Suppose that in Example 3.3, σ was 0.15 instead of 0.2 and we decided to adjust the machine if a sample of 16 had a mean weight below 7.9 or above 8.1 (same as before). (a) What is the probability of...

Suppose that in Example 3.3, σ was 0.15 instead of 0.2 and we decided to adjust the machine if a sample of 16 had a mean weight below 7.9 or above 8.1 (same as before).

(a) What is the probability of a type I error now?

(b) Draw the operating characteristic curve using the rejection region obtained in part (a).

Example 3.3

A company that packages salted peanuts in 8-oz. jars is interested in maintaining control on the amount of peanuts put in jars by one of its machines. Control is defined as averaging 8 oz. per jar and not consistently over- or underfilling the jars. To monitor this control, a sample of 16 jars is taken from the line at random time intervals and their contents weighed. The mean weight of peanuts in these 16 jars will be used to test the hypothesis that the machine is indeed working properly. If it is deemed not to be doing so, a costly adjustment will be needed.²