1 The times taken for students at a college to run 200 m have a normal distribution with mean \(\mu \mathrm { s }\). The times, \(x\) 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.
$$\bar { x } = 25.6 \quad \sum ( x - \bar { x } ) ^ { 2 } = 78.5$$
- 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 \(\mu < k\). - Find the greatest possible value of \(k\).