The concentration of a particular catalyst in a chemical batch is modelled by a normal distribution with an unknown mean μ \mu\,μ mg/L and a known standard deviation σ \sigma\,σ mg/L.
A random sample of 80 batches was analysed, resulting in a 98% confidence interval for μ \mu\,μ of (12.42,13.58)(12.42, 13.58)(12.42,13.58).
Using this confidence interval, conduct a hypothesis test to determine whether μ=12.5\mu = 12.5μ=12.5. State your null and alternative hypotheses, the significance level used, and your conclusion.
A second random sample of 150 batches is taken, and the mean concentration is found to be 13.2 mg/L.
Calculate a 90% confidence interval for μ \mu\,μ based on this second sample. Show your intermediate calculations, including the value of σ \sigma\,σ derived from the first sample.
Eight independent random samples, each consisting of 150 batches, are used to construct eight separate 90% confidence intervals for μ\muμ.
Calculate the probability that at least 7 of these 8 intervals will contain the true population mean μ\muμ.
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.