A large number of people attended a course to improve the speed of their logical thinking. The times taken to complete a particular type of logic puzzle at the beginning of the course and at the end of the course are recorded for each person. The time taken, in minutes, at the beginning of the course is denoted by \(x\) and the time taken, in minutes, at the end of the course is denoted by \(y\). For a random sample of 9 people, the results are summarised as follows.
$$\Sigma x = 45.3 \quad \Sigma x ^ { 2 } = 245.59 \quad \Sigma y = 40.5 \quad \Sigma y ^ { 2 } = 195.11 \quad \Sigma x y = 218.72$$
Ken attended the course, but his time to complete the puzzle at the beginning of the course was not recorded. His time to complete the puzzle at the end of the course was 4.2 minutes.
- By finding, showing all necessary working, the equation of a suitable regression line, find an estimate for the time that Ken would have taken to complete the puzzle at the beginning of the course.
The values of \(x - y\) for the sample of 9 people are as follows.
$$\begin{array} { l l l l l l l l l }
0.2 & 0.8 & 0.5 & 1.0 & 0.2 & 0.6 & 0.2 & 0.5 & 0.8
\end{array}$$
The organiser of the course believes that, on average, the time taken to complete the puzzle decreases between the beginning and the end of the course by more than 0.3 minutes. - Stating suitable hypotheses and assuming a normal distribution, test the organiser's belief at the \(2 \frac { 1 } { 2 } \%\) significance level.