Answer to Question #271655 in Statistics and Probability for Matilda

Question

A researcher is interested in learning how strong is the association between the sodium intake and systolic blood pressure of 12 individuals and how well can he predict blood pressure from sodium intake. He obtained the following data from these 12 individuals:

Person

1

2

3

4

5

6

7

8

9

10

Sodium

6.8

7.0

6.9

7.2

7.3

7.0

7.0

7.5

7.3

7.1

6.5

6.4

BP

154

167

162

173

190

158

166

195

189

186

148

140

a.) Obtain the equation of the regression line and estimate the blood pressure of a person if his sodium intake is 6.6.

b.) Test H0: b= 0 at the 0.01 level of significance.

c.) Use the analysis of varianve technique to test if there is a significant linear relationship between sodium intake and blood pressure.

d.) Compare the results of (b) and (c)

Answer 1

a)

equation of the regression line:

$y=a+bx$

x is sodium intake

y is blood pressure

$a=\frac{\sum y\sum x^2-\sum x\sum xy}{n\sum x^2-(\sum x)^2}=-195.7$

$b=\frac{n\sum xy-\sum x\sum y}{n\sum x^2-(\sum x)^2}=52.1$

$y=52.1x-195.7$

$y(6.6)=52.1\cdot6.6-195.7=148.16$

b)

$H_0:b=0$ , The slope of the regression line is equal to zero.

$H_0:b\neq0$ , The slope of the regression line is not equal to zero.

$t=b/SE$

$SE=\sqrt{\frac{\sum(y_i-\tilde{y}_i)^2}{(n-2)\sum(x_i-\overline{x})^2}}$

where y_i is the value of the dependent variable for observation i, ŷ_i is estimated value of the dependent variable for observation i, x_i is the observed value of the independent variable for observation i, x is the mean of the independent variable, and n is the number of observations.

ŷ_i : $y(6.8)=159,y(7.0)=169,y(6.9)=164,y(7.2)=179,y(7.3)=185$

$y(7.5)=195,y(7.1)=174,y(6.5)=143,y(6.4)=138$

$\sum(y_i-\tilde{y}_i)^2=25+3\cdot4+4+36+2\cdot25+64+25+4=230$

$\sum(x_i-\overline{x})^2=1.14$

$SE=\sqrt{\frac{230}{10\cdot1.14}}=4.5$

$t=52.1/4.5=11.578$

$df=n-2=10$

critical value:

$t_{crit}=3.169$

Since $t_{crit}<|t|$ , we reject null hypothesis. The slope of the regression line is not equal to zero.

c)

$H_0:b=0$ ​, The slope of the regression line is equal to zero (there is no linear relationship between sodium intake and blood pressure.)

$H_0:b\neq0$ ​, The slope of the regression line is not equal to zero (there is linear relationship between sodium intake and blood pressure)

ANOVA Formulas:

degrees of freedom:

between groups: $df_1=k-1$

where k is number of groups

within groups: $df_1=n-k$

where n is total number of objects

total: $df=n-1$

sum of squares between groups:

$SS_B=\sum n_i (\overline{x}_i-\overline{x})^2$

where n_i is number of subjects in i-th group

sum of squares within groups:

$SS_W=\sum (n_i -1) s_i^2$

where s_i is standard deviation of i-th group

mean square between groups:

$MS_B=SS_B/(k-1)$

mean square within groups:

$MS_W=SS_W/(n-k)$

F-statistic:

$F=MS_B/MS_W$

Since p-value = 0, we reject null hypothesis. There is linear relationship between sodium intake and blood pressure.

d)

Tests from b) and c) have same result: there is linear relationship between sodium intake and blood pressure.