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Chapter 12: Q3E (page 722)

Short Answer

Expert verified

(A) We do not reject the null hypothesis, hence the value of β2=0

(B) We reject the null hypothesis at 95% significance level, thus, the value of β2=0

(C) Variable X2might not be related to dependent variable y even if mathematically β^2>β^3

Step by step solution

01

Step-by-Step Solution Step 1: Testing the significance of β2

Therefore, value ofβ2=0

02

Testing the significance of  β3

For, α=0.05the critical value of t0.025=2.042 using the formulae table is H0rejected if . t>t0.025Since, 3.2068 > 2.042, we reject the null hypothesis at 95% significance level.

Therefore, value of β3Is not equal to zero .

03

Predicting the model 

The null hypothesis H0;β2=0 is not rejected while the null hypothesis H0:β3=0 is rejected because the variableX2 might not have a relationship with Y . The hypothesis testing implies that variable X2 might not be predicting the overall model in a better way even if the mathematical value of coefficient of the variable calculated using the method of least square is higher than the other variable.

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Most popular questions from this chapter

Consider fitting the multiple regression model

E(y)= β0+β1x1+ β2x2+β3x3+ β4x4 +β5x5

A matrix of correlations for all pairs of independent variables is given below. Do you detect a multicollinearity problem? Explain

Question:How is the number of degrees of freedom available for estimating σ2(the variance ofε ) related to the number of independent variables in a regression model?

Role of retailer interest on shopping behavior. Retail interest is defined by marketers as the level of interest a consumer has in a given retail store. Marketing professors investigated the role of retailer interest in consumers’ shopping behavior (Journal of Retailing, Summer 2006). Using survey data collected for n = 375 consumers, the professors developed an interaction model for y = willingness of the consumer to shop at a retailer’s store in the future (called repatronage intentions) as a function of = consumer satisfaction and = retailer interest. The regression results are shown below.

(a) Is the overall model statistically useful for predicting y? Test using a=0.05

(b )Conduct a test for interaction at a= 0.05.

(c) Use the estimates to sketch the estimated relationship between repatronage intentions (y) and satisfaction when retailer interest is x2=1 (a low value).

(d)Repeat part c when retailer interest is x2= 7(a high value).

(e) Sketch the two lines, parts c and d, on the same graph to illustrate the nature of the interaction.

Assertiveness and leadership. Management professors at Columbia University examined the relationship between assertiveness and leadership (Journal of Personality and Social Psychology, February 2007). The sample represented 388 people enrolled in a full-time MBA program. Based on answers to a questionnaire, the researchers measured two variables for each subject: assertiveness score (x) and leadership ability score (y). A quadratic regression model was fit to the data, with the following results:

a. Conduct a test of overall model utility. Useα=0.05 .

b. The researchers hypothesized that leadership ability increases at a decreasing rate with assertiveness. Set up the null and alternative hypotheses to test this theory.

  1. Use the reported results to conduct the test, part b. Give your conclusion(atα=0.05 )in the words of the problem.

Question: Suppose you fit the interaction model y=β0+β1x1+β2x2+β3x1x2+ε to n = 32 data points and obtain the following results:SSyy=479, SSE=21, β^3=10, and sβ^3=4

a. Find R2and interpret its value.

b. Is the model adequate for predicting y? Test at α=.05

c. Use a graph to explain the contribution of the x1 , x2 term to the model.

d. Is there evidence that x1and x2 interact? Test at α=.05 .
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