A scientific robotic probe collects soil samples from a lunar crater. The mass M M\,M grams of each sample is modeled as a normal distribution M∼N(μ,σ2)M \sim \mathrm{N}(\mu, \sigma^2)M∼N(μ,σ2). Based on an initial batch of data, a 90% confidence interval for μ \mu\,μ is calculated as (122.35,127.65)(122.35, 127.65)(122.35,127.65).
Determine a 99% confidence interval for μ\muμ.
The total yield of a specific mineral extract from a sample is given by the linear transformation Y=20MY = 20MY=20M. Using your answer from part (a), find a 99% confidence interval for the mean mineral yield μY \mu_Y\,μY of these samples.
If six independent random samples are taken and a 99% confidence interval for μ \mu\,μ is constructed for each, find the probability that at least five of these intervals will contain the true value of μ\muμ.
