From the diagram, estimate the correlation coefficient - Leaving Cert Mathematics - Question 7 - 2010

The outlier can be identified around the age of 47.

• Age = 47 years
• Max. heart rate = 137 bpm

Using the line of best fit on the diagram, the estimated maximum heart rate for a 44-year-old person is approximately 176 bpm.

To calculate the slope:

Using points (10, 200) and (90, 144), we find:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{144 - 200}{90 - 10} = \frac{-56}{80} = -0.7$ Thus, the slope of the line of best fit is -0.7.

From the slope-intercept form derived:

Starting with: $y - y_1 = m(x - x_1)$ This can be rearranged to: $y = -0.7x + 207$ Thus, the equation in the required form is:

$MHR = 207 - 0.7 \times (age)$

For younger adults, the old rule tends to give a greater MHR estimate. For example:

• For a 20-year-old:
  • Old Rule: MHR = 220 - 20 = 200 bpm
  • New Rule: MHR = 207 - 0.7 * 20 = 193 bpm

For older adults, generally the new rule gives a greater MHR value than the old rule. For example:

• For a 70-year-old:
  • Old Rule: MHR = 220 - 70 = 150 bpm
  • New Rule: MHR = 207 - 0.7 * 70 = 158 bpm

For a 65-year-old:

• Using the old rule:

  • MHR = 220 - 65 = 155 bpm
  • 75% of old MHR = 0.75 * 155 = 116 bpm

• Using the new rule:

  • MHR = 207 - 0.7 * 65 = 169 bpm
  • 75% of new MHR = 0.75 * 169 = 126.75 bpm

Thus, he should exercise at a slightly higher intensity than he has been doing, aiming for about 127 bpm.