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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

Question: Personality traits and job performance. Refer to the Journal of Applied Psychology (Jan. 2011) study of the relationship between task performance and conscientiousness, Exercise 12.54 (p. 747). Recall that the researchers used a quadratic model to relate y = task performance score (measured on a 30-point scale) to x1 = conscientiousness score (measured on a scale of -3 to +3). In addition, the researchers included job complexity in the model, where x2 = {1 if highly complex job, 0 if not}. The complete model took the form

E(y)=ฮฒ0+ฮฒ1x1+ฮฒ2x12+ฮฒ3x2+ฮฒ4x1x2+ฮฒ5x12x2herex2=1E(y)=ฮฒ0+ฮฒ1x1+ฮฒ2x12+ฮฒ3(1)+ฮฒ4x1(1)+ฮฒ5(1)2(1)E(y)=(ฮฒ0+ฮฒ3)+(ฮฒ1+ฮฒ4)x1+(ฮฒ2+ฮฒ5)(x1)2

a. For jobs that are not highly complex, write the equation of the model for E1y2 as a function of x1. (Substitute x2 = 0 into the equation.)

b. Refer to part a. What do each of the bโ€™s represent in the model?

c. For highly complex jobs, write the equation of the model for E(y) as a function of x1. (Substitute x2 = 1 into the equation.)

d. Refer to part c. What do each of the bโ€™s represent in the model?

e. Does the model support the researchersโ€™ theory that the curvilinear relationship between task performance score (y) and conscientiousness score (x1) depends on job complexity (x2)? Explain.

Question: Write a regression model relating the mean value of y to a qualitative independent variable that can assume two levels. Interpret all the terms in the model.

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 12.10 (p. 723). The researchers modelled a movieโ€™s opening weekend box office revenue (y) as a function of tweet rate (x1 ) and ratio of positive to negative tweets (x2) using a first-order model.

a) Write the equation of an interaction model for E(y) as a function of x1 and x2 .

b) In terms of theฮฒ in the model, part a, what is the change in revenue (y) for every 1-tweet increase in the tweet rate (x1 ) , holding PN-ratio (x2)constant at a value of 2.5?

c) In terms of the in the model, part a, what is the change in revenue (y) for every 1-tweet increase in the tweet rate (x1 ) , holding PN-ratio (x2)constant at a value of 5.0?

d) In terms of theฮฒ in the model, part a, what is the change in revenue (y) for every 1-unit increase in the PN-ratio (x2) , holding tweet rate (x1 )constant at a value of 100?

e) Give the null hypothesis for testing whether tweet rate (x1 ) and PN-ratio (x2) interact to affect revenue (y).

Personality traits and job performance. When attempting to predict job performance using personality traits, researchers typically assume that the relationship is linear. A study published in the Journal of Applied Psychology (Jan. 2011) investigated a curvilinear relationship between job task performance and a specific personality traitโ€”conscientiousness. Using data collected for 602 employees of a large public organization, task performance was measured on a 30-point scale (where higher scores indicate better performance) and conscientiousness was measured on a scale of -3 to +3 (where higher scores indicate a higher level of conscientiousness).

a. The coefficient of correlation relating task performance score to conscientiousness score was reported as r = 0.18. Explain why the researchers should not use this statistic to investigate the curvilinear relationship between task performance and conscientiousness.

b. Give the equation of a curvilinear (quadratic) model relating task performance score (y) to conscientiousness score (x).

c. The researchers theorized that task performance increases as level of conscientiousness increases, but at a decreasing rate. Draw a sketch of this relationship.

d. If the theory in part c is supported, what is the expected sign ofฮฒ2in the model, part b?

e. The researchers reportedฮฒ^2=0.32with an associated p-value of less than 0.05. Use this information to test the researchersโ€™ theory atฮฑ=0.05

Suppose you fit the model y =ฮฒ0+ฮฒ1x1+ฮฒ1x22+ฮฒ3x2+ฮฒ4x1x2+ฮตto n = 25 data points with the following results:

ฮฒ^0=1.26,ฮฒ^1= -2.43,ฮฒ^2=0.05,ฮฒ^3=0.62,ฮฒ^4=1.81sฮฒ^1=1.21,sฮฒ^2=0.16,sฮฒ^3=0.26, sฮฒ^4=1.49SSE=0.41 and R2=0.83

  1. Is there sufficient evidence to conclude that at least one of the parameters b1, b2, b3, or b4 is nonzero? Test using a = .05.

  2. Test H0: ฮฒ1 = 0 against Ha: ฮฒ1 < 0. Use ฮฑ = .05.

  3. Test H0: ฮฒ2 = 0 against Ha: ฮฒ2 > 0. Use ฮฑ = .05.

  4. Test H0: ฮฒ3 = 0 against Ha: ฮฒ3 โ‰  0. Use ฮฑ = .05.
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