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.
