A precision engineering plant fills high-pressure gas cylinders to a volume, V V\,V mL, which follows a normal distribution N(μ,122)N(\mu, 12^2)N(μ,122). To monitor the consistency of the filling process, a random sample of n n\,n cylinders is selected and tested at the 1% level of significance using the following hypotheses:
H0:μ=1500andH1:μ<1500 H_0 : \mu = 1500 \quad \text{and} \quad H_1 : \mu < 1500 H0:μ=1500andH1:μ<1500Find, in terms of nnn, the critical region of the test.
The probability of a Type II error, when the true mean is μ=1490\mu = 1490μ=1490, is less than 0.05.
Find the minimum value of nnn.
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.