43. A manufacturer of plumbing xtures has developed a new type of washerless faucet. Let p  P(a randomly selected faucet of this type will develop a leak within 2 years under normal use). The manufacturer has decided to proceed with production unless it can be determined that p is too large; the borderline acceptable value of p is speci ed as .10. The manufacturer decides to subject n of these faucets to accelerated testing (approximating 2 years of normal use).

With X  the number among the n faucets that leak before the test concludes, production will commence unless the observed X is too large. It is decided that if p  .10, the probability of not proceeding should be at most .10, whereas if p  .30 the probability of proceeding should be at most .10. Can n  10 be used? n  20? n  25? What is the appropriate rejection region for the chosen n, and what are the actual error probabilities when this region is used?