Quality control (2.2, 5.3, 6.3) Many manufacturing companies use statistical techniques to ensure that the products they make meet standards. One common way to do this is to take a random sample of products at regular intervals throughout the production shift.

Assuming that the process is working properly, the mean measurement –x from a random sample varies according to a Normal distribution with mean m –x and standard deviation s–x. For each question that follows, assume that the process is working properly.

(a) What’s the probability that at least one of the next two sample means will fall more than 2s–x from the target mean m –x? Show your work.

(b) What’s the probability that the first sample mean that is greater than m –x + 2s–x is the one from the fourth sample taken?

Plant managers are trying to develop a criterion for determining when the process is not working properly. One idea they have is to look at the 5 most recent sample means. If at least 4 of the 5 fall outside the interval (m –x − s–x, m –x + s–x), they will conclude that the process isn’t working.

(c) Find the probability that at least 4 of the 5 most recent sample means fall outside the interval, assuming the process is working properly. Is this a reasonable criterion? Explain.