A precision engineering firm manufactures turbine blades. The depth of a specific cooling groove is required to follow a normal distribution with standard deviation σ=0.85\sigma = 0.85σ=0.85 mm. An engineer suspects that the calibration of a new cutting machine is inconsistent, leading to higher variability in the groove depths than specified. A random sample of 15 blades is tested, yielding the following summary data:
∑x=750,∑x2=37514.7 \sum x = 750, \quad \sum x^2 = 37514.7 ∑x=750,∑x2=37514.7Stating your hypotheses clearly, and using a 5% level of significance, test the engineer's suspicion.
The engineer decides that for future machines, they will use a larger sample size of n=26n = 26n=26 and a significance level of 1% with the same hypotheses.
Using statistical tables, show that the critical region for the sample variance S2S^2S2 is S2>1.281S^2 > 1.281S2>1.281 (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 machine's cuts is actually σ=1.30\sigma = 1.30σ=1.30.