To predict the asking price of a used Chevrolet Camaro, the following data were collected on the car

1
Example 1

• To predict the asking price of a used Chevrolet
  Camaro, the following data were collected on the
  cars age and mileage. Data is stored in
  CAMARO1.Determine the regression equation and
  answer additional questions stated later.
• Solution
• Run the regression tool from Excel Data
  analysis. Click to see the output next

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The regression equation
The regression equation Price
17499.1-1131.64Age-72.31MileageBe careful about
the interpretation of the intercept (17499).Do
not argue that this is the price of a used car
with no mileagewhen its age is zero. Although
such cars may exist (a car purchased and
returned within a week with almost no
mileage)might need to be re-sold as a used car.
Yet, such values of Age and Mileage were not
covered by the sample range!!.
CAMARO1
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The model usefulness
CAMARO1

• Does the overall model contribute significantly
  to predicting the asking price of a used
  Chevrolet Camaro? Use .01 for the significance
  level
• Answer Observe the Significance F. This is
  the p value for the F Test of the hypothesesH0
  b1 b2 0H1 At least one b ¹0. Since the p
  value is practically zero, it is smaller than
  alpha. The null hypothesis is rejected, and
  therefore at least one b ¹0. The variable
  associated with this b is linearly related to the
  price, and the model is useful, thus contributes
  to predicting the asking price.

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Models fit

• How well does the model fit the data? Would you
  expect the predictions to be accurate with this
  model?
• Solution
• Observing the coefficient of determination (R2),
  81 of the variation in car prices are explained
  by this model. This is quite high, and we can
  expect accurate predictions.

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Predicting y

• Predict the value of the asking price for a
  5-years old car, with 70,000 miles on the
  odometer, with 95 confidence.
• Solution
• To obtain an interval estimate for the prediction
  of a single car asking price when Age5, and
  Mileage70, we look for the prediction interval.
  From Data Analysis Plus we have 2622.222,
  10936.38.
• The general form of the interval is
  , where D is determined from the data.
  Specifically 17499.1-1131.64(5)-72.
  31(70) 6779.303. So the interval is 6779.303
  D, For the Data Analysis Plus procedure go to
  the worksheet Prediction Interval in CAMARO1.

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Estimating the mean y

• Predict the value of the mean asking price for
  all 5-years old cars, with 70,000 miles on the
  odometer, with 95 confidence.
• Solution
• To obtain an interval estimate for the mean
  asking price of all cars for which Age5 and
  Mileage70, we look for the confidence interval.
  From Data Analysis Plus we have 5756.028,
  7802.577For details go to the worksheet
  Prediction Interval in CAMARO1.

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Testing linear relationship

• Are both variables (Age and Mileage each one in
  the presence of the other one), serve as good
  predictors of Asking Price? Test at alpha.025.
• Solution
• Perform a t-test for the b coefficient of each
  variable. The hypotheses tested are H0 bAge0
  vs. H1 bAge¹ 0 for which the p value is .002
  H0 bMileage0 vs. H1 bMileage¹ 0 for which the
  p value is .0104. In both cases the null
  hypothesis is rejected, therefore, both have
  linear relationship to the asking price at 2.5
  significance level.

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Problem 2

• The previous model for the prediction of the
  asking price of used Chevrolet Camaro, is now
  extended by adding two new independent variables
  car condition (Excellent, Average, Poor), and the
  type of the seller who sells the car (Dealer,
  Individual). The data for this case is stored in
  CAMARO2 (see next slide).
• Develop the linear regression model for this case
  and answer several questions formulated next.
• Solution
• The two new variables describe the values of
  qualitative data (the state of a car and the type
  of the seller). Thus, they are dummy variables,
  take on the values 0 and 1.

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Using dummy variables

• Solution continued
• There are three possible car condition values, so
  we need two dummy variables. Let us select the
  variables Average and Poor.
• In describing the two values of the car
  condition, these variables are used as follows
• Average Poor
• An Excellent condition car 0 0
• An Average condition car 1 0
• A Poor condition car 0 1
• In a similar manner we use one dummy variable to
  describe who sold the car. Let us define Dealer
  1 if the car was sold by a dealer. Dealer 0 if
  sold by an individual.

