A factory produces packets of biscuits. The total mass of biscuits in a packet has a normal distribution with mean \(\mu\). A random sample of 12 packets is taken and the mass of the contents of each packet, \(x\) g, is recorded. The results are summarised as follows.

\(\sum x=2390 \quad \sum x^2=476117\)

(a) Find a \(99\%\) confidence interval for \(\mu\).

A test of the null hypothesis \(\mu=k\) is carried out on this sample using a \(5\%\) significance level. The test does not support the alternative hypothesis \(\mu\lt k\).

(b) Find the greatest possible value of \(k\).