Ch20 Exercises

7/28/2019 Ch20 Exercises

1/10

4/21/2008

20 Exercises

Mix and Match

1. Reciprocal a. log y = b0 + b1 log x2.

Opposite of log b. 1.01 x

3. Elasticity c. exp(x)4. Constant slope d. 1/y5. Changing slope

e.

6. Linear pattern f. -1/27. Diminishing marginal return

g.

8. Simple variation h. %y/%x9. 1% increase

i.

10. Unusual price elasticity j. y = b0 + b1x

True/FalseIf you believe that a statement is false, briefly say why you think it isfalse.11.Regression equations only represent linear trends in data.12.If the correlation between x and y is larger than 0.5, then a linear

equation is appropriate to describe the association.13.To identify the presence of curvature, it can be helpful to begin by

fitting a line and plotting the residuals from the linear equation.


2/10

4/21/2008 20 Curves

E20-2

14.The residual standard deviation se when fitting the model 1/y = b0 +b1x has the same units as 1/y.

15.The reciprocal transformation is commonly applied to rates, such asmiles per gallon or dollars per square foot.

16.If the equation in a model has a log transformation, the interpretationof the slope should be avoided.

17.Transformations such as logs or reciprocals are determined by theposition of outliers in a scatterplot.

18.The slope of a regression model which uses log x as a predictor isknown as an elasticity.

19.Transformations in regression affect the R2 of the fitted model.20.When returned to the original scale, the fit of a model with a

transformed variable produces a curve set of predictions.

Think About It21.If diamonds have a linear relationship with small fixed costs, which

costs more: a single one-carat diamond or two carat diamonds?Are these the same?

22.According to the better model for the association between weightand mileage, which will save you more gas: getting rid of the 50pounds of junk that you leave in the trunk of your compact car orremoving the 50 pound extra seat from the family SUV? Are thesethe same?

23.Can you think of any lurking factors behind the relationship betweenweight and fuel consumption among car models?

24.The text mentions changes in the prices of rival brands as a lurkingfactor that might lead to incorrect actions when interpreting theequation that relates price and quantity. Can you think of another?

25.If the elasticity is close to zero, how are the price and quantityrelated?

26.If quantity sold increases with price , what would be the elasticity?27.If an equation uses the reciprocal of the explanatory variable, as in y

= b0 + b1 1/x, then what does the intercept b0 tell you?

28.If an equation uses the log of the explanatory variable, as in y = b0 +b1 log x, then what does the intercept b0 tell you?

29.This plot shows the residuals from the linear equation relatingmileage to weight. Whats causing the visible diagonal stripes thatare parallel to the shown arrow?


3/10

4/21/2008 20 Curves

E20-3

-15

-10

-5

0

5

10

Residual

2 3 4 5 6

Weight (000 lbs)

30.If the prices in the equation between price and quantity are

expressed in a different currency (such as euros at 1.2 per dollar),how does the elasticity change?

You Do It

31.Wal-MartThis data contains quarterly operating income (in millions of dollars)of Wal-Mart from 1990 through the end of 2005 (64 quarters).Operating income is the difference between net sales and the cost ofitems sold, including both the merchandise and the labor to make thesale.a) Create a scatterplot for operating income on Date. Is the patternlinear?b) Fit a linear trend to the operating income. If we accept the fit ofthis equation, interpret the slope and intercept.c) Plot the residuals from this linear trend on Date. Do the residualsshow simple variation, or do you see patterns in both the mean andvariation of the residuals?d) Create a scatterplot for operating income on date, but use a logscale for the y axis. Compare this view of the data to the initial plotof operating income versus date.e) Fit the equation log (Operating Income) = b0 + b1Date. Interpret theslope in the fit of this equation?f) What pattern remains in the residuals from the fit of the log ofoperating income on date? Can you explain whats happening?g) Which equation offers the better summary of the trend inoperating income at Wal-Mart? Whats the basis for your choice?

32.TargetThis data contains quarterly operating income (in millions of dollars)of Target Stores from 1990 through the end of 2005 (64 quarters).Operating income is the difference between net sales and the cost ofthings sold, including both the merchandise and the labor to makethe sale.a) Create a scatterplot for operating income on Date. Is the patternlinear?


