3. When the manufacturing process is working properly, NeverReady batteries have lifetimes that follow a slightly right-skewed distribution with = 7 hours. A quality control supervisor selects a simple random sample of n batteries every hour and measures the lifetime of each. If she is convinced that the mean lifetime of all batteries produced that hour is less than 7 hours at the 5% significance level, then all those batteries are discarded.

a. State appropriate hypotheses for the quality control supervisor to test.

b. The supervisor wants to conduct a test using samples of only n=4 so as to avoid wasting batteries. Explain briefly why a sample size of 30 would be necessary.

c. Suppose you draw a random sample of 30 batteries and find that the mean battery life for the sample is 6.8 hours and the standard deviation of the sample is 0.6 hours. Conduct a hypothesis test at the 0.05 level to determine if there is sufficient evidence to conclude that the mean lifetime of all batteries is less than 7 hours.

d. Describe a Type I and a Type II error in this situation.