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.
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.