# Linear and Nonlinear Models

A toy company has been marketing souvenir toys in conjunction with various professional sports teams in a number of cities. Over the past few years, this experience has provided some data on the effect of advertising on sales revenues because the advertising expenditures have tended to be different in each case. Although the lengths of advertising campaigns vary, the vast majority of the sales occur within the first three months, which is the time period covered in the data. A summary is shown in the table.

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Advertising 18 12 5 4 24 26 15 12 16 10 15 12

($M)

Revenues 105 72 40 44 135 122 88 77 105 67 98 54

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a. Explore the possibility of a linear relation between advertising expenditures and three-month sales revenue. Does there seem to be a strong linear relationship, based on a scatter plot and the correlation?

b. Build a linear model to represent the relation between advertising and revenue. What revenues would the model predict for expenditures of $2 million and $30 million?

c. As an alternative to the linear model in (b), build a nonlinear model using the power function y+ABx. What revenues would the model predict for expenditures of $2 million and $30 million? What are the advantages and disadvantages of a nonlinear model for this relationship?

*Please show all work, use excel and identify the value and units measured.

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#### Solution Preview

a. Yes, there seems to be a very strong linear relationship based on the scatter plot and the high value of coefficient of correlation of 0.9481.

b. The least squares regression line is: y = 4.363x + 22.47

For an ...

#### Solution Summary

Linear and nonlinear models are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.