Chap 012

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Chapter 12

Chapter 12 Simple Regression

Simple Regression

True / False Questions True / False Questions

1.

1. A scatteA scatter plot is user plot is used to visuald to visualize the asize the associatsociation (or lacion (or lack ofk of association) beteen to !uantitative variables.

association) beteen to !uantitative variables. "

"rue rue ##alsealse 2.

2. "he "he corcorrelrelation ation coe$coe$ciencientt r r measures the strength of the linear measures the strength of the linear relationship beteen to variables.

relationship beteen to variables. "

"rue rue ##alsealse %.

%. &&earsonearson's cor's correlrelation ation coe$ccoe$cient (ient (r r ) re!uires that both variables be) re!uires that both variables be interval or ratio data.

interval or ratio data. "

"rue rue ##alsealse 

.. ff r r * .++ and * .++ and nn * 1,- then the correlation is significant at  * ./+ in a * 1,- then the correlation is significant at  * ./+ in a to0tailed test.

to0tailed test. "

"rue rue ##alsealse +.

+. A samA sample ple corcorrerelatlationion r r * ./ indicates a * ./ indicates a stronger linear relatistronger linear relationshiponship than

than r r * 0.,/. * 0.,/. "

"rue rue ##alsealse ,.

,. A commA common souron source of spurce of spurious corious correlrelation beation beteeteenn X X and and Y Y is hen a is hen a third unspecied variable

third unspecied variable Z Z aects both aects both X X and and Y Y .. "

(2)

(3)

3.

3. "he "he corcorrelrelation ation coe$coe$ciencientt r r ala4s has the same sign as ala4s has the same sign as bb11 in in Y Y * * bb// 5 5

b b11 X X .. "

6. "he tted "he tted intercept intercept in a rin a regression egression has little has little meaning if meaning if no data valueno data valuess near

near X X * / have been * / have been observed.observed. "

7. "he leas"he least s!uaret s!uares regrs regression liession line is obtainene is obtained hen the sum of thed hen the sum of the s!uared residuals is minimized.

s!uared residuals is minimized. "

"rue rue ##alsealse 1/

1/ ..

n a simple regression- if the coe$cient for

n a simple regression- if the coe$cient for X X is positive and is positive and signicantl4 dierent from zero- then an increase in

signicantl4 dierent from zero- then an increase in X X is associated is associated ith an increase in the mean (i.e.- the

ith an increase in the mean (i.e.- the e8pected value) ofe8pected value) of Y Y .. "

"rue rue ##alsealse 11

11 ..

n least0s!uares regression- the residuals

n least0s!uares regression- the residuals ee11-- ee22- . . . -- . . . - eenn ill ala4s ill ala4s have a zero mean.

have a zero mean. "

12 ..

9hen using the least s!uares method- the

9hen using the least s!uares method- the column of residuals ala4scolumn of residuals ala4s sums to zero.

sums to zero. "

"rue rue ##alsealse 1%

1% ..

n the model

n the model SalesSales * 2,6 5 3.%3 * 2,6 5 3.%3 Ads Ads- an additional :1 spent on ads- an additional :1 spent on ads ill increase sales b4 3.%3 percent.

ill increase sales b4 3.%3 percent. "

"rue rue ##alsealse 1

1 ..

f

f RR22_{* .%, in the model}_{* .%, in the model} _Sales_Sales_{* 2,6 5 3.%3}_{* 2,6 5 3.%3}_Ads_Ads_ith_ith _n_n_{* +/- the to0}_{* +/- the to0} tailed test for

tailed test for corrcorrelation atelation at αα * ./+ ould sa4 that there is a signicant * ./+ ould sa4 that there is a signicant corr

correlation elation beteenbeteen SalesSales and and Ads. Ads.

"

(4)

1+ 1+ ..

f

f RR22_{* .%, in the model}_{* .%, in the model} _Sales_Sales_{* 2,6 5 3.%3}_{* 2,6 5 3.%3}_Ads_Ads_{- then}_{- then}_Ads_Ads_{e8plains %,}_{e8plains %,} percent of the variation in

percent of the variation in Sales.Sales.

"

"rue rue ##alsealse 1,

1, ..

"he ordinar4 le

"he ordinar4 least s!uares reast s!uares regression line ala4gression line ala4s passes through thes passes through the point .

point . "

13 ..

"he least s!uares r

"he least s!uares regression line givegression line gives unbiased estimates ofes unbiased estimates of β β// and and

β β11.. "

16 ..

n a

n a simple regression- the correlation coe$cientsimple regression- the correlation coe$cient r r is the s!uare root of is the s!uare root of

R R22_.. "

17 ..

f

f SSRSSR is 16// and is 16// and SSSSEE is 2//- then is 2//- then RR22_{is .7/.}_{is .7/.} "

"rue rue ##alsealse 2/

2/ ..

"he idth of a predic

"he idth of a prediction interval for an individual vtion interval for an individual value ofalue of Y Y is less is less than standard

than standard errorerrorssee.. "

21 ..

f

f SSESSE is near zero in a is near zero in a regrregression- the statistician ill conclude that theession- the statistician ill conclude that the proposed model probabl4 has too poor a t to be useful.

proposed model probabl4 has too poor a t to be useful. "

22 ..

#or a regression ith 2// observations- e e8pect that

#or a regression ith 2// observations- e e8pect that about 1/about 1/ residuals ill e8ceed to standard errors.

residuals ill e8ceed to standard errors. "

(5)

2% .

Condence intervals for predicted Y are less precise hen the residuals are ver4 small.

"rue #alse 2

.

Cause0and0eect direction beteen X and Y ma4 be determined b4 running the regression tice and seeing hether Y * β/ 5 β1 X or X * β1 5 β/Y has the larger R2.

"rue #alse 2+

.

"he ordinar4 least s!uares method of estimation minimizes the estimated slope and intercept.

