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Chapter 12: 139SE (page 807)
Write a model relating E(y) to one qualitative independent variable that is at four levels. Define all the terms in your model.
Mathematically, the model in one qualitative independent variable with 4 levels can be written as .
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Suppose the mean value E(y) of a response y is related to the quantitative independent variables x1and x2
a) Identify and interpret the slope for
b) Plot the linear relationship between E(y) andfor role="math" localid="1649796003444" , whererole="math" localid="1649796025582"
c) How would you interpret the estimated slopes?
d) Use the lines you plotted in part b to determine the changes in E(y) for eachrole="math" localid="1649796051071" .
e) Use your graph from part b to determine how much E(y) changes whenrole="math" localid="1649796075921" androle="math" localid="1649796084395" .
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 trait鈥攃onscientiousness. 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 ofin the model, part b?
e. The researchers reportedwith an associated p-value of less than 0.05. Use this information to test the researchers鈥� theory at
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):
a. Explain why the valuehas no practical interpretation.
b. Explain why the valueshould not be Interpreted as a slope.
c. Examine the value ofto determine the nature of the curvature (upward or downward) in the sample data.
d. The researchers used the model to estimate 鈥渢hat just after the year 2021 the fleet of cars with catalytic converters will completely disappear.鈥� 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.)
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 movie鈥檚 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 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 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 the 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).
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鈥� 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鈥檚 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.
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