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Chapter 12: Q120E (page 803)

Question: Identify the problem(s) in each of the residual plots shown below.

Short Answer

Expert verified

Answer

a. The mean of residual is not equal to 0 here. Plot of the residuals for the straight-line model reveals a nonrandom pattern. The residuals exhibit a curved shape. The indication is that the mean value of the random error, within the ranges of x (small, medium, large) may not be equal to 0. Such a pattern usually indicates that curvature needs to be added to the model.

b. The variance of the error is not constant which can be seen in the graph. The range in values of the residuals increases as y increases, thus indicating that the variance of the random error becomes larger as the estimate of E(y) increases in value.

c. The residuals appear to be randomly distributed around the 0 line. However, the residuals seem to be between +/- 3s range which indicates that the model is good fit for the data.

d. The error terms should be normally distributed. But, from the graph it is visible that the error terms are not normally distributed. It appears to be a positively distributed data.

Step by step solution

01

Problem in graph a 

The mean of residual is not equal to 0 here. Plot of the residuals for the straight-line model reveals a nonrandom pattern. The residuals exhibit a curved shape. The indication is that the mean value of the random error, within the ranges of x (small, medium, large) may not be equal to 0. Such a pattern usually indicates that curvature needs to be added to the model.

02

Problem in graph b

The variance of the error is not constant which can be seen in the graph. The range in values of the residuals increases as y increases, thus indicating that the variance of the random error becomes larger as the estimate of E(y) increases in value.

03

Problem in graph c

The residuals appear to be randomly distributed around the 0 line. However, the residuals seem to be between +/- 3s range which indicates that the model is good fit for the data.

04

Problem in graph d

The error terms should be normally distributed. But, from the graph it is visible that the error terms are not normally distributed. It appears to be a positively distributed data.

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Most popular questions from this chapter

Catalytic converters in cars. A quadratic model was applied to motor vehicle toxic emissions data collected in Mexico City (Environmental Science & Engineering, Sept. 1, 2000). The following equation was used to predict the percentage (y) of motor vehicles without catalytic converters in the Mexico City fleet for a given year (x): β^2

a. Explain why the valueβ^0=325790has no practical interpretation.

b. Explain why the valueβ^1=-321.67should not be Interpreted as a slope.

c. Examine the value ofβ^2to determine the nature of the curvature (upward or downward) in the sample data.

d. The researchers used the model to estimate “that just after the year 2021 the fleet of cars with catalytic converters will completely disappear.” Comment on the danger of using the model to predict y in the year 2021. (Note: The model was fit to data collected between 1984 and 1999.)

Goal congruence in top management teams. Do chief executive officers (CEOs) and their top managers always agree on the goals of the company? Goal importance congruence between CEOs and vice presidents (VPs) was studied in the Academy of Management Journal (Feb. 2008). The researchers used regression to model a VP’s attitude toward the goal of improving efficiency (y) as a function of the two quantitative independent variables level of CEO (x1)leadership and level of congruence between the CEO and the VP (x2). A complete second-order model in x1and x2was fit to data collected for n = 517 top management team members at U.S. credit unions.

a. Write the complete second-order model for E(y).

b. The coefficient of determination for the model, part a, was reported asR2=0.14. Interpret this value.

c. The estimate of theβ-value for the(x2)2term in the model was found to be negative. Interpret this result, practically.

d. A t-test on theβ-value for the interaction term in the model,x1x2, resulted in a p-value of 0.02. Practically interpret this result, usingα=0.05.

Question: Risk management performance. An article in the International Journal of Production Economics (Vol. 171, 2016) investigated the factors associated with a firm’s supply chain risk management performance (y). Five potential independent variables (all measured quantitatively) were considered: (1) firm size, (2) supplier orientation, (3) supplier dependency, (4) customer orientation, and (5) systemic purchasing. Consider running a stepwise regression to find the best subset of predictors for risk management performance.

a. How many 1-variable models are fit in step 1 of the stepwise regression?

b. Assume supplier orientation is selected in step 1. How many 2-variable models are fit in step 2 of the stepwise regression?

c. Assume systemic purchasing is selected in step 2. How many 3-variable models are fit in step 3 of the stepwise regression?

d. Assume customer orientation is selected in step 3. How many 4-variable models are fit in step 4 of the stepwise regression?

e. Through the first 4 steps of the stepwise regression, determine the total number of t-tests performed. Assuming each test uses an a = .05 level of significance, give an estimate of the probability of at least one Type I error in the stepwise regression.

Forecasting movie revenues with Twitter. Refer to the IEEE International Conference on Web Intelligence and Intelligent Agent Technology (2010) study on using the volume of chatter on Twitter.com to forecast movie box office revenue, Exercise 11.27 (p. 657). Recall that opening weekend box office revenue data (in millions of dollars) were collected for a sample of 24 recent movies. In addition to each movie’s tweet rate, i.e., the average number of tweets referring to the movie per hour 1 week prior to the movie’s release, the researchers also computed the ratio of positive to negative tweets (called the PN-ratio).

a) Give the equation of a first-order model relating revenue (y)to both tweet rate(x1)and PN-ratio(x2).

b) Which b in the model, part a, represents the change in revenue(y)for every 1-tweet increase in the tweet rate(x1), holding PN-ratio(x2)constant?

c) Which b in the model, part a, represents the change in revenue (y)for every 1-unit increase in the PN-ratio(x2), holding tweet rate(x1)constant?

d) The following coefficients were reported:R2=0.945andRa2=0.940. Give a practical interpretation for bothR2andRa2.

e) Conduct a test of the null hypothesis, H0;β1=β2=0. Useα=0.05.

f) The researchers reported the p-values for testing,H0;β1=0andH0;β2=0 as both less than .0001. Interpret these results (use).

Question: Do blondes raise more funds? Refer to the Economic Letters (Vol. 100, 2008) study of whether the color of a female solicitor’s hair impacts the level of capital raised, Exercise 12.75 (p. 756). Recall that 955 households were contacted by a female solicitor to raise funds for hazard mitigation research. In addition to the household’s level of contribution (in dollars) and the hair color of the solicitor (blond Caucasian, brunette Caucasian, or minority female), the researcher also recorded the beauty rating of the solicitor (measured quantitatively, on a 10-point scale).

  1. Write a first-order model (with no interaction) for mean contribution level, E(y), as a function of a solicitor’s hair color and her beauty rating.
  2. Refer to the model, part a. For each hair color, express the change in contribution level for each 1-point increase in a solicitor’s beauty rating in terms of the model parameters.
  3. Write an interaction model for mean contribution level, E(y), as a function of a solicitor’s hair color and her beauty rating.
  4. Refer to the model, part c. For each hair color, express the change in contribution level for each 1-point increase in a solicitor’s beauty rating in terms of the model parameters.
  5. Refer to the model; part c. Illustrate the interaction with a graph.
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