A high-precision engineering firm monitors the depth of laser-etched serial numbers on surgical instruments. Historical data indicates that the standard deviation of the etch depth is σ=0.35\sigma = 0.35σ=0.35 micrometers. A quality control engineer suspects that the laser has become unstable, leading to more inconsistent etch depths than usual. The etch depth of the instruments can be assumed to follow a normal distribution. A random sample of 13 instruments is taken and the depths, xxx, are measured, providing the following summary statistics:
∑x=62.4,∑x2=301.1 \sum x = 62.4, \quad \sum x^2 = 301.1 ∑x=62.4,∑x2=301.1Stating your hypotheses clearly, and using a 5% level of significance, test the engineer's suspicion.
The engineer decides that for future monitoring, they will use a larger sample size of n=26n = 26n=26 and a significance level of 5% with the same hypotheses.
Using statistical tables, show that the critical region for the sample variance S2 S^2\,S2 is S2>0.184S^2 > 0.184S2>0.184 (to 3 decimal places).
Calculate the probability of a Type II error for the test in part (b) if the true standard deviation of the etch depth has actually increased to σ=0.55\sigma = 0.55σ=0.55 micrometers.
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.