"rue #alse 2,

.

;sing the ordinar4 least s!uares method ensures that the residuals ill be normall4 distributed.

"rue #alse 23

.

f 4ou have a strong outlier in the residuals- it ma4 represent a dierent causal s4stem.

"rue #alse 26

.

A negative correlation beteen to variables X and Y usuall4 4ields a negative p0value for r .

"rue #alse 27

.

n linear regression beteen to variables- a signicant relationship e8ists hen the p0value of the t test statistic for the slope is greater than α.

"rue #alse %/

.

"he larger the absolute value of the t statistic of the slope in a simple linear regression- the stronger the linear relationship e8ists beteen X

and Y .

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%1 .

n simple linear regression- the coe$cient of determination (R2_{) is} estimated from sums of s!uares in the A<=>A table.

"rue #alse %2

.

n simple linear regression- the p0value of the slope ill ala4s e!ual the p0value of the F statistic.

"rue #alse %%

.

An observation ith high leverage ill have a large residual (usuall4 an outlier).

"rue #alse %

.

A prediction interval for Y is narroer than the corresponding condence interval for the mean of Y .

"rue #alse %+

.

9hen X is farther from its mean- the prediction interval and condence interval for Y become ider.

"rue #alse %,

.

"he total sum of s!uares (SST ) ill never e8ceed the regression sum of s!uares (SSR).

"rue #alse %3

.

?@igh leverage? ould refer to a data point that is poorl4 predicted b4 the model (large residual).

"rue #alse %6

.

"he studentized residuals permit us to detect cases here the regression predicts poorl4.

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%7 .

A poor prediction (large residual) indicates an observation ith high leverage.

"rue #alse /

.

Ill-conditioned refers to a variable hose units are too large or too small (e.g.- :2-%-+,3).

"rue #alse 1

.

A simple decimal transformation (e.g.- from 16-271 to 16.271) often improves data conditioning.

"rue #alse 2

.

"o0tailed t-tests are often used because an4 predictor that diers signicantl4 from zero in a to0tailed test ill also be signicantl4 greater than zero or less than zero in a one0tailed test at the same α. "rue #alse

% .

A predictor that is signicant in a one0tailed t-test ill also be signicant in a to0tailed test at the same level of signicance α. "rue #alse

 .

=mission of a relevant predictor is a common source of model misspecication.

"rue #alse +

.

"he regression line must pass through the origin. "rue #alse

, .

=utliers can be detected b4 e8amining the standardized residuals. "rue #alse

(8)

3 .

n a simple regression- there are n 0 2 degrees of freedom associated ith the error sum of s!uares (SSE).

"rue #alse 6

.

n a simple regression- the F statistic is calculated b4 taking the ratio of

MSR to the MSE.

"rue #alse 7

.

"he coe$cient of determination is the percentage of the total variation in the response variable Y that is e8plained b4 the predictor X .

"rue #alse +/

.

A dierent condence interval e8ists for the mean value of Y for each dierent value of X .

"rue #alse +1

.

A prediction interval for Y is idest hen X is near its mean. "rue #alse

+2 .

n a to0tailed test for correlation at α * ./+- a sample correlation coe$cient r * /.2 ith n * 2+ is signicantl4 dierent than zero. "rue #alse

+% .

n correlation anal4sis- neither X nor Y is designated as the independent variable.

"rue #alse +

.

A negative value for the correlation coe$cient (r ) implies a negative value for the slope (b1).

"rue #alse ++

.

@igh leverage for an observation indicates that X is far from its mean. "rue #alse

(9)

+, .

Autocorrelated errors are not usuall4 a concern for regression models using cross0sectional data.

"rue #alse +3

.

"here are usuall4 several possible regression lines that ill minimize the sum of s!uared errors.

"rue #alse +6

.

9hen the errors in a regression model are not independent- the regression model is said to have autocorrelation.

"rue #alse +7

.

n a simple bivariate regression- F calc * t calc2. "rue #alse

,/ .

Correlation anal4sis primaril4 measures the degree of the linear relationship beteen X and Y .

"rue #alse

Multiple Choice Questions

,1 .

"he variable used to predict another variable is called the

A. response variable. B. regression variable. C. independent variable. . dependent variable.

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,2 .

"he standard error of the regression

A. is based on s!uared deviations from the regression line. B. ma4 assume negative values if b1 D /.

C. is in s!uared units of the dependent variable.

. ma4 be cut in half to get an appro8imate 7+ percent prediction interval.

,% .

A local trucking compan4 tted a regression to relate the travel time (da4s) of its shipments as a function of the distance traveled (miles). "he tted regression is Time * 03.12, 5 /./21 Distance- based on a

sample of 2/ shipments. "he estimated standard error of the slope is /.//+%. #ind the value of t calc to test for zero slope.

A. 2., B. +./2 C. ./ . %.1+ , .

A local trucking compan4 tted a regression to relate the travel time (da4s) of its shipments as a function of the distance traveled (miles). "he tted regression is Time * 03.12, 5 ./21 Distance- based on a

sample of 2/ shipments. "he estimated standard error of the slope is /.//+%. #ind the critical value for a right0tailed test to see if the slope is positive- using α * ./+.

A. 2.1/1

B. 2.++2

C. 1.7,/

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,+ .

f the attendance at a baseball game is to be predicted b4 the e!uation

Attendance * 1,-+// 0 3+ Temperatre- hat ould be the predicted

attendance if Temperatre is 7/ degreesE

A. ,-3+/ B. 7-3+/ C. 12-2+/ . 1/- /2/ ,, .

A h4pothesis test is conducted at the + percent level of signicance to test hether the population correlation is zero. f the sample consists of 2+ observations and the correlation coe$cient is /.,/- then the computed test statistic ould be

A. 2./31. B. 1.7,/. C. %.+73. . 1.,+. ,3 .

9hich of the folloing is not a characteristic of the F-test in a simple regressionE

A. t is a test for overall t of the model. B. "he test statistic can never be negative.

C. t re!uires a table ith numerator and denominator degrees of freedom.

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,6 .