CAMARO2
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The linear regression equation
The linear regression equationPrice
17357.38-1131.93Age-33.242Mileage-
-2556.44Avg-3275.3Poor775.64Dealer
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Interpreting the coefficients bi

• Interpret the coefficient estimates bi of each
  variable and test the strength of their
  predicting power.
• Solution
• bAge -1131.93. In this model, For each
  additional year the asking price drops by 1132,
  keeping the rest of the variables unchanged.
• bMile -33.24. In this model, for each additional
  1000 miles the asking price drops by 33.24,
  keeping the rest of the variables unchanged.
• bAvg -2556.44. In this model, the asking price
  for a car whose condition is average is 2556.44
  lower than the asking price for a car whose
  condition is excellent, keeping the rest of the
  variables unchanged.
• bPoor -3275.3. In this model, the asking price
  for a car whose condition is poor is 3275.3
  lower than the asking price for a car whose
  condition is excellent, keeping the rest of the
  variables unchanged.
• bDeal 775.64. In this model the asking price
  for a car sold by a dealer is 775.64 higher than
  this sold by an individual, keeping the rest of
  the variables unchanged.

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The role of the dummy variable coefficients

• Let us compare the asking price equations of two
  cars, with the same age, mileage, and condition,
  one sold by a dealer, the other one by an
  individualPrice(Dealer)b0b1Ageb2Mileageb3Av
  g.b4Poor b5(Dealer1)
  b0b1Ageb2 Mileageb3Avg.b4Poor
  b5Price(Individual)b0b1Ageb2Mileageb3Avg.b
  4Poor b5(Dealer0)
  b0b1Ageb2Mileageb3Avg.b4Poor
• Conclusion When the only difference between cars
  is the type of sellers who sell them, the base
  line equation was selected to be the
  Price(Individual) equation, and then b5 is the
  average difference in asking price between them.

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The role of the dummy variable coefficients

• Let us compare the asking price equations of
  three cars, that differ in their overall
  condition but have the same age, mileage, and are
  sold by the same type of a sellerPrice(Excellen
  t)b0b1Ageb2 Mileageb3(Avg.0)b4(Poor0)
  b5(Dealer) b0b1Ageb2
  Mileageb5(Dealer)Price(Avg.)b0b1Ageb2Mileage
  b3(Avg.1)b4(Poor0) b5(Dealer)
  b0b1Ageb2 Mileageb5(Dealer) b3
  Price(Poor)b0b1Ageb2Mileageb3(Avg.0)b4(Poo
  r1) b5(Dealer)
  b0b1Ageb2 Mileageb5(Dealer) b4
• Conclusion When the only difference between cars
  is the car condition, the base line equation was
  selected to be the Price(Excellent) equation, and
  then b3 and b4 are the average differences in
  asking price between an excellent condition car
  and the other two cars.

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Prediction power of independent variable (are
there linear relationships?)

• Testing the prediction power.
• Formulate the t-test for each b. Observing the p
  values we have
• For bAge the p value.00036. Age is a strong
  predictor
• For bMileage the p value.17. Mileage is not a
  good predictor, not having linear relationship
  with price.
• For bAverage the p value.0098. There is
  sufficient evidence to infer at 1 significance
  level that the asking price of a car whose
  condition is average is different from the asking
  price of a car whose condition is excellent.

In fact, the argument is even stronger. Since the
t-statistic is negative (-2.79), the rejection
region is at the left hand tail of the
distribution, so we have sufficient evidence to
claim that bavarage
price of an Avg. Condition car is on the
average 2556 lower than the asking price of an
Excellent condition car.
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Prediction power of independent variable (are
there linear relationships?)

• Testing the prediction power - continued.
• For bPoor the p value .006. There is a very
  strong evidence to believe that the asking price
  for a Poor Condition car is different than the
  asking price for an Excellent condition car.
  Specifically, a Poor condition car is sold for
  3275.3 less than an Excellent condition car.
• For bDealer the p value .40. There is
  insufficient evidence to infer at 2.5
  significant level that on the average the asking
  price for a car sold by a dealer is different
  than the asking price for a car sold by an
  individual.

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Prediction power of independent variable (are
there linear relationships?)

• Predict the asking price of the following cars
• 4 years old, 45000 miles, Average condition, sold
  by an individual.
• Price17357 1131.9(4) 33.242(45) 2556.4(1)
  775.64(0)