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

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

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... this is the price of a used car with no mileage. when its age is ' ... In describing the two values of the car condition, these variables are used as follows: ... – PowerPoint PPT presentation

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Title: 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

2
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
3
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.

4
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.

5
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.

6
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.

7
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.

8
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.

9
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
10
The linear regression equation
The linear regression equationPrice
17357.38-1131.93Age-33.242Mileage-
-2556.44Avg-3275.3Poor775.64Dealer
11
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.

12
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.

13
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.

14
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.
15
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.

16
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)
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