Answer to Question #265041 in Statistics and Probability for kristine

Question

The following are data for 12 individual‟s daily sodium intake and their

systolic blood pressure readings. 

Person Sodium  BP 

1       6.8   154 

2       7.0   167 '

3       6.9   162 

4       7.2   175  

5       7.3   190 

6       7.0   158 

7       7.0   166 

8       7.5   195 

9       7.3    189 

10      7.1   186 

11     6.5    148 

12     6.4    140 

A researcher investigator is interested in learning how strong

the association is between these variables and how well we can

predict blood pressure from sodium intake. 

a. Calculate the value of r and the regression equation for the data

b. What would be a likely blood pressure for a person with sodium of

6.3? How about sodium of 7.6?

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c} & X & Y & XY & X^2 & Y^2\\ \hline & 6.8 & 154 & 1047.2 & 46.24 & 23716\\ & 7.0 & 167 & 1169 & 49 & 27889\\ & 6.9 & 162 & 1117.8 & 47.61 & 26244\\ & 7.2 & 175 & 1260 & 51.84 & 30625\\ & 7.3 & 190 & 1387 & 53.29 & 36100\\ & 7.0 & 158 & 1106 & 49 & 24964\\ & 7.0 & 166 & 1162 & 49 & 27556\\ & 7.5 & 195 & 1462.5 & 56.25 & 38025\\ & 7.3 & 189 & 1379.7 & 53.29 & 35721\\ & 7.1 & 186 & 1320.6 & 50.41 & 34596\\ & 6.5 & 148 & 962 & 42.25 & 21904\\ & 6.4 & 140 & 896 & 40.96 & 19600\\ Sum= & 84 & 2030 & 14269.8 & 589.14 & 346940\\ \end{array}$$ $$\bar{X}=\dfrac{1}{n}\sum_iX_i=\dfrac{84}{12}=7$$

$$\bar{Y}=\dfrac{1}{n}\sum_iY_i=\dfrac{2030}{12}=169.16666666667$$

$$SS_{XX}=\sum_i(X_i-\bar{X})^2=1.1400000000001$$

$$SS_{YY}=\sum_i(Y_i-\bar{Y})^2=3531.6666666667$$

$$SS_{XY}=\sum_i(X_i-\bar{X})(Y_i-\bar{Y})=59.800000000001$$

$$m=slope=\dfrac{SS_{XY}}{SS_{XX}}$$

$$=\dfrac{59.800000000001}{1.1400000000001}=52.45614$$

$$n=\bar{Y}-m\bar{X}$$

$$=169.16666666667-52.45614(7)$$

$$=−198.02632$$

Therefore, we find that the regression equation is:

$$Y=−198.02632+52.45614X$$

Correlation coefficient:

$$r=\dfrac{59.800000000001}{\sqrt{1.1400000000001}\sqrt{3531.6666666667}}$$

$$=0.94245127$$

a. $r=0.94245127$

We have strong positive correlation.

The regression equation is:

$$Y=−198.02632+52.45614X$$

b.

$$Y=−198.02632+52.45614(6.3)=$$

$$=132.447362$$

Systolic blood pressure readings are $132.447362.$

c.

$$Y=−198.02632+52.45614(7.6)=$$

$$=200.640344$$

Systolic blood pressure readings are $200.640344.$