QUESTION 1
The Senator of Azenator State, is worried about the rising numbers in high blood pressure related deaths in his
Jurisdiction and wants an end to this canker. A reputable medical research officer has claimed that, the
situation is probably as a result of the ageing population of his State. The blood pressures, Y (mmHg), and
Ages, X (years) of 10 hospital patients were sampled from Azenator State and summarized below.
Patient
A
B
C
DE
F
G
HI
J
Age (X) in Years
20 25 50
3045
60
10
15 35
70
BP(Y) in (mmHg) 80 85 125
90 100 135 80 70 100 140
NB: Approximate to 2 decimal places
Use the table to answer the questions that follow,
Calculate the product moment correlation coefficient for the data and interpret your result.
(3 Marks)

The Pearson product-moment correlation coefficient (PMCC) is a numerical value between -1 and 1 that shows the strength of the linear relationship between two variables.

Given the result of 10 hospital patients which were sampled below:

[tex]\left|\begin{array}{c|ccccccccccccc}$Patient&A&B&C&D&E&F&G&H&I&J\\$Age (X) in Years&20&25&50&30&45&60&10&15&35&70\\$B.P.(Y) in mmHg&80&85&125&90&100&135&80&70&100&140\end{array}\right|[/tex]

To determine the product moment correlation coefficient for the data, we follow the steps below:

Step 1: Find the following as it was done in the table below.

[tex]\sum{X}~,~ \sum{Y} ~,~ \sum{X \cdot Y} ~,~ \sum{X^2}~,~and \sum{Y^2}[/tex]

Step 2: Take the sum of every column to obtain:

[tex]\sum{X}=360 ~,~ \sum{Y}=1005 ~,~ \sum{X \cdot Y}=40425 ~,\\~ \sum{X^2}=16500 ~,~ \sum{Y^2}=106475[/tex]

Step 3: Use the following formula to find the PMCC.

[tex]r=\dfrac{n\cdot\sum{XY} - \sum{X}\cdot\sum{Y}}{\sqrt{\left[n \sum{X^2}-\left(\sum{X}\right)^2\right] \cdot \left[n \sum{Y^2}-\left(\sum{Y}\right)^2\right]}}[/tex]

[tex]=\dfrac{10\cdot 40425 - 360\cdot1005}{\sqrt{\left[10*16500}-\left(360\right)^2\right] \cdot \left[10*106475-\left(1005\right)^2\right]}}[/tex]

[tex]=\dfrac{42450}{\sqrt{1937265000}}\\=0.9645\\r\approx 0.96$ (correct to 2 decimal places)[/tex]

Interpretation

When r is closer to 1, there is a strong positive relationship between the variables.

A value of 0 indicates that there is no relationship between the variables.

Values close to -1 signal a strong negative relationship between the two variables.

Since r=0.96 which is closer to 1, we conclude that there is a strong relationship between the age and the high blood pressure related deaths in the state.