Question: The heat evolved in calories per gram of a cement mixture is approximately normally distributed. The mean is thought to be 100 and the standard deviation is 2. We wish to test \({{H}_{0}}:\mu =100\) versus \({{H}_{1}}:\mu \ne 100\) with a sample of n = 9 specimens.

(a) If the acceptance region in defined as \(98.5\le \bar{X}\le 101.5\), find the type I error probability \(\alpha \).

(b) Find \(\beta \) for the case that the true mean heat evolved is 103

(c) Find \(\beta \) for the case where the true mean heat evolved is 105. This value of \(\beta \) is smaller than the one found in part (b), why?