Ch24 Answers

7/28/2019 Ch24 Answers

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4/20/2007

24 Answers

Mix and Match

1. f2. e3. c4. g5. h6. j7. a8. d9. b10.iTrue/False

11.True.12.False

The value of se typically decreases, but it does not have to. R2 must increase.

13.FalseIts called a marginal slope because it includes the effects of other explanatory

variables.14.True15.False

It might be smaller, but it does not have to be smaller. It depends on the size andsign of any indirect effects.

16.FalseThe marginal and partial slopes need not even have the same sign, much less bothbe close to zero.

17.True18.True19.False

We should only conclude that at least some deviation from this hypothesis occurs. Itmay not be the case that both are different from zero. Perhaps only one of themdiffers from zero.

20.True21.False

Its primary use is locating the effects of leveraged outliers.


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22.TrueThink About It

23.Most likely we have some collinearity. Busy areas attract a lot of fast food outletsbecause sales are high (positive correlation). Among densely populated areas,

however, the number of competitors reduces sales of a store (negative partial slope).Youd like to have the densely populated area to yourself. The more competitorsthat are around, the lower your sales for a give population density.

24.The two explanatory variables, test score and education, are evidently redundant.Once you know one, the other adds little value. Both are positively correlated(evidently), so either has a positive correlation with performance. But once youknow the educational background, the score on the qualifying test adds littleadditional value.

25.a) Estimated Salary = b0 + 5 Age + 2 Test Scoreb) The indirect effect is 10 $M/Point = 2 years/point * 5 $M/year, larger than thedirect effect.c) The marginal effect is the direct plus indirect effect, or 10 + 2 = 12 $M/point.d) Youre not going to be much older, so we need the partial effect. Raising the testscore by 5 points nets $10,000 annually. Its probably worth it if youre going to staywith the company long enough to earn it back.

26.a) No, not without the intercept.b) Positive. The marginal slope is -0.1 + 0.7*0.2 = 0.04c) A young person with lots of money to spend.

27.a) The correlation of something with itself is 1.b) You cannot, not without knowing the variance of x1.c) The partial and marginal slopes will be the same because the two explanatoryvariables are evidently uncorrelated. There can be no indirect effect.

28.a) Yes. R2 is at least as large as 0.74082 > 0.54.b) The same as the correlation, 0.7408. The correlations become covariances whenstandardized, so we have the covariances and variances.c) They differ because the two explanatory variables are correlated.

29.a) The fitted value is87 + 0.3 * 250 + 1.5 * 100 =312, or $312,000 revenue per month87 + 0.3 * 200 + 1.5 * 75 = 259.5, or $259,500 revenue per month

Expand to the second location.b) The intercept, $87,000, resembles a fixed cost. The intercept estimates fixedrevenue that is present regardless of the distance to the destination or thepopulation. Perhaps its money earned from air freight or other services providedby the airline. Without a confidence interval, we cannot be sure if the value is reallyfar from zero. It might be a large extrapolation.c) Among comparably populated cities, flights to those that are 100 miles fartheraway produce 0.3 100 = $30,000 more revenue per month, on average.


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d) If we compare revenue from flights to cities that are equally distant from the hub,average monthly revenue to larger cities is higher by about $1.5 per person.

30.a) The estimated margin from the location near the office complex isEst Margin = 54 - 0.0073 * 2250 + 0.0216 * 400 = 46.215%

whereas at the more isolated complex the margin is

Est Margin = 54 - 0.0073 * 300 + 0.0216 * 50 = 52.89%Choose the more isolated site.b) The intercept, 54% operating margin, is a baseline value added to the estimatedmargin for every hotel. Without seeing the scale of the other variables, we cannot tellif the intercept is an extrapolation if interpreted as the predicted value for a hotel ina very isolated location with no competitors or offices.c) The negative slope indicates that on average, sites with more competing roomshave lower operating margins (at a slope of about 0.0073% per additional competingroom.d) The slope for office shows that sites near offices earn higher margins. Onaverage, with about a 0.02 gain in margin per additional 1000 square feet.

31.a,b) The filled in table isEstimate SE t-statistic p-value

Intercept 87.3543 55.0459 1.5869 0.10Distance 0.3428 0.0925 3.7060


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You Do It

35.Diamondsa) The plots show the discrete properties of the data: we only have several fixedlengths and widths. Width is very highly related to price. The two xs are not verycorrelated. The plots look straight enough (particularly that for width).

