SCENARIO 18-4
You decide to predict gasoline prices in different cities and towns in the United States for your term project.Your dependent variable is price of gasoline per gallon and your explanatory variables are per capita income,the number of firms that manufacture automobile parts in and around the city,the number of new business starts in the last year,population density of the city,percentage of local taxes on gasoline,and the number of people using public transportation.You collected data of 32 cities and obtained a regression sum of squares SSR= 122.8821.Your computed value of standard error of the estimate is 1.9549.
-Referring to Scenario 18-4,what is the value of the coefficient of multiple determination?
A)0.2225
B)0.4576
C)0.5626
D)0.6472

Question

Quizplus · Accepted Answer

The coefficient of multiple determination (R-squared) represents the proportion of the total variation in the dependent variable that is explained by the regression equation. It ranges from 0 to 1, with higher values indicating a better fit of the model to the data. To obtain it, we need to divide the regression sum of squares by the total sum of squares (SST).

The formula for R-squared is:

R-squared = SSR/SST

We are given that SSR = 122.8821, but we don't have SST explicitly. However, we can use the standard error of the estimate (SEE) to calculate it indirectly, since SEE is the square root of the mean squared error (MSE), and MSE is the ratio of the sum of squares of residuals (SSE) to the degrees of freedom of error (DFE), which is n-k-1 (where n is the sample size and k is the number of independent variables).

The formula for MSE is:

MSE = SSE/DFE

And the formula for SST is:

SST = SSR + SSE

Substituting the above formulas, we get:

R-squared = SSR/SST = SSR/(SSR+SSE) = 122.8821/(122.8821+31*1.9549^2) = 0.5626

Therefore, the value of the coefficient of multiple determination is 0.5626, or approximately 0.563.

As for the best choice, it depends on what the researcher wants to do with the data and what criteria they use to evaluate the model. Some possible choices are:

- If the goal is to explain or predict the variation in gasoline prices based on the given independent variables, then the model with an R-squared of 0.563 is doing a moderate job, but there is still a lot of unexplained variability. Other variables that were not included in the model may be important, or there may be nonlinear or interaction effects that were not accounted for. The researcher may want to consider adding more variables or transforming the existing ones to improve the fit and parsimony of the model.
- If the goal is to compare the relative importance of the independent variables in predicting gasoline prices, then the researcher could look at the magnitude and sign of the regression coefficients and their standard errors. For example, they may find that per capita income and percentage of local taxes have a positive effect on gasoline prices, while public transportation has a negative effect, but the precision and significance of these estimates may vary across cities. The researcher may want to calculate confidence intervals and conduct hypothesis tests to see if the coefficients are significantly different from zero or each other, and use this information to make policy recommendations or business decisions. 
- If the goal is to communicate the results of the analysis to a general audience, then the researcher may want to use visual aids such as scatterplots, regression lines, and residual plots to show the patterns and deviations in the data, as well as the limitations and assumptions of the model. They may also want to discuss the policy implications and trade-offs of the findings, and acknowledge the uncertainties and sources of error in the analysis.

SCENARIO 18-4 You Decide to Predict Gasoline Prices in Different Cities and Cities