Question

2. The table shows the body mass index (BMI) X, and the systolic blood pressure (SBP), Y
in the mmttg, for each of a random sample of 10 men, aged between 35 years and 40 years,
from a particular population.
X 13 23 29 35 17 34 25 20 31 27
Y 103 115 124 126 108 120 113 117 118 119



a) Calculate the coefficient of correlation of the distribution. Interpret your results.
b) Compute the coefficient of determination and interpret your results.
c) Find the Simple regression line of Y on X.
d) Use your equation to estimate the SBP of a man from this population who is aged
38 years and who has a BMI of 30.

Answer 1

a)

$$\begin{matrix} & X & Y & XY & X^2 & Y^2 \\ & 13 & 103 & 1339 & 169 & 10609 \\ & 23 & 115 & 2645 & 529 & 13225 \\ & 29 & 124 & 3596 & 841 & 15376 \\ & 35 & 126 & 4410 & 1225 & 15876 \\ & 17 & 108 & 1836 & 289 & 11664 \\ & 34 & 120 & 4080 & 1156 & 14400 \\ & 25 & 113 & 2825 & 625 & 12769 \\ & 20 & 117 & 2340 & 400 & 13689 \\ & 31 & 118 & 3658 & 961 & 13924 \\ & 27 & 119 & 3213 & 729 & 14161 \\ sum= & 254 & 1163 & 29942 & 6924 & 135693 \\ \end{matrix}$$

$$S_{XX}=\displaystyle\sum_{i=1}^nX_i^2-\dfrac{1}{n}(\displaystyle\sum_{i=1}^nX_i)^2$$ $$=6924-\dfrac{1}{10}(254)^2=472.4$$

$$S_{YY}=\displaystyle\sum_{i=1}^nY_i^2-\dfrac{1}{n}(\displaystyle\sum_{i=1}^nY_i)^2$$ $$=135693-\dfrac{1}{10}(1163)^2=436.1$$

$$S_{XY}=\displaystyle\sum_{i=1}^nX_iY_i-\dfrac{1}{n}(\displaystyle\sum_{i=1}^nX_i)(\displaystyle\sum_{i=1}^nY_i)$$ $$=29942-\dfrac{1}{10}(254)(1163)=401.8$$

$$r=\dfrac{S_{XY}}{\sqrt{S_{XX}}\sqrt{S_{YY}}}=\dfrac{401.8}{\sqrt{472.4}\sqrt{436.1}}=0.885242$$

$r>0.7$ Strong positive correlation.

b)

$$r^2=\dfrac{S_{XY}}{S_{XX}S_{YY}}=\dfrac{401.8}{472.4(436.1)}=0.783653$$

78.37 percent of the variance in Y is explained by regression line..

c)

$$slope=m=\dfrac{S_{XY}}{S_{XX}}=\dfrac{401.8}{472.4}=0.85055$$

$$\bar{X}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nX_i=\dfrac{1}{10}(254)=25.4$$

$$\bar{Y}=\dfrac{1}{n}\displaystyle\sum_{i=1}^nY_i=\dfrac{1}{10}(1163)=116.3$$

$$b=\bar{Y}-m\bar{X}=116.3-0.85055(25.4)=94.69603$$

The regression equation is:

$$y=94.69603+0.85055x$$

d)

$$y=94.69603+0.85055(30)=120.2$$

A man from this population who is aged 38 years and who has a BMI of 30 has the systolic blood pressure of 120.2 mmttg.