The manager of a newly opened franchise for auto brake relining wanted to estimate what proportion π of his customers would also need their steering aligned. If π was high enough, he planned to offer alignment as an additional service.
(i) He randomly sampled 40 cars, and inspection showed 8 of them did indeed need alignment. What is the 95% confidence interval for π?
(ii) Fortunately, the experience of 86 other franchised shops was available to the manager on request. After eliminating the 11 shops that clearly serviced a different clientele from his own, the remaining 75 shops showed targets of π ranging from 0.15 to 0.63. In fact, a relative frequency histogram of these values of π showed that it was closely approximated by the function 280π³(1−π)⁷. What is the Bayesian 95% interval for π?