Car Value Case_Study FINAL

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INTRODUCTION When trying to decide what car to buy, real value is not necessarily determined by how much you spend on the initial purchase. Instead, cars that are reliable and don’t cost much to own often represent the best values. But, no matter how reliable or inexpensive a car may cost to own, it must also perform well.

To measure value, consumer report developed a statistic referred to as a value score. The value score is based upon five-year owner costs, overall road-test scores, and predicted reliability ratings. Five-year owner costs are based on the expenses incurred in the first five years of ownership, including depreciation, fuel, maintenance and repairs, and so on.

Using a national average of 12,000 miles per year, an average cost per mile driven is used as the measure of five-year owner costs. Road-test scores are the results of more than 50 tests and evaluations and are based upon a 100-point scale, with higher scores indicating better performance, comfort, convenience, and fuel economy. The highest road-test score obtained in the tests conducted by consumer report was a 99 for a Lexus LS 460L. Predicted-reliability ratings (15 Poor, 25 Fair, 35 Good, 45 Very Good, and 55 Excellent) are based on data from consumer report Annual Auto Survey.

A car with a value score of 1.0, in table 1 is regarded “average value.” A car with a value score of 2.0 is twice as good a value as a car with a value score of 1.0; a car with a value score of 0.5 is considered half as good as average; and so on.

DATA AND GENERAL ANALYSIS A. GENERAL DATA ANALYSIS AND OBSERVATIONS Using the descriptive statistics function in Excel, we determined the mean, standard deviation, range and other key statistics for each of the 4 independent variables and the dependent variable Value Score.

The count for all variables is 21.

Table 1. Numerical Summaries of the Data

Variable Price ($) Cost/Mile Road-Test Score Predicted Reliability Value Score

Mean 27272.3333 0.6457 81.1429 3.8095 1.4790

Standard Standard Sample Error Median Mode Deviation Variance 814.9666 28090 3734.6461 13947581.83 0.0135 0.67 0.67 0.0620 0.0038 2.2171 83 81 10.1601 103.2286 0.1483 0.0470

4 1.43

4 1.73

0.6796 0.2153

0.4619 0.0463

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Range Min Max Sum 13195 21800 34995 572719 0.18 0.56 0.74 13.56 43 52 95 1704 2 0.85

3 1.05

5 1.9

80 31.06

B. REGRESSION ANALYSES TO DEVELOP ESTIMATED REGRESSION EQUATIONS For simple linear regression, the least squares estimates are denoted as b0 and b1. Using these estimates, an estimated regression equation is constructed: ŷ = b0 + b1x. Before we solved for each of the estimations, we showed the data graphically in a scatter diagram. The independent variable is located on the x-axis and the dependent variable is located on the y-axis. We then used the regression function in Excel to obtain the below statistical output summaries.

1. Estimated regression equation that could be used to predict the value score given the price of the car. ŷ = value score, b0 = 1.8538, and b1x = -1E-05x; therefore the estimated equation is: value score = 1.8538 + -1E-05x

Intercept Price ($)

Standard Lower Coefficients Error t Stat P-value 95% 1.8538 0.3534 5.2465 0.0000 1.1143 -1.374E-05 1.2842E-05 -1.0700969 0.2979821 -4.06214E-05

Upper Lower 95% 95.0% 2.5934 1.1143 1.31E-05 -4.06214E-05

Upper 95.0% 2.5934 1.31E-05

2. Estimated regression equation that could be used to predict the value score given the fiveyear owner costs (cost/mile). ŷ = value score, b0 = 2.5837, and b1x = -1.7108x; therefore the estimated equation is: value score = 2.5837 + -1.7108x

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Intercept Cost/Mile

Standard Coefficients Error 2.5837 0.4494 -1.7108 0.6929

Upper Lower Upper 95% 95.0% 95.0% t Stat P-value Lower 95% 5.7494 0.0000 1.6432 3.5243 1.6432 3.5243 -2.4690 0.0232 -3.1611 -0.2605 -3.1611 -0.2605

3. Estimated regression equation that could be used to predict the value score given the roadtest score. ŷ = value score, b0 = 2.6375, and b1x = 0.0104x; therefore the estimated equation is: value score = 2.6375 + 0.0104x

Standard Coefficients Error Intercept 0.6375 0.3465 Road-Test Score 0.0104 0.0042

t Stat P-value 1.8398 0.0815 2.4470 0.0243

Lower Upper Lower Upper 95% 95% 95.0% 95.0% -0.0877 1.3627 -0.0877 1.3627 0.0015 0.0192 0.0015 0.0192

4. Estimated regression equation that could be used to predict the value score given the predicted reliability.

ŷ = value score, b0 = 0.9157, and b1x = 0.1479x; therefore the estimated equation is: value score = 0.9157 + 0.1479x

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Coefficients Intercept 0.9157 Predicted Reliability 0.1479

Standard Error 0.2485 0.0643

t Stat P-value 3.6852 0.0016 2.3015 0.0329

Lower Upper Lower Upper 95% 95% 95.0% 95.0% 0.3956 1.4357 0.3956 1.4357 0.0134 0.2824 0.0134 0.2824

C. MULTIPLE REGRESSION MODEL USED TO PREDICT THE VALUE FOR A CAR To develop a multiple regression model, we completed the following steps: • Used the F test to assess overall significance • Used the t test to assess significance of each individual variable • Leveraged Excel’s Regression function to develop our multiple regression model • Applied our model to the data received to assess accuracy of our model • Used our multiple regression model predict a value score 1. Testing for overall significance: The F Test The F test is used to determine whether a significant relationship exists between the dependent variable, value score, and the set of all the independent variables. The hypothesis is: • • •

H0: B1 = B2 = B3 = B4 = 0 Ha: One or more of the parameters is not equal to zero The F test statistic: F = MSR/MSE

Using the regression function in Excel we determine that the p-value associated with the F test is 1.46-07 which is < our level of significance (.05) so we can reject H0 and conclude that a significant relationship exists between the value score and the independent variables.

