The times taken for students at a college to run 200 m have a normal distribution with mean \(\mu \mathrm{s}\). The times, \(x \mathrm{~s}\), are recorded for a random sample of 10 students from the college. The results are summarised as follows, where \(\bar{x}\) is the sample mean.
\(\bar{x}=25.6 \quad \sum(x-\bar{x})^{2}=78.5\)
(a) Find a 90\% confidence interval for \(\mu\).

A test of the null hypothesis \(\mu=k\) is carried out on this sample, using a \(10 \%\) significance level. The test does not support the alternative hypothesis \(\mu\lt k\).
(b) Find the greatest possible value of \(k\).