Question

A survey is conducted on 700 Californians older than 30 years of age. The study wants to obtain inference on the relationship between years of education and yearly income in dollars. The response variable is income and the explanatory variable is years of education.
A simple linear regression model is fit, and the output from R is below.

lm(formula = Income ~ Education, data = CA)

Coefficients:
Estimate Std. Error t value Pr(>|t|)

(Intercept) 25200.20 1488.94 16.93 3.08e-10 ***

Education 2905.35 112.61 25.80 1.49e-12 ***

a)Write out the estimated linear equation. What is the estimated expected income of a Californian that has 12 years of education (high school level)?

b)Does the intercept have a useful interpretation in this study? Why or why not.

c)Interpret the slope estimate in context of the model. Now say you have two people, where one has 4 years more education than the other. What is the estimated difference in expected income?

d)The p-value to test the null hypothesis that the slope on Education is 0 (H0 : β1 = 0 vs Ha : β1 ̸= 0), is approximately 0. What can you say about Education being a significant explanatory variable or covariate when explaining Income?

Answer 1

Result:

A survey is conducted on 700 Californians older than 30 years of age. The study wants to obtain inference on the relationship between years of education and yearly income in dollars. The response variable is income and the explanatory variable is years of education.
A simple linear regression model is fit, and the output from R is below.

lm(formula = Income ~ Education, data = CA)

Coefficients:
                 Estimate   Std. Error t value Pr(>|t|)

(Intercept) 25200.20 1488.94   16.93    3.08e-10 ***

Education 2905.35 112.61    25.80      1.49e-12 ***

a)Write out the estimated linear equation. What is the estimated expected income of a Californian that has 12 years of education (high school level)?

estimated linear equation: Income = 25200.20+2905.35* education

when education is 12 years,

predicted Income = 25200.20+2905.35* 12

=60064.40

b)Does the intercept have a useful interpretation in this study? Why or why not.

This intercept have a useful interpretation in this study. When a person have no education ( ie. education is 0 years) the expected income is 25200.20.

c)Interpret the slope estimate in context of the model. Now say you have two people, where one has 4 years more education than the other. What is the estimated difference in expected income?

Slope estimate is 2905.35. when education increases by 1 year the income increases by 2905.35.

when one has 4 years more education than the other, the estimated difference in expected income is 4*2905.35 =11621.40.

d)The p-value to test the null hypothesis that the slope on Education is 0 (H0 : β1 = 0 vs Ha : β1 ̸= 0), is approximately 0. What can you say about Education being a significant explanatory variable or covariate when explaining Income?

Since the p value is approximately 0 which is less than the significance level of 0.05, the coefficient if significant. This shows that Education being a significant explanatory variable or covariate when explaining Income.