A researcher's F8cel results are shon belo using Femlab (labor force participation rate among females) to tr4 to predict !ancer (death rate per 1//-/// population due to cancer) in the +/ ;.S. states.

9hich of the folloing statements is not trueE

A. "he standard error is too high for this model to be of an4 predictive use.

B. "he 7+ percent condence interval for the coe$cient of Femlab is 0.27 to 0/.26.

C. Signicant correlation e8ists beteenFemlab and !ancer at α * . /+.

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,7 .

A researcher's results are shon belo using Femlab (labor force

participation rate among females) to tr4 to predict !ancer (death rate per 1//-/// population due to cancer) in the +/ ;.S. states.

9hich statement is valid regarding the relationship beteen Femlab

and !ancer E

A. A rise in female labor participation rate ill cause the cancer rate to decrease ithin a state.

B. "his model e8plains about 1/ percent of the variation in state cancer rates.

C. At the ./+ level of signicance- there isn't enough evidence to sa4 the to variables are related.

. f 4our sister starts orking- the cancer rate in 4our state ill decline.

(14)

3/ .

A researcher's results are shon belo using Femlab (labor force

participation rate among females) to tr4 to predict !ancer (death rate per 1//-/// population due to cancer) in the +/ ;.S. states.

9hat is the R2_{for this regressionE}

A. .7/16 B. ./762 C. .6%7+ . .1,/+ 31 .

A nes netork stated that a stud4 had found a positive correlation beteen the number of children a orker has and his or her earnings last 4ear. Gou ma4 conclude that

A. people should have more children so the4 can get better Hobs.

B. the data are erroneous because the correlation should be negative. C. causation is in serious doubt.

. statisticians have small families. 32

.

9illiam used a sample of ,6 large ;.S. cities to estimate the

relationship beteen !rime (annual propert4 crimes per 1//-///

persons) and Income (median annual income per capita- in dollars). @is estimated regression e!uation as !rime * 26 5 /./+/ Income. 9e can conclude that

A. the slope is small so Income has no eect on !rime. B. crime seems to create additional income in a cit4.

C. ealth4 individuals tend to commit more crimes- on average.

. the intercept is irrelevant since zero median income is impossible in a large cit4.

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3% .

Iar4 used a sample of ,6 large ;.S. cities to estimate the relationship beteen !rime (annual propert4 crimes per 1//-/// persons) and

Income (median annual income per capita- in dollars). @er estimated

regression e!uation as !rime * 26 5 /./+/ Income. f Income

decreases b4 1///- e ould e8pect that !rime ill

A. increase b4 26. B. decrease b4 +/. C. increase b4 +//. . remain unchanged. 3 .

Amelia used a random sample of 1// accounts receivable to estimate the relationship beteen Da"s (number of da4s from billing to receipt of pa4ment) and Si#e (size of balance due in dollars). @er estimated regression e!uation as Da"s * 22 5 /.//3 Si#e ith a correlation coe$cient of .%//. #rom this information e can conclude that

A. 7 percent of the variation inDa"s is e8plained b4 Si#e. B. autocorrelation is likel4 to be a problem.

C. the relationship beteen Da"s and Si#e is signicant. . larger accounts usuall4 take less time to pa4.

3+ .

&rediction intervals for Y are narroest hen

A. the mean of X is near the mean of Y . B. the value of X is near the mean of X .

C. the mean of X diers greatl4 from the mean of Y . . the mean of X is small.

3, .

f n * 1+ and r * .27,- the corresponding t 0statistic to test for zero correlation is

A. 1.31+.

B. 3.6,2.

C. 2./6.

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33 .

;sing a to0tailed test at α * ./+ for n * %/- e ould reHect the h4pothesis of zero correlation if the absolute value of r e8ceeds

A. .2772. B. .%,/7. C. ./2+/. . .2//. 36 .

"he ordinar4 least s!uares (=JS) method of estimation ill minimize

A. neither the slope nor the intercept. B. onl4 the slope.

C. onl4 the intercept.

. both the slope and intercept. 37

.

A standardized residual ei * 02.2/+ indicates

A. a rather poor prediction.

B. an e8treme outlier in the residuals. C. an observation ith high leverage. . a likel4 data entr4 error.

6/ .

n a simple regression- hich ould suggest a signicant relationship beteen X and Y E

A. Jarge p0value for the estimated slope B. Jarge t statistic for the slope

C. Jarge p0value for the F statistic . Small t 0statistic for the slope 61

.

9hich is indicative of an inverse relationship beteen X and Y E

A. A negative F statistic

B. A negative p0value for the correlation coe$cient C. A negative correlation coe$cient

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62 .

9hich is not correct regarding the estimated slope of the =JS regression lineE

A. t is divided b4 its standard error to obtain itst statistic. B. t shos the change in Y for a unit change in X .

C. t is chosen so as to minimize the sum of s!uared errors. . t ma4 be regarded as zero if its p0value is less than α. 6%

.

Simple regression anal4sis means that

A. the data are presented in a simple and clear a4. B. e have onl4 a fe observations.

C. there are onl4 to independent variables. . e have onl4 one e8planator4 variable. 6

.

"he sample coe$cient of correlation does not have hich propert4E

A. t can range from 01.// up to 51.//. B. t is also sometimes called &earson's r . C. t is tested for signicance using a t 0test. . t assumes that Y is the dependent variable. 6+

.

9hen comparing the 7/ percent prediction and condence intervals for a given regression anal4sis

A. the prediction interval is narroer than the condence interval. B. the prediction interval is ider than the condence interval. C. there is no dierence beteen the size of the prediction and

condence intervals.

. no generalization is possible about their comparative idth. 6,

.