0

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Price

($)

15 20 25 30

Length (Inch)

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Price

($)

1 1.5 2 2.5 3 3.5 4 4.5

Width (mm)

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Length

(Inch)

1 1.5 2 2.5 3 3.5 4 4.5

Width (mm)

b) The largest correlation (0.95) is between price and width. Evidently width tellsyou more about how much gold than the length.

Price ($) Length (Inch) Width (mm)Price ($) 1.0000 0.1998 0.9544Length (Inch) 0.1998 1.0000 0.0355Width (mm) 0.9544 0.0355 1.0000

c) The fit of this model has R2 = 0.94 and se = $57 with these coefficients

Term Estimate Std Error t Ratio Prob>|t|Intercept -405.635 62.11863 -6.53


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-100

-50

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P

rice

($)Residual

0 200 400 600

Price ($) Predicted

e) We formed the volume of the chain as the length (in mm) times the width2. Thisin a way gets at the amount of gold in the chain, though not perfectly. The residualshave some pattern left, but theres not the clear trend as before, and now we canidentify some outliers (a bargain and an expensive chain) that were hidden.

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-50

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50

Price

($)Residual

0 200 400 600 800 1000

Price ($) Predicted

f) Heres the fit for the improved model. With the added volume, the other twoexplanatory variables, particularly the length, lose importance. The model looks

much straighter with a much smaller se near $17. Theres still a problem in theresiduals, but they are much smaller. Our proxy for gold isnt perfect for the heavierchains.

R2 0.994674se 17.0672

Term Estimate Std Error t Ratio Prob>|t|Intercept 55.118884 34.43198 1.60 0.1225Length (inch) 0.0451975 0.971144 0.05 0.9633Width (mm) -30.59663 16.27885 -1.88 0.0724Volume (cu mm) 0.0930388 0.005845 15.92


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1000

2000

3000

Sales(Dollars)

1000 2000 3000 4000 5000

Volume (Gallons)

1000

2000

3000

Sales(Dollars)

0 100 200 300 400 500 600 700 800

Car Washes

1000

2000

3000

4000

5000

Volume(Gallons)

0 100 200 300 400 500 600 700 800

Car Washes

b) The largest correlation is between volume of gas and sales. Car washes areslightly correlated with both of these, but not very much. Seems as though sales atthe car wash are not very predictive of either gas volume or sales at the store.

Sales(Dollars) Volume(Gallons) Car Washes

Sales (Dollars) 1.0000 0.6496 0.1700

Volume (Gallons) 0.6496 1.0000 0.1242Car Washes 0.1700 0.1242 1.0000

c) The fitted model isR2 0.430022se 245.6717

Term Estimate Std Error t Ratio Prob>|t|Intercept 1112.1759 77.8611 14.28


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-800

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Sales(Dollar

s)Residual

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Sales (Dollars) Predicted

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Count

.01 .05.10 .25 .50 .75 .90.95 .99

-3 -2 -1 0 1 2 3

Normal Quantile Plot

e) The slope for car washes indicates that among stations with comparable levels ofgasoline sales, those that sell more car washes generate higher sales in the connectedconvenience store. The size of the effect is small, however, with added salesamounting to between nothing and $0.47 in added daily sales (on average) per

additional wash.To get the interval, the calculations are0.2326914 - 2 * 0.1166, 0.2326914 + 2 * 0.1166 -.0005 to .4659

and round to 2 decimals. The reportedp-value is slightly less than 0.05 because theprecise cutoff with this number of cases is 1.96 rather than our approximate 2. Noticein the rounding, however, it does not matter. The lower endpoint is basically zero.

37.Downloada) The file sizes increased steadily over the day, meaning that these two explanatoryvariables are closely associated. The scatterplots of transfer time on file size andtime of day seem reasonably linear, though their may be some bending in the plot of

transfer time on the time of day.

