The pH levels of food items prepared in a certain way are normally distributed with a standard deviation of σ = 0.82. An experimenter estimates the mean pH level by averaging the pH levels of a random sample of n items.
(a) If n = 5, what is the probability that the experimenter's estimate is within 0.5 of the true mean value?
(b) If n = 10, what is the probability that the experimenter's estimate is within 0.5 of the true mean value?
(c) What sample size n is needed to ensure that there is a probability of at least 99% that the experimenter's estimate is within 0.5 of the true mean value?