A manufacturer of high-precision turbine components monitors the mass of steel ball bearings produced by a specific automated lathe. The mass of a bearing, W W\,W grams, is modeled by a normal distribution with W∼N(μ,0.0452)W \sim N(\mu, 0.045^2)W∼N(μ,0.0452). The target mass for these bearings is exactly 12.000 g.
Seven bearings are selected at random from the production line, and their masses are recorded as follows:
11.962,12.015,11.974,11.988,11.956,12.003,11.970 11.962, \quad 12.015, \quad 11.974, \quad 11.988, \quad 11.956, \quad 12.003, \quad 11.970 11.962,12.015,11.974,11.988,11.956,12.003,11.970(i) Calculate a 99% confidence interval for μ\muμ, giving your limits to 3 decimal places.
(ii) Based on this interval, determine whether there is evidence to suggest the lathe is failing to meet its target mass.
In a subsequent quality control check involving a sample of n n\,n bearings, a sample mean of 11.984 g was obtained. A 95% confidence interval for μ \mu\,μ was constructed, and the resulting upper limit was found to be strictly less than the target mass of 12.000 g. Calculate the minimum possible value of nnn.
