8
\begin{enumerate}[label=(\alph*)]
\item $\mathrm { VO } _ { 2 \max }$ is a measure of athletic fitness. Since $\mathrm { VO } _ { 2 \max }$ is fairly time-consuming and expensive to measure, an exercise scientist wants to predict $\mathrm { VO } _ { 2 _ { \text {max } } }$ from data such as times for running different distances. The scientist uses these data for a random sample of 15 athletes to predict their $\mathrm { V } \mathrm { O } _ { 2 \text { max } }$ values, denoted by $y$, in suitable units. She also obtains accurate measurements of the $\mathrm { V } \mathrm { O } _ { 2 \text { max } }$ values, denoted by $x$, in the same units.

The scatter diagram in Fig. 8.1 shows the values of $x$ and $y$ obtained, together with the equation of the regression line of $y$ on $x$ and the value of $r ^ { 2 }$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{ce557137-f9eb-4c09-a7e3-e4ec626109dc-08_750_1324_660_317}
\captionsetup{labelformat=empty}
\caption{Fig. 8.1}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item Use the regression line to estimate the predicted $\mathrm { VO } _ { 2 \text { max } }$ of an athlete whose accurately measured $\mathrm { VO } _ { 2 \text { max } }$ is 50 .
\item Comment on the reliability of your estimate.
\item The equation of the regression line of $x$ on $y$ is $x = 0.7565 y + 10.493$.

Find the coordinates of the point at which the two regression lines meet.
\item State what the point you found in part (iii) represents.
\end{enumerate}\item It is known that there is negative correlation between $\mathrm { VO } _ { 2 \text { max } }$ and marathon times in very good runners (those whose best marathon times are under 3 hours). The exercise scientist wishes to know whether the same applies to runners who take longer to run a marathon. She selects a random sample of 20 runners whose best marathon times are between $3 \frac { 1 } { 2 }$ hours and $4 \frac { 1 } { 2 }$ hours and accurately measures their $\mathrm { VO } _ { 2 \text { max } }$.

Fig. 8.2 is a scatter diagram of accurately measured $\mathrm { VO } _ { \text {2max } }$, $v$ units, against best marathon time, $t$ hours, for these runners.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{ce557137-f9eb-4c09-a7e3-e4ec626109dc-09_671_1064_648_319}
\captionsetup{labelformat=empty}
\caption{Fig. 8.2}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item Explain why the exercise scientist comes to the conclusion that a test based on Pearson's product moment correlation coefficient may be valid.

Summary statistics for the 20 runners are as follows.

$$\sum t = 80.37 \quad \sum v = 970.86 \quad \sum t ^ { 2 } = 324.71 \quad \sum v ^ { 2 } = 47829.24 \quad \sum t v = 3886.53$$
\item Find the value of Pearson's product moment correlation coefficient.
\item Carry out a test at the $5 \%$ significance level to investigate whether there is negative correlation between accurately measured $\mathrm { VO } _ { 2 _ { \text {max } } }$ and best marathon time for runners whose best marathon times are between $3 \frac { 1 } { 2 }$ hours and $4 \frac { 1 } { 2 }$ hours.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Statistics Major 2021 Q8 [16]}}