A manufacturer of toothpicks wonders what the mean width of a toothpick is under a new manufacturing method. How many toothpicks must the manufacturer measure to be 95% confident that the sample mean is no farther from the true mean than 0.50 mm? Assume normality, the toothpicks produced under the old method had a standard deviation of 0.4 mm. A random sample of 150 batteries has a mean lifetime of 20 hours. Suppose the standard deviation for battery life in the population of all batteries is 2.5 hours. Give a 95% CI for the mean battery life. Before agreeing to purchase a large order of polyethylene sheaths for a particular type of high- pressure oil-filled submarine power cable, a company wants to see conclusive evidence that the true standard deviation of sheath thickness is less than .05 mm. What hypotheses should be tested, and why? In this context, what are the type I and type II errors?