9hich is not true of the coe$cient of determinationE

A. t is the s!uare of the coe$cient of correlation.

B. t is negative hen there is an inverse relationship beteen X and Y . C. t reports the percent of the variation inY e8plained b4 X .

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63 .

f the tted regression is Y * %.+ 5 2.1 X (R2_{* .2+-} _n_{* 2+)- it is}

incorrect to conclude that

A. Y increases 2.1 percent for a 1 percent increase in X . B. the estimated regression line crosses the Y a8is at %.+. C. the sample correlation coe$cient must be positive. . the value of the sample correlation coe$cient is /.+/. 66

.

n a simple regression Y * b/ 5 b1 X here Y * number of robberies in a cit4 (thousands of robberies)- X * size of the police force in a cit4

(thousands of police)- and n * + randoml4 chosen large ;.S. cities in 2//6- e ould be least likel4 to see hich problemE

A. Autocorrelated residuals (because this is time0series data)

B. @eteroscedastic residuals (because e are using totals uncorrected for cit4 size)

C. <onnormal residuals (because a fe larger cities ma4 ske the residuals)

. @igh leverage for some observations (because some cities ma4 be huge)

67 .

9hen homoscedasticit4 e8ists- e e8pect that a plot of the residuals versus the tted Y 

A. ill form appro8imatel4 a straight line. B. crosses the centerline too man4 times. C. ill 4ield a urbin09atson statistic near 2. . ill sho no pattern at all.

(19)

7/ .

9hich statement is not correctE

A. Spurious correlation can often be reduced b4 e8pressing X and Y in per capita terms.

B. Autocorrelation is mainl4 a concern if e are using time0series data. C. @eteroscedastic residuals ill have roughl4 the same variance for

an4 value of X .

. Standardized residuals make it eas4 to identif4 outliers or instances of poor t.

71 .

n a simple bivariate regression ith 2+ observations- hich statement is most nearl" correct E

A. A non0standardized residual hose value is ei * .22 ould be considered an outlier.

B. A leverage statistic of /.1, or more ould indicate high leverage. C. Standardizing the residuals ill eliminate an4 heteroscedasticit4. . <on0normal residuals impl4 biased coe$cient estimates- a maHor

problem. 72

.

A regression as estimated using these variables Y * annual value of reported bank robber4 losses in all ;.S. banks (:millions)- X * annual value of currenc4 held b4 all ;.S. banks (:millions)- n * 1// 4ears (1712 through 2/11). 9e ould not anticipate

A. autocorrelated residuals due to time0series data.

B. heteroscedastic residuals due to the ide variation in data magnitudes.

C. nonnormal residuals due to skeed data as bank size increases over time.

. a negative slope because banks hold less currenc4 hen the4 are robbed.

(20)

7% .

A tted regression for an e8am in &rof. @ardtack's class shoed Score

* 2/ 5 3 Std" - here Score is the student's e8am score and Std" is the student's stud4 hours. "he regression 4ielded R2_{* /.+/ and} _SE_* 6. Bob studied 7 hours. "he !uick 7+ percent prediction interval for Bob's grade is appro8imatel4

A. ,7 to 73. B. 3+ to 71. C. ,3 to 77. . 3, to 7/. 7 .

9hich is not an assumption of least s!uares regressionE

A. <ormal X values B. <on0autocorrelated errors C. @omoscedastic errors . <ormal errors 7+ .

n a simple bivariate regression ith ,/ observations there ill be KKKKK residuals. A. ,/ B. +7 C. +6 . +3 7, .

9hich is correct to nd the value of the coe$cient of determination (R2_)E

A. SSRLSSE

B. SSRLSST

(21)

73 .

"he critical value for a to0tailed test of $/ β1 * / at α * ./+ in a simple regression ith 22 observations is

A. M1.32+ B. M2./6, C. M2.+26 . M1.7,/ 76 .

n a sample of size n * 2%- a sample correlation of r * .// provides su$cient evidence to conclude that the population correlation

coe$cient e8ceeds zero in a right0tailed test at

A. α * ./1 but not α * ./+. B. α * ./+ but not α * ./1. C. both α * ./+ and α * ./1. . neither α * ./+ nor α * ./1. 77 .

n a sample of n * 2%- the Student's t test statistic for a correlation of r

* .+// ould be

A. 2.++7.

B. 2.617.

C. 2.,,.

. can't sa4 ithout knoing α. 1//

.

n a sample of n * 2%- the critical value of the correlation coe$cient for a to0tailed test at α * ./+ is

A. M.+2

B. M.12

C. M.+//

(22)

1/1 .

n a sample of n * 2%- the critical value of Student's t for a to0tailed test of signicance for a simple bivariate regression at α * ./+ is

A. M2.227 B. M2.617 C. M2.,, . M2./6/ 1/2 .

n a sample of n * /- a sample correlation of r * .// provides su$cient evidence to conclude that the population correlation coe$cient e8ceeds zero in a right0tailed test at

A. α * ./2+ but not α * ./+. B. α * ./+ but not α * ./2+. C. both α * ./2+ and α * ./+. . neither α * ./2+ nor α * ./+. 1/% .

n a sample of n * 2/- the Student's t test statistic for a correlation of

r * .// ould be

A. 2.11/

B. 1.,+

C. 1.6+2

. can't sa4 ithout knoing if it's a to0tailed or one0tailed test.

