The distribution of pH levels for all community swimming pools in a large county is approximately Normal with mean 7.5 and standard deviation 0.2. According to swimming pool studies, the safest pH levels for water in swimming pools are between 7.2 and 7.8. (a) One community swimming pool in the county will be selected at random. What is the probability that the selected pool has a pH level that is not considered safe?

The county health inspector will select a random sample of 4 community swimming pools in the county to investigate the pH levels.

(b) Describe the sampling distribution of the sample mean for samples of size 4 (shape, center, and spread).

(c) Consider the situation in which the health inspector find the sample mean of the 4 pools to be outside the safe pH levels. As a result, the inspector declares that the population mean is not 7.5. However, if the population mean really is 7.5, the inspector will have made an error. Such an error is called a Type I error. Find the probability that the inspector will make a Type I error with the sample of 4 pools. Show your work.