will always select doors 1, 2 and 3 with probabili- ties 0.5,0.3,0.2 (respectively), what is the expected number of days until he reaches freedom? (b) assuming that the prisoner is always equally likely to choose among those doors that he has not used, what is the expected number of days until he reaches freedom? (in this version, if the prisoner initially tries door 1, for example, then when he returns to the cell, he will now select only from doors 2 and 3.) (c) for parts (a) and (b), find the variance of the number of days until the prisoner reaches freedom. hint for part (b): define ni to be the number of additional days the prisoner spends after initially choosing door i and returning to his cell.