1/ .

n a sample of n * 2/- the critical value of the correlation coe$cient for a to0tailed test at α * ./+ is

A. M.+63

B. M.12

C. M.

(23)

1/+ .

n a sample of n * 23- the critical value of Student's t for a to0tailed test of signicance for a simple bivariate regression at α * ./+ is

A. M2./,/ B. M2./+2 C. M2.676 . M2./3 1/, .

n a sample of size n * %,- a sample correlation of r * 0.+/ provides su$cient evidence to conclude that the population correlation

coe$cient diers signicantl4 from zero in a to0tailed test at

A. α * ./1 B. α * ./+ C. both α * ./1 and α * ./+. . neither α * ./1 nor α * ./+. 1/3 .

n a sample of n * %,- the Student's t test statistic for a correlation of

r * 0.+/ ould be

A. 02.11/.

B. 02.7%6.

C. 02./%/.

. can't sa4 ithout knoing α. 1/6

.

n a sample of n * %,- the critical value of the correlation coe$cient for a to0tailed test at α * ./+ is

A. M.%27

B. M.%63

C. M.2%

(24)

1/7 .

n a sample of n * %,- the critical value of Student's t for a to0tailed test of signicance of the slope for a simple regression at α * ./+ is

A. 2.7%6 B. 2.32 C. 2./%2 . 2./3 11/ .

A local trucking compan4 tted a regression to relate the travel time (da4s) of its shipments as a function of the distance traveled (miles). "he tted regression is Time * 03.12, 5 /./21 Distance. f Distance

increases b4 +/ miles- the e8pected Time ould increase b4

A. 1./3 da4s

B. 3.1% da4s

C. 2.1 da4s

(25)

111 .

A local trucking compan4 tted a regression to relate the cost of its shipments as a function of the distance traveled. "he F8cel tted regression is shon.

Based on this estimated relationship- hen distance increases b4 +/ miles- the e8pected shipping cost ould increase b4

A. :26,. B. :1%. C. :1/. . :%/1. 112 .

f SSR is 2+72 and SSE is ,/6- then

A. the slope is likel4 to be insignicant. B. the coe$cient of determination is .61. C. the SST ould be smaller than SSR. . the standard error ould be large.

(26)

11% .

#ind the sample correlation coe$cient for the folloing data.

A. .6711 B. .712 C. .7622 . .7++, 11 .

#ind the slope of the simple regression * b/ 5 b1 % .

A. 1.6%%

B. %.27

C. /.3,2

(27)

11+ .

#ind the sample correlation coe$cient for the folloing data.

A. .3271 B. .63%, C. .7116 . .7+,% 11, .

#ind the slope of the simple regression * b/ 5 b1 % .

A. 2.+7+

B. 1.1/7

C. 02.221

(28)

113 .

A researcher's results are shon belo using n * 2+ observations.

"he 7+ percent condence interval for the slope is

A. N 0%.262- 01.26O. B. N 0.%7- 0/.213O. C. N1.116- +./2,O. . N 0/.776- 5/.776O. 116 .

A researcher's regression results are shon belo using n * 6 observations.

"he 7+ percent condence interval for the slope is

A. N1.%%%- 2.26O. B. N1.,/2- 2./,O. C. N1.2,6- 2.%76O. . N1.116- 2.7O.

(29)

117 .

Bob thinks there is something rong ith F8cel's tted regression. 9hat do 4ou sa4E

A. "he estimated e!uation is obviousl4 incorrect. B. "he R2_{looks a little high but otherise it looks =P.} C. Bob needs to increase his sample size to decide. . "he relationship is linear- so the e!uation is credible.

(30)

12/ .

&edro became interested in vehicle fuel e$cienc4- so he performed a simple regression using 7% cars to estimate the model !it"M&' * β/ 5

β1 (eig)t here (eig)t is the eight of the vehicle in pounds. @is results are shon belo. 9rite a brief anal4sis of these results- using hat 4ou have learned in this chapter. s the intercept meaningful in this regressionE Iake a prediction of !it"M&' hen (eig)t * %///-and also hen (eig)t * ///. o these predictions seem believableE f 4ou could make a car 1/// pounds lighter- hat change ould 4ou predict in its !it"M&'E

(31)

121 .

Iar4 noticed that old coins are smoother and more orn. She eighed %1 nickels and recorded their age- and then performed a simple regression to estimate the model (eig)t * β/ 5 β1 Age here eight is the eight of the coin in grams and Age is the age of the coin in 4ears. @er results are shon belo. 9rite a brief anal4sis of these results- using hat 4ou have learned in this chapter. Iake a prediction of (eig)t hen Age * 1/- and also hen Age * 2/. 9hat does this tell 4ouE s the intercept meaningful in this regressionE

(32)

Chapter 12 Simple Regression Anser Pe4

True / False Questions

1. A scatter plot is used to visualize the association (or lack of association) beteen to !uantitative variables.

TRUE

"he scatter plot shos association beteen to !uantitative variables.

AA!S*+ Anal"tic *looms+ Remember

Diclt"+  Eas" /earning 0b1ecti2e+ 3-4 !alclate and test a correlation coecient 5or signi6cance.

Topic+ 7isal Displa"s and !orrelation Anal"sis

2. "he correlation coe$cient r measures the strength of the linear relationship beteen to variables.

TRUE

A correlation coe$cient measures linearit4 beteen to variables.

(33)

%. &earson's correlation coe$cient (r ) re!uires that both variables be interval or ratio data.

TRUE

Correlation assumes !uantitative data ith at least interval measurements.

. f r * .++ and n * 1,- then the correlation is significant at  * ./+ in a to0tailed test. TRUE t calc * r N(n 0 2)L(1 0 r 2)O1L2 * (.++)N(1, 0 2)L(1 0 .++2)O1L2 * 2., Q t ./2+ * 2.1+ for d.5. * 1, 0 2 * 1. AA!S*+ Anal"tic *looms+ Appl" Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4 !alclate and test a correlation coecient 5or signi6cance. Topic+ 7isal Displa"s and !orrelation Anal"sis

+. A sample correlation r * ./ indicates a stronger linear relationship than r * 0.,/.

FALSE

"he sign onl4 indicates the direction- not the strength- of the linear relationship.

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+  Eas" /earning 0b1ecti2e+ 3-4 !alclate and test a correlation coecient 5or signi6cance.

(34)

,. A common source of spurious correlation beteen X and Y is hen a third unspecied variable Z aects both X and Y .

