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Chapter 12: Q98E (page 777)

Question: Suppose you fit the regression modelE(y)=尾0+尾1x1+尾2x2+尾3x1+尾4x12+尾5x22to n = 30 data points and wish to test H0: 尾3 = 尾4 = 尾5 = 0

a. State the alternative hypothesis Ha.

b. Give the reduced model appropriate for conducting the test.

c. What are the numerator and denominator degrees of freedom associated with the F-statistic?

d. Suppose the SSE鈥檚 for the reduced and complete models are SSER = 1,250.2 and SSEC = 1,125.2. Conduct the hypothesis test and interpret the results of your test. Test using 伪 = .05.

Short Answer

Expert verified

Answer

a. The alternate hypothesis would be at least one of the 尾 parameters under test is nonzero.

b. The reduced model under consideration here is y=尾0+尾1x1+尾2x2.

c. In the numerator there鈥檚 (k-g) and in the denominator there are [n-(k+1)] degrees of freedom where (k 鈥� g) = Number of b parameters specified in H0 (i.e., number of 尾 parameters tested), k + 1 = Number of 尾 parameters in the complete model (including 尾0), and n = Total sample size.

d. At 95% confidence interval there is enough evidence to not reject H0. Hence, at least one of 尾 parameters are nonzero.

Step by step solution

01

Alternate hypothesis

The alternate hypothesis would be at least one of the 尾 parameters under test is nonzero.

02

Reduced model

The reduced model under consideration here is y=尾0+尾1x1+尾2x2.

03

Degrees of freedom

In the numerator there鈥檚 (k-g) and in the denominator there are [n-(k+1)] degrees of freedom

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Most popular questions from this chapter

Question: Ambiance of 5-star hotels. Although invisible and intangible, ambient conditions such as air quality , temperature , odor/aroma , music , noise level , and overall image may affect guests鈥� satisfaction with their stay at a hotel. A study in the Journal of Hospitality Marketing & Management (Vol. 24, 2015) was designed to assess the effect of each of these ambient factors on customer satisfaction with the hotel . Using a survey, researchers collected data for a sample of 422 guests at 5-star hotels. All variables were measured as an average of several 5-point questionnaire responses. The results of the multiple regression are summarized in the table on the next page.

  1. Write the equation of a first-order model for hotel image as a function of the six ambient conditions.
  2. Give a practical interpretation of each of the b-estimates shown.
  3. A 99% confidence interval for is (.350, .576). Give a practical interpretation of this result.
  4. Interpret the value of adjusted .
  5. Is there sufficient evidence that the overall model is statistically useful for predicting hotel image ? Test using a = .01.

Question: Suppose you fit the interaction model y=尾0+尾1x1+尾2x2+尾3x1x2+蔚 to n = 32 data points and obtain the following results:SSyy=479,鈥�SSE=21,鈥�尾^3=10, and s尾^3=4

a. Find R2and interpret its value.

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

Service workers and customer relations. A study in Industrial Marketing Management (February 2016) investigated the impact of service workers鈥� (e.g., waiters and waitresses) personal resources on the quality of the firm鈥檚 relationship with customers. The study focused on four types of personal resources: flexibility in dealing with customers(x1), service worker reputation(x2), empathy for the customer(x3), and service worker鈥檚 task alignment(x4). A multiple regression model was employed used to relate these four independent variables to relationship quality (y). Data were collected for n = 220 customers who had recent dealings with a service worker. (All variables were measured on a quantitative scale, based on responses to a questionnaire.)

a) Write a first-order model for E(y) as a function of the four independent variables. Refer to part

Which 尾 coefficient measures the effect of flexibility(x1)on relationship quality (y), independently of the other

b) independent variables in the model?

c) Repeat part b for reputation(x2), empathy(x3), and task alignment(x4).

d) The researchers theorize that task alignment(x4)鈥渕oderates鈥� the effect of each of the other x鈥檚 on relationship quality (y) 鈥� that is, the impact of eachx, x1,x2, orx3on y depends on(x4). Write an interaction model for E(y) that matches the researchers鈥� theory.

e) Refer to part d. What null hypothesis would you test to determine if the effect of flexibility(x1)on relationship quality (y) depends on task alignment(x4)?

f) Repeat part e for the effect of reputation(x2)and the effect of empathy(x3).

g) None of the t-tests for interaction were found to be 鈥渟tatistically significant鈥�. Given these results, the researchers concluded that their theory was not supported. Do you agree?

Question: Accuracy of software effort estimates. Periodically, software engineers must provide estimates of their effort in developing new software. In the Journal of Empirical Software Engineering (Vol. 9, 2004), multiple regression was used to predict the accuracy of these effort estimates. The dependent variable, defined as the relative error in estimating effort, y = (Actual effort - Estimated effort)/ (Actual effort) was determined for each in a sample of n = 49 software development tasks. Eight independent variables were evaluated as potential predictors of relative error using stepwise regression. Each of these was formulated as a dummy variable, as shown in the table.

Company role of estimator: x1 = 1 if developer, 0 if project leader

Task complexity: x2 = 1 if low, 0 if medium/high

Contract type: x3 = 1 if fixed price, 0 if hourly rate

Customer importance: x4 = 1 if high, 0 if low/medium

Customer priority: x5 = 1 if time of delivery, 0 if cost or quality

Level of knowledge: x6 = 1 if high, 0 if low/medium

Participation: x7 = 1 if estimator participates in work, 0 if not

Previous accuracy: x8 = 1 if more than 20% accurate, 0 if less than 20% accurate

a. In step 1 of the stepwise regression, how many different one-variable models are fit to the data?

b. In step 1, the variable x1 is selected as the best one- variable predictor. How is this determined?

c. In step 2 of the stepwise regression, how many different two-variable models (where x1 is one of the variables) are fit to the data?

d. The only two variables selected for entry into the stepwise regression model were x1 and x8. The stepwise regression yielded the following prediction equation:

Give a practical interpretation of the 尾 estimates multiplied by x1 and x8.

e) Why should a researcher be wary of using the model, part d, as the final model for predicting effort (y)?

Suppose you have developed a regression model to explain the relationship between y and x1, x2, and x3. The ranges of the variables you observed were as follows: 10 鈮� y 鈮� 100, 5 鈮� x1 鈮� 55, 0.5 鈮� x2 鈮� 1, and 1,000 鈮� x3 鈮� 2,000. Will the error of prediction be smaller when you use the least squares equation to predict y when x1 = 30, x2 = 0.6, and x3 = 1,300, or when x1 = 60, x2 = 0.4, and x3 = 900? Why?
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