10

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TransferTime(sec)

20 30 40 50 60 70 80 90 100

File Size (MB)

10

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TransferTime(sec)

0 1 2 3 4 5

Time


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20

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60

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FileSize(MB)

0 1 2 3 4 5

Time

b) The marginal and partial slopes for the file size will be very different. We will noteasily be able to separate their influence from one another. The file size and time ofday are virtually redundant, so the indirect effect of file size will be very large.

c) The multiple regression isR2 0.624569

se 6.283617

Term Estimate Std Error t Ratio Prob>|t|Intercept 7.1388209 2.885703 2.47 0.0156File Size (MB) 0.3237435 0.179818 1.80 0.0757Time (hours since 8 am) -0.185726 3.16189 -0.06 0.9533

d) Somewhat, but not completely. The residual plot suggests slightly more variationfor larger file sizes. The effect is fairly subtle and is also evident in a time plot of theresiduals. There is also a slight negative dependence over time, with the residualsoscillating back in forth from positive to negative. Again, the effect is not too strong

(albeit significant by the Durbin-Watson test, D = 2.67). The residuals appear nearlynormal with no evidence of bending patterns.

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-5

0

5

10

15

TransferTime(sec)Residual

15 20 25 30 35

Transfer Time (sec) Predicted

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5 10 15

Count

.01 .05.10 .25 .50 .75 .90.95 .99

-3 -2 -1 0 1 2 3


e) No. The outcomes of these tests are weird. The overall F-statistic isapproximately F = (0.624/(1-0.624))*(77/2) 64 is very significant (being muchlarger than 4). On the other hand, the t-statistics as seen in the tabular summary areboth less than 2. Thus, we can reject H0: 1 = 2 = 0, but cannot reject either H0: 1 = 0or H0: 2 = 0.

f) The key difference is the increase in the se of the slope. The confidence intervalfor the partial slope for file size from the multiple regression is 0.3237435 - 2 *0.179818 to 0.3237435 + 2 * 0.179818, or about -.04 to 0.68 seconds per MB a huge


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range that includes zero. The marginal slope is 0.3133 - 2 * 0.0275 to 0.3133 + 2 *0.0275, or about .2583 to .3683 seconds per MB. The estimates (slopes) are about thesame, but the range in the multiple regression is much larger.

g) The direct effect of file size (from the multiple regression) is indirect effect of filesize is 0.32 sec/MB. The indirect effect (from the simple regressions) is

(0.0562 hours since 8am/MB)* (-0.186 sec/hour after 8am) = -.0104532 sec/MBis very small. The path diagram only tells you about the difference between theindirect and direct effect (slope in the simple and multiple regression), not thechange in the standard errors.

38.Home pricesa) Some of the homes are large and expensive, making these leveraged outliers. Therelationships appear linear. One particularly large home has 7 bath bet they havesomeone else do the cleaning. The two explanatory variables are related, as youwould expect.

0

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Price($M)

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Sq Feet

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Price($M)

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Sq

Feet

1 2 3 4 5 6 7

Num Bath Rms

b)

R2 0.533512

se 81.03068



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assumptions. The concern remains the presence of the leveraged outlier. Theresiduals are nearly normal.

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Price($M)Residual

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Price ($M) Predicted

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Count

.01 .05.10 .25 .50 .75 .90.95 .99

-3 -2 -1 0 1 2 3


d) Yes. The overall F-statistic is F = (0.5335/(1-0.5335))* (150-1-2)/2 84 which ismuch larger than 4 needed to assure statistical significance.

e) The confidence interval for the marginal slope is82.3267 - 2 * 9.4291, 82.3267 + 2 * 9.4291 = 63.4685 to 101.1849

or about 63 to 101 thousand dollars per bathroom. For the partial slope, the CI is14.7939 - 2 * 11.7472, 14.7939 + 2 * 11.7472 = -8.7005 to 38.2883

or about -9 to 38 thousand dollars per bathroom. The range of the intervals iscomparable, but the estimates are rather different. The estimates change because ofthe correlation between the two explanatory variables (evident in a) which impliesa large indirect effect.

f) Shes unlikely to recover the value of the conversion from the sale price. Thevalue of converting space (the partial slope; the conversion to a bathroom does notincrease the size of the home) is from -9 to 38, and her cost of 40 thousand liesoutside this range. Dont do it (unless she just wants another bathroom).

39.Production costsa) The scatterplots are OK: roughly linear with a few troublesome outliers. Thesejobs feature expensive material costs, but relatively typical labor hours and averagecosts.

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AverageC

ost($/unit)

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ost($/unit)

.1 .2 .3 .4 .5 .6 .7 .8

Labor Hours ($/unit)


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1

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3

4

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8

Mat

erialCost($/unit)

.1 .2 .3 .4 .5 .6 .7 .8

Labor Hours ($/unit)

b) The estimated multiple regression is

R2 0.336022se 7.337964



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40.Leasesa) Other than the outliers (which are rather expensive for their size and age, markedhere with xs further below) the plots look reasonably linear, though not very strongassociation. The two explanatory variables appear unrelated, so there will be similarmarginal and partial slopes.

