\(\mathrm { VO } _ { 2 \max }\) 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]
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 .
Comment on the reliability of your estimate.
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.
State what the point you found in part (iii) represents.
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]
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.
$$\sum t = 80.37 \quad \sum v = 970.86 \quad \sum t ^ { 2 } = 324.71 \quad \sum v ^ { 2 } = 47829.24 \quad \sum t v = 3886.53$$
Find the value of Pearson's product moment correlation coefficient.
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.