Question

5.6 The price of a car is thought to depend on the horsepower of the engine and the country where the car is made. The variable Country has four categories USA, Japan, Germany, and Others. To include the variable Country in a re gression equation, three indicator variables are created, one for USA, another for Japan, and the third for Germany. In addition, there are three interaction variables between the horsepower and each of the three Country categories (HP*USA. HP*Japan, and HP*Germany). Some regression outputs when fitting three models to the data is shown in Table 5.16. The usual regression assumptions hold. (a) Compute the correlation coefficient between the price and the horsepower. (b) What is the least squares estimated price of an American car with a 100 horsepower engine? (c) Holding the horsepower fixed, which country has the least expensive car? Why? (d) Test whcther thcre is an interaction between Country and horsepoweir Specify the null and alternative hypotheses, test statistics, and conclusions. (c) Given the horsepower of the car, test whether the Country is an important predictor of the price of a car. Specify the null and alternative hypotheses, test statistics, and conclusions (f) Would you recommend that the number of categories of Country be re duced? If so, which categories can be joined together to form one cate- gory? (g) Holding the horsepower fixed, write down the formula for the test statistic for testing the equality of the price of American and Japanese cars?

Answer 1

ANS::

as for given data

Note: JoeJoepp's solutions are very wrong - as they usually are. He clearly does NOT know stats that well. :(

(a)

Correlation between price and horsepower that does NOT incorporate country
--> look at Model #1
--> SST = SSR + SSE = 4604.7 + 1604.44 = 6209.14
--> R^2 = SSR/SST = 4604.7/6209.14 = 0.7416 or 74.16%
--> r = (sign of the slope) x sqrt(R^2)
--> r = + sqrt(.7416)
--> r = +0.861

--> Answer: r = +0.861

(b)

Note: JoeJoepp's solution to this problem is wrong.

LS estimate of an American car with a 100 horsepower engine
--> look at Model #2 that incorporates country
--> Predicted Price = -4.117 + 0.174 HP - 3.162 USA - 3.818 Japan + 0.311 Germany
--> Predicted Price = -4.117 + 0.174(100) - 3.162(1) - 3.818(0) + 0.311(0)
--> Predicted Price = 10.121
--> Answer: 10.121

Note: The units of y (although not stated) are probably in $1,000's. So, the correct answer could be

$10,121.

(c)

--> look at Model #2 that incorporates country
--> Since the outcome for Others is left out, the dummy variables (aka indicator variables)

coefficients are relative to Others.
--> The coefficient for USA indicates that cars from the USA, on average, are predicted to sell for

$3,162 LESS (since the coefficient is negative) than cars from Others, holding HP constant.
--> The coefficient for Japan indicates that cars from Japan, on average, are predicted to sell for

$3,818 LESS (since the coefficient is negative) than cars from Others, holding HP constant.
--> The coefficient for Germany indicates that cars from Germany, on average, are predicted to sell for

$311 MORE (since the coefficient is positive) than cars from Others, holding HP constant.
--> Answer: Japan

(d)

Note: JoeJoepp's solution to this problem is wrong. He is doing 3 t-tests, when instead you need to do

a partial F-test.

Note:
* The global F-test tests all of the predictors at once.
* The t-tests tests the predictors individually.
* The partial F-test tests a group of predictors at once.

--> If we want to test whether the interaction terms are significant, we have to do a partial F-test.
--> Model #3: Full model (with interaction terms): SSE = 1319.85, MSE = 16.0957
--> Model #2: Reduced model (without interaction terms): SSE = 1390.31
--> # of predictors being tested: 3 <-- we are testing the 3 interaction terms

Ho: The interaction terms are NOT significant
Ha: At least 1 of the interaction terms ARE significant

F = (change in SSE/# of predictors being tested)/MSE(FM)
F = ((1390.31-1319.85)/3)/16.0957
F = 1.4592

F-critical value: dfnumerator = 3 (we are testing 3 predictors) and dfdenominator = 82 (since we have

MSE of the full model in the denonimator). Using Excel, the exact p-value is:
=F.DIST.RT(1.4592,3,82)
=0.231754934

Since the p-value = 0.232 > alpha = 0.05, do not reject Ho.
--> Based on the sample data, there is INsufficient evidence to indicate an interaction betweeen County

and horsepower.

Note: So, Model #2 (the model without the interaction terms) is clearly better and Model #3.

(e)

Note: JoeJoepp's solution to this problem is wrong. He is doing 3 t-tests, when instead you need to do

a partial F-test.

Note: Now that we decided that we didn't need the interaction terms, question (e) wants us to determine

whether we need the dummy variables.

--> Model #2: Full model (with HP and the dummy variables): SSE = 1390.31, MSE = 16.3566
--> Model #1: Reduced model (with only horsepower): SSE = 1604.44
--> # of predictors being tested: 3 <-- we are testing the 3 dummy variables

Ho: The dummy variables for Country are NOT significant
Ha: At least one of the dummy varibles for Country ARE significant

F = (change in SSE/# of predictors being tested)/MSE(FM)
F = ((1604.44 - 1390.31)/3)/16.3566
F = 4.3638

F-critical value: dfnumerator = 3 (we are testing 3 predictors) and dfdenominator = 85 (since we have

MSE of the full model in the denonimator). Using Excel, the exact p-value is:
=F.DIST.RT(4.3638,3,85)
=0.00658462

Since the p-value = 0.007 < alpha = 0.05, we CAN reject Ho.
--> Based on the sample data, there is sufficient evidence to indicate that at least one of the dummy

variables for Country are significant.

Note: So, Model #2 is better than Model #1.

(f)

Note: JoeJoepp's solution to this problem is wrong.

Recall: Since the outcome for Others is left out, the dummy variables (aka indicator variables)

coefficients are relative to Others.

UsA --> p-value = .0216 < alpha = .05 --> Cars made in the USA are significantly cheaper (because the

coefficient for USA is negative) than those made in Other countries. Since USA is significant, we do

NOT want to combine USA and Other.

Japan --> p-value = .0061 < alpha = .05 --> Cars made in the Japan are significantly cheaper (because

the coefficient for Japan is negative) than those made in Other countries. Since Japan is significant,

we do NOT want to combine USA and Other.

Germany --> p-value = .8682 > alpha = .05 --> Cars made in the Germany are NOT significantly more

expensive than those made in Other countries. Since Germany is NOT significant, we should join Germany

and Others.

(g)

Note: JoeJoepp's solution to this problem is wrong.

After combining Germany with Other, Country now has 3 outcomes, i.e. USA, Japan and Other. So, now we need 3-1 = 2 dummy variables. This time leave off Japan* and the coefficients will be relative to Japan. Now, to test for a significant difference in price between American and Japanese cars, our t-test statistic would be:

t = coefficient for USA / s.e. for USA

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