Ten Corvettes between 1 and 6 years old were randomly selected from last year’s sales records in Virginia Beach, Virginia. The following data were obtained, where x denotes age, in years, and y denotes sales price, in hundreds of dollars. x y a. b. c. d. e. f. g. h. i. j. k. l. m. 6 125 6 115 6 130 4 160 2 219 5 150 4 190 5 163 1 260 2 260
Graph the data in a scatterplot to determine if there is a possible linear relationship. Compute and interpret the linear correlation coefficient, r. Determine the regression equation for the data. Graph the regression equation and the data points. Identify outliers and potential influential observations. Compute and interpret the coefficient of determination, r2. Obtain the residuals and create a residual plot. Decide whether it is reasonable to consider that the assumptions for regression analysis are met by the variables in questions. At the 5% significance level, do the data provide sufficient evidence to conclude that the slope of the population regression line is not 0 and, hence, that age is useful as a predictor of sales price for Corvettes? Obtain and interpret a 95% confidence interval for the slope, β, of the population regression line that relates age to sales price for Corvettes. Obtain a point estimate for the mean sales price of all 4-year-old Corvettes. Determine a 95% confidence interval for the mean sales price of all 4-year-old Corvettes. Find the predicted sales price of Jack Smith’s 4-year-old Corvette. Determine a 95% prediction interval for the sales price of Jack Smith’s 4-year-old Corvette.
Note that the following steps are not required for all analyses…only perform the necessary steps to complete your problem. Use the above steps as a guide to the correct SPSS steps.
Enter the age values into one variable and the corresponding sales price values into another variable (see figure, below).
Select Graphs Legacy Dialogs Scatter/Dot… (select Simple then click the Define button) with the Y Axis variable (Price) and the X Axis variable (Age) entered (see figures, below). Click “Titles…” to enter a descriptive title for your graph, and click “Continue”. Click “OK”.
Your output should look similar to the figure below.
Graph the data in a scatterplot to determine if there is a possible linear relationship. The points seem to follow a somewhat linear pattern with a negative slope.
Select Analyze Correlate Bivariate… (see figure, below).
Select “Age” and “Price” as the variables, select “Pearson” as the correlation coefficient, and click “ “OK” (see the left figure, below).
Compute and interpret the linear correlation coefficient, r. The correlation coefficient is –0.9679 (see the right figure, above). This value of r suggests a strong negative linear correlation since the value is negative and close to –1. Since the above value of r suggests a strong negative linear correlation, the data points should be clustered closely about a negatively sloping regression line. This is consistent with the graph obtained above. Therefore, since we see a strong negative linear relationship between Age and Price, linear regression analysis can continue.
Since we eventually want to predict the price of 4-year-old Corvettes (parts j–m), enter the number “4” in the “Age” variable column of the data window after the last row. Enter a “.” for the corresponding “Price” variable value (this lets SPSS know that we want a prediction for this value and not to include the value in any other computations) (see left figure, below).
Select Analyze Regression Linear… (see right figure, above).
Select “Price” as the dependent variable and “Age” as the independent variable (see upperleft figure, below). Click “Statistics”, select “Estimates” and “Confidence Intervals” for the regression coefficients, select “Model fit” to obtain r2, and click...