Problem Statement
A high school track coach recorded the weekly training hours ($x$) and the 100-meter sprint times in seconds ($y$) of 8 athletes:
$$\begin{array}{c|cccccccc} x \text{ (hours)} & 4 & 6 & 7 & 9 & 10 & 12 & 14 & 15 \\ \hline y \text{ (seconds)} & 13.2 & 12.8 & 12.5 & 12.1 & 11.9 & 11.4 & 11.2 & 10.9 \end{array}$$
**(a)** [2 marks] Calculate the Pearson product-moment correlation coefficient $r$, and describe the strength and direction of the linear relationship.
**(b)** [2 marks] Write down the equation of the regression line of $y$ on $x$ in the form $y = ax + b$.
**(c)** [2 marks] Use your regression equation to estimate the sprint time for an athlete who trains 8 hours per week, and comment on the reliability of this estimate.
Verified Solution & Marking Scheme
Part (a) [M1]
Enter paired data into GDC LinReg(ax+b)
Part (a) [A1]
$r \approx -0.995$. This indicates a very strong negative linear correlation between training hours and sprint times.
Part (b) [A1]
Gradient $a \approx -0.201$
Part (b) [A1]
$y$-intercept $b \approx 13.94 \implies y = -0.201x + 13.9$ (to 3 s.f.)
Part (c) [M1]
Substitute $x = 8$: $y = -0.2008(8) + 13.94 = 12.33\text{ seconds}$
Part (c) [R1]
Reliable because $x = 8$ lies within the experimental range $[4, 15]$ (interpolation) and $|r| = 0.995$ is exceptionally close to 1.