a) The diameter of a certain shaft is normally distributed with a mean expected to be 2.79 cm and a standard deviation of 0.01 cm. The specification limits are 2.77 ± 0.03. If 1000 shafts are...

a) The diameter of a certain shaft is normally distributed with a mean expected to be 2.79 cm and a standard deviation of 0.01 cm. The specification limits are 2.77 ± 0.03. If 1000 shafts are produced, how many can we expect will be unacceptable? If the sample mean for 1000 shafts is 2.786 cm, what are the 99% confidence limits for the true mean diameter?

b) What is the probability that a single diameter measurement will deviate from the true mean by at least ± 0.02 cm?

c) Estimate the required size of the random sample in future measurements so that the 95% confidence interval for the population mean will not be wider than from the sample mean less 0.01 cm, to the sample mean plus 0.01 cm.