TRUE

Both X and Y could be inuenced b4 Z .

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+  Eas" /earning 0b1ecti2e+ 3-4 !alclate and test a correlation coecient 5or signi6cance.

3. "he correlation coe$cient r ala4s has the same sign as b1 in Y * b/ 5 b1 X .

TRUE

"he t 0test for the slope in simple regression gives the same result as the t 0test for r .

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+  Eas" /earning 0b1ecti2e+ 3-49 Fit a simple regression on an E%cel scatter plot.

Topic+ Regression Terminolog"

6. "he tted intercept in a regression has little meaning if no data values near X * / have been observed.

TRUE

&redicting Y for X * / makes little sense if the observed data have no values near X * /.

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+  Eas" /earning 0b1ecti2e+ 3-43 Interpret t)e slope and intercept o5 a r egression e:ation.

(35)

7. "he least s!uares regression line is obtained hen the sum of the s!uared residuals is minimized.

TRUE

"he =JS method minimizes the sum of s!uared residuals.

Diclt"+  Eas" /earning 0b1ecti2e+ 3-49 Fit a simple regression on an E%cel scatter plot.

Topic+ 0rdinar" /east S:ares Formlas

1/. n a simple regression- if the coe$cient for X is positive and

signicantl4 dierent from zero- then an increase in X is associated ith an increase in the mean (i.e.- the e8pected value) of Y .

TRUE

"he conditional mean ofY depends on X (unless the slope is eectivel4 zero).

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+  Eas" /earning 0b1ecti2e+ 3-43 Interpret t)e slope and intercept o5 a r egression e:ation.

Topic+ Simple Regression

11. n least0s!uares regression- the residuals e1- e2- . . . - en ill ala4s have a zero mean.

TRUE

"he residuals must sum to zero if the =JS method is used- so their mean is zero.

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-43 Interpret t)e slope and intercept o5 a r egression e:ation. Topic+ 0rdinar" /east S:ares Formlas

(36)

12. 9hen using the least s!uares method- the column of residuals ala4s sums to zero.

TRUE

"he residuals must sum to zero if the =JS method is used.

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-43 Interpret t)e slope and intercept o5 a r egression e:ation. Topic+ 0rdinar" /east S:ares Formlas

1%. n the model Sales * 2,6 5 3.%3 Ads- an additional :1 spent on ads ill increase sales b4 3.%3 percent.

FALSE

"he slope coe$cient is in the same units as Y (dollars- not percent-in this case).

AA!S*+ Anal"tic *looms+ Appl" Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-43 Interpret t)e slope and intercept o5 a r egression e:ation. Topic+ Simple Regression

1. f R2_{* .%, in the model} _Sales_{* 2,6 5 3.%3}_Ads_ith _n_{* +/- the to0} tailed test for correlation at α * ./+ ould sa4 that there is a

signicant correlation beteen Sales and Ads. TRUE t calc * r N(n 0 2)L(1 0 r 2)O1L2 * (.,/)N(+/ 0 2)L(1 0 .%,)O1L2 * +.17, Q t ./2+ * 2./11 for d.5. * +/ 0 2 * 6. AA!S*+ Anal"tic *looms+ Appl" Diclt"+ ; $ard /earning 0b1ecti2e+ 3-4 !alclate and test a correlation coecient 5or signi6cance. Topic+ 7isal Displa"s and !orrelation Anal"sis

(37)

1+. f R2_{* .%, in the model} _Sales_{* 2,6 5 3.%3}_Ads_{- then}_Ads_e8plains %, percent of the variation in Sales.

TRUE

9e can interpret R2_{as the fraction of variation in} _Y_{e8plained b4}_X (e8pressed as a percent).

AA!S*+ Anal"tic *looms+ Appl" Diclt"+  Eas" /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test.

1,. "he ordinar4 least s!uares regression line ala4s passes through the point .

TRUE

"he =JS formulas re!uire the line to pass through this point.

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-43 Interpret t)e slope and intercept o5 a r egression e:ation. Topic+ Regression Terminolog"

13. "he least s!uares regression line gives unbiased estimates of β/ and

β1.

TRUE

"he e8pected values of the =JS estimators b/ and b1 are the true parameters β/ and β1.

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-49 Fit a simple regression on an E%cel scatter plot. Topic+ 0rdinar" /east S:ares Formlas

(38)

16. n a simple regression- the correlation coe$cient r is the s!uare root of R2_.

TRUE

n fact- e could use the notation r 2_{instead of} _R2_{hen talking about}

simple regression.

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. Topic+ 0rdinar" /east S:ares Formlas

17. f SSR is 16// and SSE is 2//- then R2_{is .7/.}

TRUE

R2_* _SSR_L_SST_* _SSR_L(_SSR₅ _SSE_{) * 16//L(16// 5 2//) * .7/.}

AA!S*+ Anal"tic *looms+ Appl" Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. Topic+ Tests 5or Signi6cance

2/. "he idth of a prediction interval for an individual value ofY is less than standard error se.

FALSE

"he formula for the interval idth multiplies the standard error b4 an e8pression Q 1.

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4? Distingis) bet@een con6dence and prediction inter2als 5or Y.

(39)

21. f SSE is near zero in a regression- the statistician ill conclude that the proposed model probabl4 has too poor a t to be useful.

FALSE

SSF is the sum of the s!uare residuals- hich ould be smaller if the t is good.

AA!S*+ Anal"tic *looms+ Appl" Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. Topic+ Tests 5or Signi6cance

22. #or a regression ith 2// observations- e e8pect that about 1/ residuals ill e8ceed to standard errors.

TRUE

f the residuals are normal- 7+. percent (17/ of 2//) ill lie ithin M2se (so 1/ outside).

