Question

Following are data on price, curb weight, horsepower, time to go from 0 to 60 miles per hour, and the speed at 1/4 mile for 16 sports and gt cars.

Sports & GT Car/ Price ($1000s)/ Curb Weight (lb.)/ Horsepower /0-60 mph (seconds) /Speed at 1/4 mile (mph):

Acura Integra Type R/ 25.035 /2577/ 195/ 7/ 90.7

Acura NSX-T/ 93.758/ 3066/ 290/ 5/ 108

BMW Z3 2.8/ 40.900/ 2844/ 189/ 6.6/ 93.2

Chevrolet Camaro Z28/ 24.865/ 3439/ 305/ 5.4/ 103.2

Chevrolet Corvette/ 50.144/ 3246/ 345/ 5.2/ 102.1

Dodge Viper/ 69.742/ 3319/ 450/ 4.4/ 116.2

Ford Mustang GT/ 23.200/ 3227/ 225/ 6.8/ 91.7

Honda Prelude Type SH/ 26.382/ 3042/ 195/ 7.7/ 89.7

Mercedes-Benz CLK320/ 44.988/ 3040/ 205/ 7.2/ 93

Mercedes-Benz SLK230/ 42.762/ 3025/ 185/ 6.6/ 92.3

Mitsubishi 3000GT VR-4/ 47.518/ 3737/ 320/ 5.7/ 99

Nissan 240SX SE/ 25.066/ 2862/ 155/ 9.1/ 84.6

Pontiac Firebird Trans Am/ 27.770/ 3455/ 305/ 5.4/ 103.2

Porsche Boxster/ 45.560/ 2822/ 201/ 6.1/ 93.2

Toyota Supra Turbo/ 40.989/ 3505/ 320/ 5.3/ 105

Volvo C70/ 41.120/ 3285/ 236/ 6.3/ 97

Develop a regression model with curb weight, horsepower, time from 0-60 mph, and the speed at 1/4 mile as 4 different independent variables to predict the car price. Typically, you should keep only significant variables in your final estimated regression model. That is, if you find statistically insignificant variables, you should remove them from the model and then re-run the model again (one at a time). However, for this particular homework problem, please keep all 4 independent variables in the final estimated equation even if some of the independent variables might not be statistically significant at the .10 alpha level.

a. Please specify the regression model?

b. At a .10 level of significance, please report your conclusions related to the hypotheses specified in part (a).

c. Again, please focus your attention on the estimated regression equation (with one dependent variable and four independent variables). Please predict the price of a sports car that takes only 7 seconds from 0 to 60 mph, along with the following features: 2,850 lbs, 200 horsepower, 100 mph 1/4 mile. In addition, please report the 95% prediction interval related to the estimated price.

Answer 1

Using MINITAB:

The COMMANDS are:

Stat >> Regression >> Regression >> Fit Regression:

File Edit Data Calc Stat Graph Editor Tools Windo Help Assistant Basic Statistics Regression ANOVA DOE Control Charts Quality Tools Reliability/Surva Multivariate Time Series Tables Nonparametrics Equivalence Tests Fitted Line Plot Regression Nonlinear Regression.. Stability Study Orthogonal Regression.. Partial Least Squares Binary Fitted Line Plot.. Binary Logistic Regression Ordinal Logistic Regression... Fit Regression Mode... Best Subsets.. Predict... Factorial Plots... Contour Plot... interaction and polynomial terms, or Surface Plot.... transform the response if needed. Overlaid Contour Plot... Fit Regression Model Model the relationship between ! categorical or continuous predictors and one response. Easily include | Response Optimizer Nominal Logistic Regression... Power and Sample Size>,be Nominal Lo Pojsson Regression

MINITAB window would be:

Regression C2 Price ($1000s) C3 Curb Weight (b. Price ($1000s) C4 Horsepower C5 0-60 mph (second: C6 Speed at 1/4 mile Responses: Continuous predictors: Curb Weight (b.)' Horsepower '0-60 mph (seconds) Speed at 1/4 mile (mph)l Categorical predictors: Coding... Stepwise... Select Storage Help Cancel

Click on Graphs:

Regression: Graphs Residuals for plots: Regular Residuals plots C Individual plots Histogram of residuals Γ Normal probability plot of residuals 「Residuals versus fits Residuals versus order Four in one Residuals versus the variables: Select Help OK Cancel

Click OK.

Then Click OK.

