An aerospace engineering firm produces micro-bearings for satellite sensors. The mass of these bearings is assumed to follow a normal distribution. A random sample of 8 bearings is taken, and their masses, in milligrams, are recorded as follows:
24.12,23.95,24.08,24.15,23.98,24.03,24.11,24.06 24.12, \quad 23.95, \quad 24.08, \quad 24.15, \quad 23.98, \quad 24.03, \quad 24.11, \quad 24.06 24.12,23.95,24.08,24.15,23.98,24.03,24.11,24.06Determine unbiased estimates for the population mean and the population variance of the masses of these micro-bearings.
The population standard deviation of the bearing masses is known from historical data to be 0.08 mg. The quality control team requires that for a random sample of size nnn, the probability that the sample mean mass lies within 0.015 mg of the true population mean mass is at least 0.99.
Calculate the minimum sample size n n\,n required to meet this specification.