A cosmetic manufacturer is formulating a new shampoo and is interested in foam height (in millimeters). The foam height is approximately normally distributed, with standard deviation 10 millimeters. The company wishes to test whether the population mean is greater than 177.0, using the results of 11 samples.

(a) Find the type I error probability

if the critical region lower bound is 169.

(b) Given the critical region in part (a), what is the probability of a type II error if the true mean foam height is 169.0?

(c) Findwhen the true mean is 195.0.