Question

Consider the following data for two variables, x and y.

x   2 3 4 5 7 7 7 8 9

y 4 5 4 6 4 6 9 5 11

a. Does there appear to be a linear relationship between x and y? Explain.(f-test, to do f-test for the overall significance)

b. Develop the estimated regression equation relating x and y.

c. Plot the standardized residuals versus yˆ for the estimated regression equation developed in part (b). Do the model assumptions appear to be satisfied? Explain.

d. Perform a logarithmic transformation on the dependent variable y. Develop an estimated regression equation using the transformed dependent variable. Do the model assumptions appear to be satisfied by using the transformed dependent variable? Does a reciprocal transformation work better in this case? Explain.

 

Answer 1

a. By plotting scatter plot between the two variables x and y we can judge whether the relationship between the two variables is linear or not

In the above plot the relation seems not perfect linear.

b)  The estimated regression equation between the two variables  is

x = 2.15 + 0.604 y

c) The standardized residual plot is given below

it does not seem the assumption of independence of residuals and Homoscedasticity of residuals is not met. As the residuals are wide in the start of x axes and shrinks with the increase in the value at x axes.

F.test for the above model is

Analysis of Variance
Source	DF	SS	MS	F	P.Value
Regression	1	17.521	17.521	4.37	0.075
Residual	7	28.035	4.005
Total	8	45.556

d) The regression equation is
x = - 1.28 + 9.40 ln(y) where ln(y) is log_ey

The model variables seems to be bit more satisfying then with out log transformation.

The F Test results are as below for log transformed model

Analysis of Variance
Source	DF	SS	MS	F	P.Value
Regression	1	17.575	17.575	4.4	0.074
Residual	7	27.98	3.997
Total	8	45.556

Now if we check with the reciprocal transformation following

The regression equation is
x = 10.4 - 24.6 rep(Y) Where rep(y) is reciprocal of variable y i,e (1/y)

It seems to be more satisfying then the precious one

F.test for reciprocal model

Analysis of Variance
Source	DF	SS	MS	F	P.Value
Regression	1	17.046	17.046	4.19	0.08
Residual	7	28.51	4.073
Total	8	45.556