Question

In: Statistics and Probability

3. Consider the following data for two variables, x and y.x   2 3 4 5 7...

3. 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.

 

Solutions

Expert Solution

data

x y
2 4
3 5
4 4
5 6
7 4
7 6
7 9
8 5
9 11

a)

Excel regression result

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.620164219
R Square 0.384603659
Adjusted R Square 0.296689895
Standard Error 2.054229935
Observations 9
ANOVA
df SS MS F Significance F
Regression 1 18.46097561 18.46097561 4.374783255 0.074793318
Residual 7 29.53902439 4.219860627
Total 8 48
Coefficients Standard Error t Stat P-value Lower 95%
Intercept 2.32195122 1.88710113 1.230432849 0.258274863 -2.140333878
x 0.636585366 0.304353556 2.091598254 0.074793318 -0.083096433

p-value = 0.0748 > 0.05 (alpha)

hence we fail to reject the null hypothesis

we conclude that there does not appear to be a linear relationship

b)

y^ = 2.32195 + 0.636585 x

c)

Predicted y Standard Residuals
3.595121951 0.21070321
4.231707317 0.39982838
4.868292683 -0.451869534
5.504878049 0.257667178
6.77804878 -1.445728649
6.77804878 -0.404905565
6.77804878 1.15632906
7.414634146 -1.256603479
8.051219512 1.5345794

residual increasing as x increases

model assumption is not satisfied

d)

ln y

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.621127
R Square 0.385799
Adjusted R Square 0.298056
Standard Error 0.304318
Observations 9
ANOVA
df SS MS F Significance F
Regression 1 0.407196 0.407196 4.396925 0.074212
Residual 7 0.648266 0.092609
Total 8 1.055462
Coefficients Standard Error t Stat P-value Lower 95%
Intercept 1.182238 0.279559 4.228938 0.003893 0.521186
x 0.094543 0.045088 2.096885 0.074212 -0.01207

ln y^ = 1.182238 + 0.094543 x

NO, model assumption is still not satisfied

1/Y = a + bx

now model assumptions are satisfied


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