Problem 7An engineer is interested in the mean length of pipes produced by a production line. The pipes are supposed to be 30 feet long, but customers are complaining that

the pipe sare too long. An initial simple random sample of 50 pipes has a mean and standard deviation of 30.25 feet and 0.6 feet, respectively. In lecture, we performed a statistical test with null hypothesis Ho : µ = 30 ft and alternative hypothesis H. : µ = 30.25 ft > 30 ft. We found that the attained significance level was p 0.0016 = 0.16%. (a) We found in lecture that, with a = 0.05, B× 0.0968. (i) (3 points) Show that, with a = 0.01, ß - 0.27. ii) (3 points) Find the probability of a Type-II error with a = 0.10. (iii) (3 points) How are a and 3 related, qualitatively, for a fixed sample size? (b) We found in lecture that the sample size needed for a = B = 0.01 was n = 126. Find the sample size needed for (i) (3 points) a = B = 0.001 (ii) (3 рoints) а — 0.01, В 3D 0.001 (ii) (3 рoints) а %— 0.001, В 3D 0.01