Question

Sales of Body           Price of Lens             Gender           Sales of Lens

                        155                 $700                             1                             122

                        101                     650                             1                             120

                        157                     725                            0                             135

                        180                     575                             1                             95

                        150                     600                             0                             100

                        201                     750                             0                             174

                        99                     560                             1                             118

                        137                    500                             0                             130

                        155                             675                             1                                  128

                        165                             550                             1                                  166

                        152                            725                             0                                  131

                        127                             750                             1                                  102

                        217                             565                             0                                  165

                        186                             670                             0                                  154

                        176                             600                             1                                  97

                        123                             585                             0                                  129

                        109                             645                             0                                  98

                        90                             575                             1                                  105

                        176                             660                             0                                  120

                        129                             650                             1                                  105

f)          Find the sample coefficient of determination for the data. Explain its meaning in the context of the problem.

g)        Find the adjusted sample coefficient of determination for the data. Explain its meaning in the context of the problem. Why does it differ, if it does, from the coefficient of determination that you found in the previous part of the problem?

Answer 1

We first run a regression analysis on the given data in minitab-

Steps for regression-

1. Enter the given values in different columns.

2. Go to “Stat” then “Regression” then “Regression” then “Fit regression model”.

3. Enter Sales of body in “response” and price of leens,gender,sales of lens in “continuous predictor”.

4. Click OK.

After running the above steps we get the following output-

Model Summary

S R-sq R-sq(adj) R-sq(pred)
31.0886 33.07% 20.52% 2.52%

f) The sample coefficient of determination for the data is 0.3307.

We know that coefficient of determination is a measure of goodness of fit for regression model. This statistic also interprets the percentage of variation in the dependent variable which is explained by the independent variable. According to our value we can say that it is not a good fit.

g) The adjusted sample coefficient of determination for the data is 0.2052.

The adjusted coefficient of determination value increases or decreases compared to the coefficient of determination if a new term improves or doesnot improve the fit of the model respectively.

Here, our value is less trhan the sample coefficient of determination so we can say that there is no improvement in the model.

The adjusted coefficient of determination adjusts for the number of terms in the regression model. Here we see that both the values are different from each other because the adjusted sample coefficient of determination adjusts for the number of terms in the model which the normal sample coefficient of determination doesn't.