A sampling process produced the following sets of (x,y) combinations: (10, 97);

Question

A sampling process produced the following sets of (x,y) combinations: (10, 97); (16, 156); (7, 98); (6, 89); (14, 161); (19, 208); (19, 236), (14, 118); (9, 94); (12, 125) What is the slope of the regression equation? (Round to two decimal places) What is the intercept of the regression equation? If the independent variable is 15, what is the expected value of the dependent variable? What proportion of the change in the dependent variable is explained by the change in the independent variable? Is this a valid linear regression model?

Explanation / Answer

R-code :

x=c(10,16,7,6, 14,19, 19, 14, 9,12)
y=c(97,156, 98,89, 161, 208, 236, 118, 94, 125)
m=lm(y~x)
summary(m)

Output:

Call:
lm(formula = y ~ x)

Residuals:
Min 1Q Median 3Q Max
-34.468 -12.905 -1.255 14.786 32.576

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 9.790 20.292 0.482 0.642403
x 10.191 1.521 6.701 0.000153 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 21.1 on 8 degrees of freedom
Multiple R-squared: 0.8488, Adjusted R-squared: 0.8299
F-statistic: 44.9 on 1 and 8 DF, p-value: 0.0001526

a) Here ,the slope of the regression equation is 10.19

b) Here, the intercept of the regression equation is  9.79

c) Here , fitted equation is,

y= 9.79  +10.191*x

Here, independent variable is 15 i.e. x=15 then y=9.79  +10.191*15=162.655

Thus, the expected value of the dependent variable is 162.655 .

d) Here, R-squared: 0.8488 which explains proportion of the change in the dependent variable is explained by the change in the independent variable.

e) Here, p-value: 0.0001526<0.05 means x is significantly contributes to explain variablility in y linearly i.e. this a valid linear regression model.

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