Answer to Question #120019 in Statistics and Probability for Arafeen

Question

The blood pressures, Y (mmHg), and
Ages, X (years) of 10 hospital patients were sampled from a senator’s State and summarized below.
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
BP(Y) in (mmHg) 80 85 125 90 100 135 80 70 100 140
Use the table to answer the questions that follow;
i) Calculate the product moment correlation coefficient for the data and interpret your result.
(ii) If the Senator decides to purchase and distribute Norvasc (a medicine that reduces blood pressure),based on your results in (i), which age group (youth or old adults) should be given priority? Briefly explain your answer.
iii) Give a reason to support fitting a regression model of the form

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c} & X & Y & XY & X^2 & Y^2 \\ \hline & 20 & 80 & 1600 & 400 & 6400 \\ & 25 & 85 & 2125 & 625 & 7225 \\ & 50 & 125 & 6250 & 2500 & 15625 \\ & 30 & 90 & 2700 & 900 & 8100 \\ & 45 & 100 & 4500 & 2025 & 10000 \\ & 60 & 135 & 8100 & 3600 & 18225 \\ & 10 & 80 & 800 & 100 & 6400 \\ & 15 & 70 & 1050 & 225 & 4900 \\ & 35 & 100 & 3500 & 1225 & 10000 \\ & 70 & 140 & 9800 & 4900 & 19600 \\ Sum=& 360 & 1005 & 40425 & 16500 & 106475 \end{array}$$

$$\bar{X}={1\over n}\displaystyle\sum_{i=1}^nX_i={360\over 10}=36$$

$$\bar{Y}={1\over n}\displaystyle\sum_{i=1}^nY_i={1005\over 10}=100.5$$

$$S_{XX}=\displaystyle\sum_{i=1}^nX_i^2-{1\over n}(\displaystyle\sum_{i=1}^nX_i)^2=$$ $$=16500-{1\over 10}(360)^2=3540$$

$$S_{YY}=\displaystyle\sum_{i=1}^nY_i^2-{1\over n}(\displaystyle\sum_{i=1}^nY_i)^2=$$ $$=106475-{1\over 10}(1005)^2=5472.5$$

$$S_{XY}=\displaystyle\sum_{i=1}^nX_iY_i-{1\over n}(\displaystyle\sum_{i=1}^nX_i)(\displaystyle\sum_{i=1}^nY_i)=$$ $$=40425-{1\over 10}(360)(1005)=4245$$

Correlation coefficient

$$r={S_{XY}\over \sqrt{S_{XX}}\sqrt{S_{YY}}}$$

$$r={4245\over \sqrt{3540}\sqrt{5472.5}}\approx0.9645$$

This is a strong positive correlation, which means that high X variable scores go with high Y variable scores (and vice versa).

$$B={S_{XY}\over S_{XX}}={4245\over 3540}\approx1.1992$$

$$A=\bar{Y}-B\bar{X}=100.5-{4245\over 3540}(36)\approx57.3305$$

$$y=57.3305+1.1992x$$

(ii) Old adults should be given priority. Older adults suffer from high blood pressure.

(iii) R-squared is a goodness-of-fit measure for linear regression models. R-squared is a statistical measure of how close the data are to the fitted regression line. It is also known as the coefficient of determination

$$r^2=0.9302$$

High coefficient of determination gives a reason to support fitting a regression model of the form

$$y=57.3305+1.1992x$$