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CostperSq

Foot

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Foot

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Feet

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Age

b)R2 0.329793se 1.438612



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d) Yes. F= (0.3298/(1-0.3298)) * (223-1-2)/2 54 which is much larger than 4, andthus statistically significant.

e) Among leases for the same amount of office space, those in older buildings appearslightly more expensive. The average cost of a lease in a 5 year old building is about3 to 5 cents more per square foot than comparable space in a 4 year old building.

Details for the confidence interval0.035269 - 2 * 0.004673, 0.035269 + 2 * 0.004673 0.026 to 0.045

f) This model does not address the location of the buildings. This lurking variablecould have a considerable impact on the slopes in this model. Perhaps thats why theolder buildings cost more its not the age of the buildings, its the location and theolder buildings are in a nice part of town.

41.R&D expensesa) The scatterplots (all on log scales) show strongly linear trends, but between y and

the explanatory variables as well as between the explanatory variables

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R&DE

xpense

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xpense

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Assets

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b)R2 0.80991se 0.869808

Term Estimate Std Error t Ratio Prob>|t|Intercept -1.203173 0.089859 -13.39


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The model would not be suitable for prediction (ie, 95% prediction intervals wouldnot have the right coverage). The CLT suggests inferences about slopes are OK, butnot for predicting individual companies.

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R&DE

xpenseResidual

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Log R&D Expense Predicted

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Count

.001.01.05.10.25.50.75.90.95.99.999

-4 -3 -2 -1 0 1 2 3 4


d) Yes, because the t-statistic (4.3) indicates that this slope is significantly differentfrom zero. Hence, the addition of this explanatory variable significantly increases

R2.e) The partial elasticity of R&D expenses with respect to net sales is

0.2284876 - 2 * 0.053194, 0.2284876 + 2 * 0.053194 = .1220996 to .3348756or about (to presentation precision) 0.12 to 0.33. Among companies of equal assets,R&D spending averages between 0.12 to 0.33 percent higher among those with 1%higher net sales.

f) Yes, its considerably smaller. The marginal elasticity is 0.79 0.04, so theconfidence intervals for the estimates do not even overlap. The simple explanationfor the difference is that the partial elasticity estimates the effect of percentagedifferences in net sales among companies with equal assets. The marginal elasticity

includes the indirect effect: the marginal elasticity includes the benefit of havingmore assets (which itself has positive partial elasticity).

42.Carsa) The calibration and residual plot show the a small amount of curvature (the fitunderpredicts the price of the small cars) as well as large changes in the variation.

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RPActual

0 10000 30000 50000 70000

Base Price MSRP Predicted P|t|Intercept -4622.225 4796.003 -0.96 0.3440Trade Bal (%GDP) 959.60593 232.7805 4.12 0.0003Muni Waste (kg/person) 62.184369 9.153925 6.79


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with larger exports have more consumption (producing more trash), and thisconsumption contributes to GDP.

f) The 95% confidence interval for the slope for municipal waste is62.1843 - 2 * 9.1539, 62.1843 + 2 * 9.1539 = $43.8765 to $80.4921

more GDP per kilogram of waste. The se rounds to 9, would be rounded to $44 to

$80. The interval does not include zero, so that 2 is not zero. This does not meancountries should produce more waste. Rather, it means that at a given tradebalance, countries with more waste per person have larger GDP per person. Themodel is not causal.

44.Hiringa) The scatterplots seem reasonably linear, though the association is weak in eachcase. The association between the two explanatory variables is particularly weak.This plot may have two clusters of employees.

7

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Log

Profit

0 1 2 3 4 5 6 7

Log Accounts

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8

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Log

Profit

0 1 2 3 4 5 6 7 8 9 10

Log Commission

0

1

2

34

5

6

7

Log

Accounts

0 1 2 3 4 5 6 7 8 9 10

Log Commission

b) Because the association between the two explanatory variables is weak, themarginal and partial elasticities should be similar.

c) The estimated model is

RSquare 0.279374Root Mean Square Error 0.671333Observations (or Sum Wgts) 464



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d) The residuals show little pattern, though negative residuals seem more dispersed(more variable) than positive residuals. The residuals are a bit skewed, but thedeviations are only in the lower extreme. As a whole, the residuals are nearlynormal.

