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.

$$\bar { x } = 25.6 \quad \sum ( x - \bar { x } ) ^ { 2 } = 78.5$$

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

\hfill \mbox{\textit{CAIE Further Paper 4 2021 Q1}}