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.
355 exam-style questions on OCR (MEI) A Level Maths 2.5 Statistical Hypothesis Testing, covering 2.5.1 Process and language of hypothesis testing, 2.5.2 When to apply 1-tail and 2-tail tests, 2.5.3 Significance level and incorrect rejection, 2.5.4 Null and alternative hypotheses (binomial), 2.5.5 Conduct a binomial hypothesis test, 2.5.6 Critical and acceptance regions (binomial), 2.5.7 Distribution of the sample mean (A-level only), 2.5.8 Hypothesis test for a single mean (A-level only), 2.5.9 Critical and acceptance regions (mean) (A-level only), 2.5.10 Correlation as closeness to a straight line (A-level only), 2.5.11 Inference using a correlation coefficient (A-level only), and 2.5 Statistical Hypothesis Testing. Each one has a worked solution and a mark scheme showing where the marks go.