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ProfitResidual

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Log Profit Predicted

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Count

.01 .05.10 .25 .50 .75 .90.95 .99

-3 -2 -1 0 1 2 3


e) The confidence interval for the partial elasticity is

0.1995083 - 2 * 0.029552, 0.1995083 + 2 * 0.029552or about (to presentation precision) 0.14 to 0.26. The marginal elasticity is largerthan this interval. Looks like there was more of an indirect effect than weanticipated.

f) The path diagram shows that the partial elasticity for the number of accounts is0.20 and the partial elasticity for early commission is 0.13. The indirect effect for thelog of the number of the accounts is

.0908 0.1325 * 0.6855 (from the regression of log commission on log accounts)Notice that this checks (up to rounding errors) : the sum of direct and indirect effectsis the marginal elasticity given in the text, 0.09 + 0.20 = 0.29.

g) To answer this question requires that you believe the MRM and treat these effectsas causal. Because there could be other factors at work, thats wishful thinking. Ifyou do choose to believe the model, then go with the program that concentrates ondeveloping accounts. The partial elasticity of the number of accounts is larger thanthe partial elasticity of the early commissions, so put the effort here.

45.Promotiona) The scatterplots are vaguely linear, with weak associations between the twopredictors and the response. The largest correlation is between the two explanatory

variables, so marginal and partial slopes will likely differ.


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0.205

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MarketShare

.02 .04 .06 .08 .10 .12 .14

Detail Voice

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MarketShare

.1 .2 .3 .4 .5 .6 .7

Sample Voice

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DetailVoice

.1 .2 .3 .4 .5 .6 .7

Sample Voice

b) The estimated model isR2 0.280169se 0.006605n 39



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-0.015

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Count

.01 .05.10 .25 .50 .75 .90.95 .99

-3 -2 -1 0 1 2 3


d) Yes. F = (0.28/(1-0.28)) * (39-1-2)/2 7 > 4, so the effect is statistically significant.

e) No. The partial effect for detailing is not significantly different from zero.

f) No. The model is not causal. The partial slope for detailing is not significantlydifferent from zero (i.e., zero is in the 95% confidence interval), but this does not

mean detailing has no effect. It only means, as in the statement of the question inpart e, that at a given level of sample share, periods with a higher share ofdetailing have not shown gains in market share. Since detailing and sampling tendto come together, it is hard to separate the two. Perhaps the best advice would be todo some experiments.

46.Applea) All three variables are correlated with each other, with common outlying events(such as October 1987). The correlations are modest in size, but reasonably linear.

-0.6

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Apple

Retur

n

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Market Return

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IBM Return

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0

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MarketReturn

-0.3 -0.2 -0.1 0 .1 .2 .3 .4

IBM Return

b) The estimated model isR2

0.216589

se 0.13255

n 300


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Term Estimate Std Error t Ratio Prob>|t|Intercept 0.0048214 0.007868 0.61 0.5405Market Return 1.3168817 0.204542 6.44


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4M Leasing

a) Without an estimated value for the residual price, the manufacturer may not beable to cover costs when the cars are returns. Perhaps it should have charged morefor mileage if this factor has a large effect on resale value.

b) We need multiple regression because it is likely that the two factors are related;namely, that older cars have been driven further. If we use marginal estimates ofthese effects, for example, well in effect double count for the age of the car when weestimate the impact of mileage on the residual value. That might lead us to chargemore than we need to cover our costs. Thats OK (from the manufacturers point ofview), but we might be losing profitable sales due to charging too much.

c) Most of the curvature we have seen in previous examples with cars (See Chapter20) come from combining very different models: for example, theres more variationin attributes among very expensive cars than among cheaper cars. Also, thenonlinear patterns that come as cars lose value (you cannot lose $10,000 for many

years and stay positive) become more evident as cars get much older.d) The plots appear straight-enough, and we can see the collinearity between thetwo proposed explanatory variables. A few outlier appear in the plots, but none ofthese seem extreme.

20000

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Price

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Mileage

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e)R2 0.510372se 3178.879n 218



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Term Estimate Std Error t Ratio Prob>|t|Mileage -0.124023 0.02375 -5.22

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Ch24 Answers