ANOVA df Regression Residual Total

4 16 20

SS 0.827095965 0.099684988 0.926780952

MS 0.206773991 0.006230312

F 33.18838602

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Significance F 1.45821E-07

2. Testing for individual significance: t testing A separate t test is conducted for each of the independent variables in the model to determine individual variable significance. In each case our hypothesis is: • • •

H0: Bx = 0 Ha: Bx not equal to 0 We can reject H0 if p-value of the test statistic t is <= our level of significance .05

Using the regression function in excel we determine the p-value associated with the t statistic for each independent variable. Through this analysis we can see that the pvalue associated with the t test for Price is > our level of significance .05, so we can conclude that Price is not significant and exclude it from the multiple regression model.

We can also see that the p-value for all other variables is <= our level of significance .05 and therefore conclude that they are significant.

Intercept Price ($) Cost/Mile Road-Test Score Predicted Reliability

Coefficients 0.945229969 -5.09399E-06 -1.836324199 0.013082378

Standard Error 0.293461218 1.0609E-05 0.601194493 0.001917875

t Stat 3.220970648 -0.480155214 -3.054459451 6.821289908

P-value 0.005337167 0.637615595 0.007567863 4.10731E-06

0.209197204

0.028848811

7.251501744

1.9358E-06

3. Development of the multiple regression model and predicting value Given that Price has been determined to not be significant, we can now make a decision on whether or not to include it in our multiple regression model.

Analysis of the data using excel regression function provides the following insights: With price included in the model we have: Y = .945 + (-5.09 -06 * price) + (-1.84 * cost/mile) + (.01 * road-test score) + (.21 * reliability)

R Square Adjusted R Square Intercept Price ($) Cost/Mile Road-Test Score Predicted Reliability

0.892439537 0.865549422 Coefficients 0.945229969 -5.09399E-06 -1.836324199 0.013082378 0.209197204

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With price excluded from the model and a new analysis from the excel regression function we have: Y = 1.02 + (-2.09 * cost/mile) + (.01 * road-test score) + .20 * reliability)

R Square Adjusted R Square

0.890889665 0.8716349 Coefficients 1.020320966 -2.089708136 0.012714873 0.203794473

Intercept Cost/Mile Road-Test Score Predicted Reliability

We note that both equations have similar coefficients of determination (R2 = .89 for both) but the adjusted multiple coefficient of determination (Adjusted R Square) is slightly improved for the regression model with price removed due to insignificance.

We determine that we would progress with this multiple regression model for future predictions.

4. Using the multiple regression model to predict a value score Using our multiple regression model, we predict the value score for the ABC-Cars-M380 model to be:

Car ABC-Cars-M380

Price ($) 32000

Cost/Mile 0.9

Road-Test Score 83

Predicted Reliability 5

Predicted Value Score 1.21

CONCLUSION From our analysis, we discovered that when modeling the provided data, the variable of Price is not significant and does not need to be included. All other variables are significant and may have an impact on the value score.

We also see that using a regression model to predict value does provide a reliable estimate. To confirm the validity of our model we used the multiple regression model to compare the actual values of our dependent variable against the predicted value and achieved the following:

Car Nissan Altima 2.5 S (4-cyl.)

Kia Optima LX (2.4)

Actual Value Score

Predicted Value Score

1.75 1.73

1.76 1.65

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Subaru Legacy 2.5i Premium Ford Fusion Hybrid Honda Accord LX-P (4-cyl.)

Mazda6 i Sport (4-cyl.)

Hyundai Sonata GLS (2.4) Ford Fusion SE (4-cyl.)

Chevrolet Malibu LT (4-cyl.)

Kia Optima SX (2.0T) Ford Fusion SEL (V6) Nissan Altima 3.5 SR (V6) Hyundai Sonata Limited (2.0T) Honda Accord EX-L (V6) Mazda6 s Grand Touring (V6) Ford Fusion SEL (V6, AWD) Subaru Legacy 3.6R Limited Chevrolet Malibu LTZ (V6) Chrysler 200 Limited (V6) Chevrolet Impala LT (3.6) Mercedes EC320D

1.73 1.70 1.62 1.60 1.58 1.55 1.48 1.43 1.42 1.42 1.39 1.36 1.34 1.32 1.29 1.20 1.20 1.05 1.90

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1.66 1.79 1.68 1.55 1.59 1.61 1.38 1.40 1.45 1.58 1.38 1.38 1.32 1.31 1.27 1.29 1.24 1.03 1.74