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.