Question

A random sample of 4 workers has been selected. The following table lists their wage per hour (dollars) and their number years of education.

Wage (dollars per hour).           Number of years of education

5                                                          8

            7                                                          10

            8                                                          12

            14                                                        13

a.   Do you expect a positive or a negative relationship between these two variables?

    b.   Find the least squares regression line by choosing appropriate dependent and independent variables (compute beta0, beta1, yhat and the residuals)

     c.   Interpret the meaning of the values of beta 0 and beta 1, calculated in part b.

     d.   Calculate the correlation coefficient (r) and the coefficient of determination (R²) and explain what they mean.

     e.   Predict the wage for a worker with 11 years of education.

Please show work

Answer 1

	Y	X	Y-Mean(Y)	X-Mean(X)	(Y-Mean(Y))(X-Mean(X))	(X-Mean(X))^2
	5	8	-3.5	-2.75	9.625	7.5625
	7	10	-1.5	-0.75	1.125	0.5625
	8	12	-0.5	1.25	-0.625	1.5625
	14	13	5.5	2.25	12.375	5.0625
Mean Y	8.5
Mean X	10.75

Ans A)

We can observe that as Education years increase the wage earned therefore from first glance it is obvious that there is positive correlation between these 2 variables

Ans B)

We take Number of years of education as Independent variable and Wages as Dependent variable

Y=Wages , X= Education Years

We have regression line to be

Y=Beta_0+eta_18*X+u.. where u is an error term

Beta_1=Sum((Y-Mean(Y))(X-Mean(X)))/Sum(X-Mean(X))^2

Beta_1=1.525

Beta_0=Mean(Y)-Beta_1*Mean(X)=8.5-1.525*10.75=-7.8938

Therefore we have regression line as

Y=-7.8938+1.525*X....(Least Square Regression Line)

Ans C)

When an Individuial has no education he would have Wages equals to -7.8938 units

When an individual increases his education years by 1 year then his wages increase by 1.525 units

Ans D)

Correlation coefficient= Covariance (X,Y)/SD(X)*SD(Y)=22.5/sqrt(14.75*45)=0.8733

Covariance (X,Y)=Sum((Y-Mean(Y))(X-Mean(X)))/(n-1)

Variance (X)=Sum(X-Mean(X))^2/(n-1)

Variance (Y)=Sum(Y-Mean(Y))^2/(n-1)

R^2=Square of Correlation coefficient =0.8733^2=0.7627

R^2 means that X can explain Y 76.27% times

When Wage of Worker is 11 years

Y=-7.8938+1.525*(11)=8.8812

Observed Y   Predicted Y Error
5 4.362 0.938
7 7.3562 -0.356
8 10.406     -2.406
14 11.931       2.069