MINITAB output:

Regression Analysis: Price ($1000 versus Curb Weight, Horsepower, 0-60 mph (se, Speed at 1/4 Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value 3.25 0.055 0.64 0.442 0.99 0.342 0.03 0.874 3.32 0.096 Regression 4 2836.37 709.094 1 138.85 138.849 1 215.54 215.535 5.79 5.788 Speed at 1/4 mile (mph 1 724.08 724.078 11 2401.83 218.348 Curb Weight (lb.) Horsepower 0-60 mph (seconds) Error Total 15 5238.20 Model Summary S R-sq R-sq (adj) R-sq (pred) 0. 00 14.7766 54.15% 37 . 47% Coefficients Term Constant Curb Weight (lb.) Horsepower 0-60 mph (seconds) Speed at 1/4 mile (mph) Coef SE Coef T-Value P-Value VIF -208 211 -0.99 0.344 -0.0140 0.0176 -0.80 0.442 1.93 -0.156 0.157-0.99 0.342 10.45 0.16 0.874 6.86 1.82 0.096 15.37 1.38 3.34 8.45 1.84 Regression Equation Price ($10003) --208-0.0140 Curb Weight (lb.) - 0.156 Horsepower 1.38 0-60 mph (seconds) + 3.34 Speed at 1/4 mile (mph)

Residual Plots for Price ($1000s) Normal Probability Plot Versus Fits 20 30 15 15 30 20 Residual Fitted Value Histogranm Versus Order 20 20 10 10 9 10 11 12 13 14 15 16 Residual Observation Order

a)

In the output above, the regression model is:

Price ($1000s) = -208 - 0.0140 Curb Weight (lb.) - 0.156 Horsepower + 1.38 0-60 mph (seconds) + 3.34 Speed at 1/4 mile (mph)

That is,

$\widehat{y}$ = -208 - 0.0140 Curb Weight (lb.) - 0.156 Horsepower + 1.38 0-60 mph (seconds) + 3.34 Speed at 1/4 mile (mph)

b)

Conclusion:

In the output above, we can see that the predictor variable of Speed at 1/4 mile (mph) is significant because of their p-value is 0.096. However, the p-value for Curb Weight (lb.) (0.442), the p-value for Horsepower (0.342), the p-value for 0-60 mph (seconds)(0.874) are greater than the alpha level of 0.10, which indicates that it is not statistically significant.

Typically, we use the coefficient p-values to determine which terms to keep in the regression model. In the model above, we should consider removing the Curb Weight (lb.) (0.442), Horsepower (0.342), 0-60 mph (seconds).

c)

For the given values of

0 to 60 mph = 7 seconds,

curb weight = 2,850 lbs,

horsepower = 200,

mph 1/4 mile = 100

The Prediction for Price ($1000s) is obtained as:

Substitute these values into the above equation

$\widehat{y} = -208 - 0.0140 (2850) - 0.156 (200) + 1.38 (7) + 3.34 (100)$

  $= 64.56$

Thus, the Prediction for Price ($1000s) of a sports car that takes only 7 seconds from 0 to 60 mph, along with the following features: 2,850 lbs, 200 horsepower, 100 mph 1/4 mile is $64.56 (in $1000).

Using the MINITAB we can find the 95% prediction interval related to the estimated price.

The COMMANDS are:

Stat >> Regression >> Regression >> Predict:

File Edit Data Calc Stat Graph Editor Iools Window Help Assistant Basic Statistics Regression ANOVA DOE Control Charts Quality Tools Reliability/Sa Multivariate Time Series Tables Nonparametrics Equivalence Tests Fitted Line Plot Regression Nonlinear Regression.. Stability Study Fit Regression Model... Best Subsets.. Predict ber Factorial Orthogonal Regression contour! Predict Predict response values based on a Partial Least Squares Surface R fitted model. Overlaid Contour Binary Fitted Line Plot.. Binary Logistic Regression Ordinal Logistic Regression... !☆ Response Optimizer 、l ta a Nominal Logistic Regression ower and Sample Size Pojsson Regression

MINITAB window would be:

Response: Price($1000s) Enter individualvalues xl Curb eight 2850 Horsepower0-60 mph (s 200 'Speed at 1/44 100 | Results. .. Storage. J 」 Select Options. yew, Model. Help or cancel

Click OK.

MINITAB Output:

Prediction for Price ($1000s) Regression Equation Price ($10003) =-208-0.0140 Curb Weight (lb.) ー0.156 Horsepower +1.38 0-60 mph (seconds) +3.34 Speed at 1/4 mile (mph) Variable Curb Weight (lb.) Horsepower 0-60 mph (seconds) Speed at 1/4 mile (mph) Setting 2850 200 100 Fit SE Fit 95% CI 95% PI 64.3549 15.0786 (31.1672, 97.5426) (17.8880, 110.822) X X denotes an unusual point relative to predictor levels used to fit the model.