Question

2. Suppose we are interested in the relationship between number of hours of exercise per week and systolic blood pressure in males 50 years of age. A random sample of 10 males 50 years of age is selected. In this analysis, the IV is number of hours of exercise per week and the DV is systolic blood pressure. The data are as follows: X = #hrs of exercise WkY=systolic BPXY 120 4 10 3 3 1 2 2 110 120 135 140 115 150 165 160 180 ΣΥ ΣΧΥ - a. Fill in the missing values for the above table, and calculater. Note that SSx = 69.60 and SSy - 5072.50. How can we describe the relationship between number of hours of exercise per week and systolic blood pressure? b. Using the same data and your calculated value for r, test whether there is a significant correlation between systolic BP and number of hours of exercise per week among all males 50 years of age, two-tailed test (= 0.05). c. Using the correlation coefficient you calculated above, determine the values for the slope and Y-intercept, and provide the simple linear regression equation. But first you must calculate the mean values for X and Y. Here is some additional information from the collected data: variance for X is 7.73 and the variance for Y is 563.611.

Answer 1

Answer:

Let X be number of hours of exercise per week and Y be systolic blood pressure in males 50 years of age.

; = 32

$\sum y_{i}=1395$

liyi = 4020

(a)

Here,

SS. 69.60

SSy = 5072.500

$SS_{xy}=\sum x_{i}y_{i}-n\bar{x}\bar{y}$

X 32 10 = 3.2 n

$\bar{Y}=\frac{\sum y_{i}}{n}=\frac{1395}{10}=139.5$

$\therefore SS_{xy}=4020-(10\times 3.2\times 139.5)=-444$

$\therefore r=\frac{SS_{xy}}{\sqrt{SS_{x}.SS_{y}}}=\frac{-444}{\sqrt{69.60\times 5072.500}}=-0.747252$

$\because r=-0.747252$

$\therefore$ there is negative relationship between number of hours of exercise per week and systolic blood pressure.

(b)

We eant to test

$H_{0}:$ There is not significant correlation between systolic BP & number of hours of exercise per week.

ie, H: p=0 Opis population correlation coefficient

VS

$H_{1}:$ There is significant correlation between systolic BP and number of hours of exercise per week

$ie,\ H_{1}:\ \rho\neq 0$

Test statistic is

$t_{cal}=\frac{r\sqrt{n-2}}{\sqrt{1-r^{2}}}\sim t_{n-2},\ \ under\ H_{0}$

Here $r=-0.747252,n=10$

$\therefore t_{cal}=\frac{(-0.747252)\sqrt{8}}{1-(-0.747252)^{2}}=-3.1805$

Decision criteria:

We reject
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at $\alpha\%$ significance level if

$|t_{cal}|\geq t_{\frac{\alpha}{2},n-2}$

Here, $|t_{cal}|\geq t_{\frac{0.05}{2},8}=2.3060$

Here, $|t_{cal}|\geq t_{\frac{\alpha}{2},n-2}$

ie 3.1805 > 2.3060

Therefore we reject at 5% level of significance.

он

Conclusion:

There is significant correlation between systolic 89 and number of hours of exercise per week

(c)

here, $\bar{x}=3.2,\bar{y}=139.5$

we have,

The estimated regression line is given by ,

$\hat{y}=\hat{a}+\hat{b}\bar{x}$

$\hat{b}=\frac{SS_{xy}}{SS_{x}}$

$\therefore \hat{b}=\frac{-44}{69.60}=-6.3793$

$\therefore \hat{a}=139.5-(-6.3793\times 3.2)=159.9138$

$\therefore$ The estimated regression line is

$\hat{y}=159.9138-6.3793x$

(d)

From above, y.intercept $\hat{a}=159.9138$

if the range of data on X includes X=0, then the intercept 'a' is the mean of distribution of the response y when x=0, if the range of x does not include zero, then 'a' has no practical interpretation

(e)

From above (c)

estimate of slope, $\hat{b}=-6.3793$

As the value of number of hours of exercise per week changes by one unit (increases) then it produces -6.3793 unit change in systolic blood pressure in males 50years of age

(f)

Here, we have to estimate the systolic BP of male aged 50 who exercises 3 hours per week. i.e , we have to find $\hat{y}\ at\ x=3hours/week$

$\therefore \hat{y}=159.9138-(6.3793x)$

  $=159.9138-(6.3793\times 3)$

$=140.7759$

$\therefore \hat{y}=140.7759$