The diameters, \(x\) millimetres, of a random sample of 200 discs made by a certain machine were recorded. The results are summarised below. \[n=200,\qquad\sum x=2520,\qquad\sum x^2=31852.\]

(a) Calculate a 95% confidence interval for the population mean diameter. [6]

Jean chose 40 random samples and used each sample to calculate a 95% confidence interval for the population mean diameter.

(b) How many of these 40 confidence intervals would be expected to include the true value of the population mean diameter? [1]