4/10


5/10

4/21/2008 20 Curves

E20-5

shelf feet. Does the relationship seem more linear?d) Fit a regression of sales on the log of the number of shelf feet.Does this model provide a better description of the pattern in thedata? What do the slope and intercept tell you?e) Compare the fit of the two models to the data. Can you rely on

summary statistics like R2

and se?35.Used AccordsCars depreciate over time. These data show the listed prices ofHonda Accords listed for sale by individuals in the PhiladelphiaInquirerin an issue during 2005. One column gives the asking price(in thousands of dollars) and a second column gives the age (inyears).a) Do you expect the resale value of a car to drop by a fixed amounteach year?b) Fit a linear equation with price as the response and age as theexplanatory variable. What do the slope and intercept tell you, if we

accept this equations description of the pattern in the data?c) Plot the residuals from the linear equation onAge. Do theresiduals suggest a problem with the linear equation?d) Fit the equation

Estimated Price = b0 + b1 logAgeDo the residuals from this fit fix the problem found in c?e) Compare the fitted values from this equation with those from thelinear model. Show both in the same scatterplot. In particular,compare what this graph has to say about the effects of increasingage on resale value.f) Compare the values of R2 and se between these two equations.Give units where appropriate. Does this comparison agree youreyour impression of the better model? Should these summarystatistics be compared?g) Interpret the intercept and slope in this equation.h) Compare the change in asking price for cars that are 1 and 2 yearsold to the difference between cars that are 11 and 12 years old. Usethe equation with the log of age as the explanatory variable. Is thisdifference the same or different?

36.Used CamrysCars depreciate over time, whether made by Honda or in this case,Toytota. These data show the listed prices of Toyota Camrys listedfor sale by individuals in the Philadelphia Inquirerin an issue during2005. One column gives the asking price (in thousands of dollars)and a second column gives the age (in years).a) Do you expect the resale value of a car to drop by a fixed amounteach year?b) Fit a linear equation with price as the response and age as theexplanatory variable. What do the slope and intercept tell you, if weaccept this equations description of the pattern in the data.


6/10

4/21/2008 20 Curves

E20-6

c) Plot the residuals from the linear equation onAge. Do theresiduals suggest a problem with the linear equation?d) Fit the equation

Estimated Price = b0 + b1 logAgeDo the residuals from this fit fix the problem found in c?

e) Compare the fitted values from this equation with those from thelinear model. Show both in the same scatterplot. In particular,compare what this graph has to say about the effects of increasingage on resale value.f) Compare the values of R2 and se between these two equations.Give units where appropriate. Does this comparison agree youreyour impression of the better model? Should these summarystatistics be compared?g) Interpret the intercept and slope in this equation.h) Compare the change in asking price for cars that are 1 and 2 yearsold to the difference between cars that are 11 and 12 years old. Use

the equation with the log of age as the explanatory variable. Is thisdifference the same or different?

37.Cellular phones in the USCellular (or mobile) phones are everywhere these days, but it has notalways been so. These data from CTIA, an organization representingthe wireless communications industry, track the number of cellularsubscribers in the US. The data is semiannual, from 1985 throughmid 2006.a) From what you have observed about the use of cellulartelephones, what do you expect the trend in the number ofsubscribers to look like?b) Create a scatterplot for the number of subscribers on the date ofthe measurement. Does the trend look like you would haveexpected?c) Fit a linear equation with the number of subscribers as theresponse and the date as the explanatory variable. What do the slopeand intercept tell you, if we accept this equations description of thepattern in the data?d) Create a scatterplot for the same data shown in the scatterplotdone in b, but for this plot, put the response on a log scale. Doesthe scatterplot suggest a curve of the form

Estimated Log Number of Subscribers = b0 + b1Dateis a good summary?e) Create a scatterplot for the percentage change in the number ofsubscribers versus the year minus 1984. (Thats like treating 1984 asthe start of the cellular industry in the US.) Does this plot suggest anyproblem with the use of the equation of the log of the number ofsubscribers on the date? What should this plot look like if a logequation of the form in d is going to be a good summary?f) Summarize the curve in this scatterplot using a curve of the form


7/10

4/21/2008 20 Curves

E20-7

Estimated Percentage Growth = b0 + b1 1/(Date-1984)Does this curve appear to be a better summary of the pattern ofgrowth in the domestic cellular industry?g) Whats the interpretation of the estimated intercept b0 in the curveestimated in e?

h) Use the equation from e to predict the number of subscribers inthe next period. Do you think this will be a better estimate thanoffered by the linear equation or logarithmic curve?

