3. We are interested in using income (x) and gender (x2, with \"1\" indicating f

Question

3. We are interested in using income (x) and gender (x2, with "1" indicating female and "O" indicating male) to predict the price of a car owner's vehicle (y). Moreover, we want to add an interaction effect between income and gender to our model (it is plausible that income could have a "moderating" or "exaggerating" effect on gender as far as the price of vehicle purchased goes, given societal pressures for conspicuous consumption for both genders). (a) Write the theoretical full interaction model and the assumptions. (b) i. Write the theoretical model for males. ii. Write the theoretical model for females. Coefficients SE Coef Term Constant INCOME GENDER W 1 INCOME GENDER W1 0.021 Coef 3.487 0.415 1.131 0.251 13.872 0.000 0.004 106.28 0.000 0.697 0.006 1.623 0.119 (c) The Minitab output above gives the results of our model based on data fromn 26 individuals. Let's conduct the t-test to see if there is significant interaction between the two predictors, income and gender. Namely we will test Ho B3 0 vs. Ha: P30 i. Find the test statistic and a range for the p-value based on the test statistic. ii. State the conclusion for this test. (d) As the Minitab output shows, the p-value for the t-test for Gender (B2) is 0.119. Does this mean that we should conclude that Gender is not significant and remove this pre dictor from our model? Explain. (e) What is the predicted price of a vehicle for a woman who has income of $60,000? (f) What is the predicted price of a vehicle for a man who has income of $60,000?

Explanation / Answer

(a) Here the full interactio model is

y^( Price of car owner) = 3.487 + 0.415 * income + 1.131 * Gender - 0.021 * (Income * Gender)

Here the assumptions are

(b) For males

Gender = 0

y^( Price of car owner) = 3.487 + 0.415 * income

for females Gender = 1

y^( Price of car owner) = 3.487 + 0.415 * income + 1.131 * 1 - 0.021 * Income

y^( Price of car owner) = 4.618 + 0.394 * income

(c)

(i) Here the given test is two tailed test.

Test staistic

t = ^3/se(^3) = -0.021/0.006 = -3.5

Here dF = 26 -2 = 24 and alpha = 0.05

p - value = TDIST (l t l > 3.5 ; 24 ; two tailed) = 0.001842 < 0.05 ; Here we can see the range of p -values is in between 0.01 to 0.001

so we will reject the null hypothesis and conclude that there is significant interacion between the two predictors.

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