The blood pressures, p mmHg, and the ages, t years, of 7 hospital patients are shown in the table below - Edexcel - A-Level Maths Statistics - Question 6 - 2010 - Paper 1

The product moment correlation coefficient, $r$, is calculated by the formula:

$r = \frac{n \sum tp - (\sum t)(\sum p)}{\sqrt{[n \sum t^{2} - (\sum t)^{2}] \cdot [n \sum p^{2} - (\sum p)^{2}]}}$

With the known values:

• $n = 7$
• $\sum t = 341$
• $\sum p = 833$,
• $\sum t^{2} = 18181$
• $\sum p^{2} = 106397$
• $\sum tp = 42948$,

Plugging in these values gives:

$r = \frac{7(42948) - (341)(833)}{\sqrt{[7(18181) - 341^{2}][7(106397) - 833^{2}]}}$

Calculating this yields:

$r = 0.701373$

The correlation coefficient, $r = 0.701373$, indicates a strong positive linear relationship between age and blood pressure. This suggests that as one variable increases, the other tends to increase as well.

To plot the scatter diagram, plot each patient’s age against their corresponding blood pressure on a graph, marking each data point clearly. The x-axis represents age and the y-axis represents blood pressure. Ensure to label the axes appropriately.

The regression line can be determined using the formula:

$p = a + bt$

Where:

• $b = \frac{S_{tp}}{S_{t}}$
• $a = \bar{p} - b \bar{t}$

Using the calculated values:

• $S_{tp}$ and $S_{t}$ from (a) get:
• After substituting, the regression equation yields:

$p = 45.5 + 1.5t$

To plot the regression line, calculate $p$ for a range of $t$ values and plot the resulting points on the scatter diagram, then draw a line through these points to represent the regression line.

Using the regression equation:

$p = 45.5 + 1.5(40) = 45.5 + 60 = 105.5$

Thus, the estimated blood pressure for a 40-year-old patient is approximately 106 mmHg.