6 A health researcher is investigating the relationship between age and maximum heart rate. A commonly quoted formula states that 'maximum heart rate \(= 220\) - age in years'. The researcher wants to check if this formula is a satisfactory model for people who work in the large hospital where she is employed. The researcher selects a random sample of 20 people who work in her hospital, and measures their maximum heart rates.

1. Explain why the researcher selects a sample, rather than using all of the people who work in the hospital. The ages, \(x\) years, and maximum heart rates, \(y\) beats per minute, of the people in the researcher's sample are summarised as follows.
   \(n = 20 \quad \sum x = 922 \quad \sum y = 3638 \quad \sum x ^ { 2 } = 47250 \quad \sum y ^ { 2 } = 664610 \quad \sum x y = 164998\) These data are illustrated below.
   \includegraphics[max width=\textwidth, alt={}, center]{5be067ff-4668-48d6-8ed2-b8dfa3e678f7-5_758_1246_1027_244}
2. 1. Draw the line which represents the formula 'maximum heart rate \(= 220 -\) age in years' on the copy of the scatter diagram in the Printed Answer Booklet.
   2. Comment on how well this model fits the data.
3. Determine the equation of the regression line of maximum heart rate on age.
4. Use the equation of the regression line to predict the values of the maximum heart rate for each of the following ages.
   • 40 years
   • 5 years
   • Comment on the reliability of your predictions in part (d).