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.
616 exam-style questions on Edexcel A Level Maths The Normal Distribution, covering 3.1 The Normal Distribution, 3.2 Finding Probabilities for Normal Distributions, 3.3 The Inverse Normal Distribution Function, 3.4 The Standard Normal Distribution, 3.5 Finding the mean and standard deviation, 3.6 Approximating a Binomial Distribution, and 3.7 Hypothesis Testing with the Normal Distribution. Each one has a worked solution and a mark scheme showing where the marks go.