A daily train journey time is normally distributed with standard deviation 5.8 minutes. The mean of a random sample of 20 journey times is 81.5 minutes.

(a) Calculate a 98% confidence interval for the population mean journey time. [3]

(b) A student says: “The times for 98% of these journeys are likely to be within the confidence interval.” Explain briefly whether this is true. [1]

Two independent 98% confidence intervals are found.

(c) Given that at least one contains the population mean, find the probability that both contain it. [2]