Simple Linear Regression
1. A car dealer wants to find the relationship between the odometer reading and the selling price of used cars. A random sample of 100 cars is selected and the data are recorded in the Excel file XM18-03. Estimate a linear relationship between price and odometer reading. Calculate the standard error of estimate and describe what it tells you about the model fit. Find the coefficient of determination. What does this statistic tell you about the model?
- (Week 8) Test to determine whether there is enough evidence to infer that a linear relationship exists between the price and the odometer reading at the 5% significance level.
- Predict the selling price of a three-year-old Ford Laser with 40 000 km on the odometer.
- Provide an interval estimate for the bidding price on a Ford Laser with 40 000 km on the odometer.
- The car dealer wants to bid on a lot of 250 Ford Lasers, where each car has been driven for about 40 000 km.
- Test the coefficient of correlation to determine if a linear relationship exists between the price and odometer reading (use α = 0.05).
2. One general belief held by observers of the business world is that taller men earn more money than shorter men. In a University of Pittsburgh study, 250 MBA graduates, all about 30 years old, were surveyed and asked to report their annual incomes (in $US to the nearest $1,000) and their heights (in inches). This information is recorded in the Excel file, Tutorial 08-Q2.xls with "Height" in the first column and "Income" in the second column.
a. Plot a scatter diagram with height on the horizontal axis and income on the vertical axis. Describe the relationship between the two variables. Is it reasonable to use a linear model?
b. Using the Regression command in Excel, perform a regression analysis to estimate the sample regression line. Write down your regression line.
c. Interpret the coefficients.
d. Comment on the goodness of fit of the model.
e. Check whether the OLS assumptions for error variable are satisfied in your model.
f. At 5% significance level, do these data provide sufficient statistical evidence to infer that higher people earn more?
3. The marketing manager for a chain of hardware stores needs more information about the effectiveness of the three types of advertising that the chain uses. These are (i) localised direct mailing (in which flyers describing sales and product features are distributed to homes in the surrounding area), (ii) newspaper advertising and (iii) local television advertising. To determine which type of advertising is most effective, the manager collects one week's data from 100 randomly selected stores. For each store, the following variables are recorded:
Weekly gross sales
Weekly expenditures on direct mailing
Weekly expenditures on newspaper advertising Weekly expenditures on television commercials
All variables are recorded in thousands of dollars and stored in columns 1 to 4, respectively, in file Tutorial 08-Q3.xls.
a. Find and write out the regression line.
b. What is the Coefficient of Determination? What is the Coefficient of Determination adjusted for degrees of freedom? What do these statistics tell you about the regression model? Using the ANOVA table, calculate R2 , the Adjusted R2 and the standard error of the estimate, Sε. Confirm that these values are the same as those shown in the computer output.
c. What does the standard error of the estimate tell you about the regression model (compare with the mean weekly gross sales, which you will need to compute)?
d. Test the overall utility of the model using an F-Test. What does the p-value of this test tell you?