38.Cellular phones in AfricaMobile phones (as cellular phones are often called outside the US)have replaced traditional landlines in parts of the developing worldwhere it has been impractical to build the infrastructure needed forlandlines. These data from the ITU (International TelecommunicationUnion) estimate the number of mobile and landline subscribers (inthousands) in Sub-Saharan Africa outside of South Africa. The datais annual, from 1995 through 2005.a) On the same axes, show timeplots of the two types of subscribersversus year together. Does either curve appear linear?b) Create a scatterplot for the number of landline subscribers on theyear of the count, then fit a linear equation with the number ofsubscribers as the response and the year as the explanatory variable.Does the R2 of this fitted equation mean that its a good summary?c) What do the slope and intercept tell you, if we accept thisequations description of the pattern in the data?d) Do the residuals from the linear equation confirm your impressionof the fit of the model?

e) Does a curve of the formEstimated Log Number of Subscribers = b0 + b1Yearprovide a better summary of the growth of the use of landlines? Usethe residuals to help decide.f) Interpret the slope in the previous equation that uses the log thenumber of subscribers. What does it tell you about the growth of thissector?g) Fit a similar logarithm equation to the growth in the number ofmobile users. The curve is not such a nice fit, but allows somecomparison of the rates of growth. How do the approximate rates ofgrowth compare?

39.Pet foods, revisitedThis exercise uses base 10 logs instead of natural logs.a) Using the pet food data of the text example, transform the priceand volume data using natural logs and then using base 10 logs.Then plot the natural log of volume on the natural log of price, andthe natural log of volume on the natural log of price. Whats thedifference in your plots?b) Fit the linear equation of the log of volume on the log of price in


8/10

4/21/2008 20 Curves

E20-8

both scales. What differences do you find between the fitted slopesand intercepts?c) Do the summary statistics R2 and se differ between the two fittedequations?d) Create a scatterplot for the loge of volume on log10 of volume, and

then fit the least squares line. Explain how this relationship explainsthe similarities and differences.e) Which log should be used to estimate an elasticity, or does itmatter?

40.MoviesThese data describe the box-office success of 407 movies releasedduring the years 1998 through 2001. For this analysis, wereinterested in the relationship between initial success at the movietheatre and subsequent sales for pay-per-view services, such as thoseoffered by cable television. Both columns in the data table are inmillions of dollars. Base 10 logs are more useful than natural logswhen dealing with large monetary quantities since they give us thenumber of digits (minus 1).a) Create a scatterplot for subsequent sales on the box office sales,and scatterplot the log10 of the subsequent sales on the log10 of thebox office sales. Can we describe the pattern in either plot using alinear equation?b) Estimate the elasticity of subsequent sales with respect to box-office sales using the least squares equation with y given by the log10of box-office sales and x given by the log10 of subsequent sales. (Note:loge x = 2.30262 log10 x.)

c) Interpret the elasticity in the context of these data.d) How would the fitted equation in b differ had we used naturallogs (base e) rather than common logs (base 10)? Does the equationfit the data any better?e) Several successful movies that grossed above $10 million at thebox office have usually large, negative residuals relative to the linearequation of log10 subsequent sales on log10 gross. Which movies arethese and do you think that they have anything in common?


9/10


10/10

4/21/2008 20 Curves

E20-10

4M Crime and Housing Prices in Philadelphia

Modern housing areas often seek to obtain, or at least convey, low rates of crimes.Homeowners and families like to live in safe areas and presumably are willing to pay apremium for the opportunity to have a safe home.

These data from Philadelphia Magazine summarize crime rates and housing prices incommunities near and including Philadelphia. The housing prices for each communityare the median selling prices for homes sold in the prior year. The crime rate variablemeasures the number of reported crimes per 100,000 people living in the community.

Exclude the data for Center City, Philadelphia from this analysis. Its a predominantlycommercial area that includes clusters of residential housing. The amount ofcommercial activity produces a very large crime rate relative to the number ofresidents.

Motivation

a) How could local political and business leaders use an equation that relates crimerates to housing values to advocate higher expenditures for police?

b) Would an equation from these data produce a causal statement relating crime ratesto housing prices? Explain.

Method

c) For modeling the association between crime rates and housing prices, explain why acommunity leader should consider crime rates the explanatory variable.

d) Do you anticipate differences in the level of crime to be linearly related todifferences in the housing prices? Explain in the context of your answer theunderlying implication of a linear relationship.

Mechanics

e) Create a scatterplot for the housing prices on crime rates. Describe the association.

Is it strong? What is the direction?f) Fit the linear equation of housing prices on crime rates. Interpret the slope, intercept,and summary statistics (R2 and se).

g) As an alternative to the linear model, fit an equation that uses housing prices as theresponse with the reciprocal (1/x) of the crime rate as the explanatory variable. Whatis the natural interpretation of the reciprocal of the crime rate?

Message

h) Which model do you think offers the better summary of the association betweencrime rates and housing prices? Use residual plots, summary statistics, andsubstantive interpretation to make your case.

i) Interpret the equation that you think best summarizes the relationship betweencrime rates and housing prices.

j) Does an increment in the crime rate from 1 to 2 per 100,000 have the same impact (onaverage) on housing prices as the change from 11 to 12 per 100,000?

Ch20 Exercises

Documents

Transcript of Ch20 Exercises