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.
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.