AA!S*+ Anal"tic *looms+ Appl" Diclt"+ 3 Medim /earning 0b1ecti2e+ 3- Identi5" nsal residals and )ig)-le2erage obser2ations. Topic+ 8nsal 0bser2ations

2%. Condence intervals for predicted Y are less precise hen the residuals are ver4 small.

FALSE

Small residuals impl4 a small standard error and thus a narro@er

prediction interval.

(40)

2. Cause0and0eect direction beteen X and Y ma4 be determined b4 running the regression tice and seeing hether Y * β/ 5 β1 X or X *

β1 5 β/Y has the largerR2.

FALSE

Cause and eect cannot be determined in the conte8t of simple regression models.

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-43 Interpret t)e slope and intercept o5 a r egression e:ation. Topic+ Simple Regression

2+. "he ordinar4 least s!uares method of estimation minimizes the estimated slope and intercept.

FALSE

=JS minimizes the sum of s!uared residuals.

2,. ;sing the ordinar4 least s!uares method ensures that the residuals ill be normall4 distributed.

FALSE

=JS produces unbiased estimates but cannot ensure normalit4 of the residuals.

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4 Test residals 5or 2iolations o5 regression assmptions. Topic+ Residal Tests

(41)

23. f 4ou have a strong outlier in the residuals- it ma4 represent a dierent causal s4stem.

TRUE

=utliers might come from a dierent population or causal s4stem.

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+ 3 Medim /earning 0b1ecti2e+ 3- Identi5" nsal residals and )ig)-le2erage obser2ations. Topic+ 0t)er Regression &roblems 0ptionalB

26. A negative correlation beteen to variables X and Y usuall4 4ields a negative p0value for r .

FALSE

"he p0value cannot be negative.

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4C Test )"pot)eses abot t)e slope and intercept b" sing t tests. Topic+ 7isal Displa"s and !orrelation Anal"sis

27. n linear regression beteen to variables- a signicant relationship e8ists hen the p0value of the t test statistic for the slope is greater than α.

FALSE

ReHect β1 * / if the p0value is less t)an α.

AA!S*+ Anal"tic *looms+ Appl" Diclt"+  Eas" /earning 0b1ecti2e+ 3-4C Test )"pot)eses abot t)e slope and intercept b" sing t tests.

(42)

%/. "he larger the absolute value of the t statistic of the slope in a simple linear regression- the stronger the linear relationship e8ists beteen X and Y .

TRUE

"he correlation coe$cient measures linearit4- regardless of its sign (5 or 0).

AA!S*+ Anal"tic *looms+ Appl" Diclt"+  Eas" /earning 0b1ecti2e+ 3-4C Test )"pot)eses abot t)e slope and intercept b" sing t tests.

Topic+ Tests 5or Signi6cance

%1. n simple linear regression- the coe$cient of determination (R2_{) is} estimated from sums of s!uares in the A<=>A table.

TRUE

R2_* _SSR_L_SST_or _R2_{* 1 0} _SSE_L_SST_.

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. Topic+ 0rdinar" /east S:ares Formlas

%2. n simple linear regression- the p0value of the slope ill ala4s e!ual the p0value of the F statistic.

TRUE

"his is true onl4 if there is one predictor (but is no longer true in multiple regression).

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. Topic+ Anal"sis o5 7ariance+ 02erall Fit

(43)

%%. An observation ith high leverage ill have a large residual (usuall4 an outlier).

FALSE

"he concepts are distinct (a high0leverage point could have a good t).

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+ 3 Medim /earning 0b1ecti2e+ 3- Identi5" nsal residals and )ig)-le2erage obser2ations. Topic+ 8nsal 0bser2ations

%. A prediction interval for Y is narroer than the corresponding condence interval for the mean of Y .

FALSE

&redicting an individual case re!uires a ider condence interval than predicting the mean.

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4? Distingis) bet@een con6dence and prediction inter2als 5or Y.

Topic+ !on6dence and &rediction Inter2als 5or Y

%+. 9hen X is farther from its mean- the prediction interval and condence interval for Y become ider.

TRUE

"he idth increases hen X diers from its mean (revie the formula).

(44)

%,. "he total sum of s!uares (SST ) ill never e8ceed the regression sum of s!uares (SSR).

FALSE

"he identit4 is SSR 5 SSE * SST .

Diclt"+  Eas" /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test.

Topic+ Anal"sis o5 7ariance+ 02erall Fit

%3. ?@igh leverage? ould refer to a data point that is poorl4 predicted b4 the model (large residual).

FALSE

A high0leverage observation ma4 have a good t (onl4 its X value determines its leverage).

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3- Identi5" nsal residals and )ig)-le2erage obser2ations. Topic+ 8nsal 0bser2ations

%6. "he studentized residuals permit us to detect cases here the regression predicts poorl4.

TRUE

Studentized residuals resemble a t 0distribution. A large studentized t 0 value (e.g.- t D 02.// or t Q 5 2.//) ould implies a poor t.

(45)

%7. A poor prediction (large residual) indicates an observation ith high leverage.

FALSE

@igh leverage indicates an unusuall4 large or small  value (not a poor prediction). A high0leverage observation ma4 have a good t or a poor t. =nl4 its X value determines its leverage.

/. Ill-conditioned refers to a variable hose units are too large or too small (e.g.- :2-%-+,3).

TRUE

n F8cel- a s4mptom of poor data conditioning is e8ponential notation (e.g.- .%F 5 /,).

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4 &er5orm r egression anal"sis @it) E%cel or ot)er so5t@are. Topic+ 0t)er Regression &roblems 0ptionalB

1. A simple decimal transformation (e.g.- from 16-271 to 16.271) often improves data conditioning.

TRUE

Peeping data magnitudes similar helps avoid e8ponential notation (e.g.- .%F 5 /,).

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4 &er5orm r egression anal"sis @it) E%cel or ot)er so5t@are. Topic+ 0t)er Regression &roblems 0ptionalB

(46)

2. "o0tailed t-tests are often used because an4 predictor that diers signicantl4 from zero in a to0tailed test ill also be signicantl4 greater than zero or less than zero in a one0tailed test at the same

α.

