The times taken to run 200 metres at the beginning of the year and at the end of the year are recorded for each member of a large athletics club. The time taken, in seconds, at the beginning of the year is denoted by \(x\) and the time taken, in seconds, at the end of the year is denoted by \(y\). For a random sample of 8 members, the results are shown in the following table.
| Member | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) |
| \(x\) | 24.2 | 23.8 | 22.8 | 25.1 | 24.5 | 24.0 | 23.8 | 22.8 |
| \(y\) | 23.9 | 23.6 | 22.8 | 24.5 | 24.2 | 23.5 | 23.6 | 22.7 |
$$\left[ \Sigma x = 191 , \quad \Sigma x ^ { 2 } = 4564.46 , \quad \Sigma y = 188.8 , \quad \Sigma y ^ { 2 } = 4458.4 , \quad \Sigma x y = 4510.99 . \right]$$
- Find, showing all necessary working, the equation of the regression line of \(y\) on \(x\).
The athletics coach believes that, on average, the time taken by an athlete to run 200 metres decreases between the beginning and the end of the year by more than 0.2 seconds. - Stating suitable hypotheses and assuming a normal distribution, test the coach's belief at the \(10 \%\) significance level.