A swimming school wants to determine whether a recently hired instructor is working out. 42 out of 80 of Instructor A's students passed the lifeguard certification test on the first try. In comparison, 64 out of 80 of more experienced Instructor B's students passed the test on the first try. We are interested in determining whether there is sufficient evidence to conclude that the experienced instructor pass rate is higher than that of the new instructor. The power of this test, if the actual population proportions are 0.6 and 0.8 for Instructor A and Instructor B respectively, is approximately:

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.