TRUE

"rue because the critical t is larger in the to0tailed test (the default in most softare).

AA!S*+ Anal"tic *looms+ Appl" Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4C Test )"pot)eses abot t)e slope and intercept b" sing t tests. Topic+ Tests 5or Signi6cance

%. A predictor that is signicant in a one0tailed t-test ill also be signicant in a to0tailed test at the same level of signicance α.

FALSE

#alse because the critical t ould be larger in a to0tailed test.

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4C Test )"pot)eses abot t)e slope and intercept b" sing t tests. Topic+ Tests 5or Signi6cance

. =mission of a relevant predictor is a common source of model misspecication.

TRUE

n a multivariate orld- simple regression ma4 be inade!uate.

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4 &er5orm r egression anal"sis @it) E%cel or ot)er so5t@are. Topic+ 0t)er Regression &roblems 0ptionalB

(47)

+. "he regression line must pass through the origin.

FALSE

"he =JS intercept estimate does not- in general- e!ual zero. 9e might be unable to reHect a zero intercept if a t 0test- but the tted intercept is rarel4 zero.

Diclt"+  Eas" /earning 0b1ecti2e+ 3-49 Fit a simple regression on an E%cel scatter plot.

,. =utliers can be detected b4 e8amining the standardized residuals.

TRUE

A poor t implies a large t 0value (e.g.- larger than M% ould be an outlier).

Diclt"+  Eas" /earning 0b1ecti2e+ 3- Identi5" nsal residals and )ig)-le2erage obser2ations.

Topic+ 8nsal 0bser2ations

3. n a simple regression- there are n 0 2 degrees of freedom associated ith the error sum of s!uares (SSE).

TRUE

"his is true in simple regression because e estimate to parameters ( β/ and β1).

Diclt"+  Eas" /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test.

(48)

6. n a simple regression- the F statistic is calculated b4 taking the ratio of MSR to the MSE.

TRUE

B4 denition- F calc * MSRMSE (obtained from the A<=>A table).

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. Topic+ Anal"sis o5 7ariance+ 02erall Fit

7. "he coe$cient of determination is the percentage of the total

variation in the response variable Y that is e8plained b4 the predictor

X .

TRUE

R2_* _SSR_L_SST_or _R2_{* 1 0} _SSE_L_SST_{lies beteen / and 1 and often is} e8pressed as a percent.

AA!S*+ Anal"tic *looms+ 8nderstand Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. Topic+ 0rdinar" /east S:ares Formlas

+/. A dierent condence interval e8ists for the mean value of Y for each dierent value of X .

TRUE

Both the interval idth and also E(Y T X ) * β/ 5 β1 X depend on the value of X .

(49)

+1. A prediction interval for Y is idest hen X is near its mean.

FALSE

"he prediction interval is narro@est hen X is near its mean. Revie the formula- hich has a term ( % i - )2 in the numerator. "he

minimum ould be hen % i  .

Topic+ !on6dence and &rediction Inter2als 5or Y

+2. n a to0tailed test for correlation at α * ./+- a sample correlation coe$cient r * /.2 ith n * 2+ is signicantl4 dierent than zero.

TRUE t calc * r N(n 0 2)L(1 0 r 2)O1L2 * (.2)N(2+ 0 2)L(1 0 .22)O1L2 * 2.217 Q t ./2+ * 2./,7 for d.5. * 2+ 0 2 * 2%. AA!S*+ Anal"tic *looms+ Appl" Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4 !alclate and test a correlation coecient 5or signi6cance. Topic+ 7isal Displa"s and !orrelation Anal"sis

+%. n correlation anal4sis- neither X nor Y is designated as the independent variable.

TRUE

n correlation anal4sis- X and Y covar4 ithout designating either as ?independent.?

(50)

+. A negative value for the correlation coe$cient (r ) implies a negative value for the slope (b1).

TRUE

"he sign of r must be the same as the sign of the slope estimateb1.

++. @igh leverage for an observation indicates that X is far from its mean.

TRUE

B4 denition- observations have higher leverage hen X is far from its mean.

AA!S*+ Anal"tic *looms+ Remember Diclt"+ 3 Medim /earning 0b1ecti2e+ 3- Identi5" nsal residals and )ig)-le2erage obser2ations. Topic+ 8nsal 0bser2ations

+,. Autocorrelated errors are not usuall4 a concern for regression models using cross0sectional data.

TRUE

9e more often e8pect autocorrelated residuals in time series data.

Diclt"+  Eas" /earning 0b1ecti2e+ 3-4 Test residals 5or 2iolations o5 regression assmptions.

(51)

+3.

+3. "her"here are e are usuallusuall4 seve4 several posral possible rsible regregressioession lines tn lines that ihat ill minll minimizimizee the sum of

the sum of s!uared erros!uared errors.rs.

FALSE FALSE

"he =JS solution for the estim

"he =JS solution for the estimatorsators bb// and and bb11 is uni!ue. is uni!ue.

AA!S*+ Anal"tic AA!S*+ Anal"tic *looms+ Remember *looms+ Remember Diclt"+  Eas" Diclt"+  Eas" /earning 0b1ecti2e+ 3-49 Fit a simple regression on an

/earning 0b1ecti2e+ 3-49 Fit a simple regression on an E%cel scatter plot.E%cel scatter plot. Topic+ 0rdinar" /east S:ares Formlas Topic+ 0rdinar" /east S:ares Formlas

+6.

+6. 9hen 9hen the ethe errrrors iors in a rn a regregressioession moden model arl are not e not indepindependeendent- thnt- thee regr

regression model is ession model is said to said to have autocorrelation.have autocorrelation.

TRUE TRUE

#or e8ample- in rst0order autocorrelation

#or e8ample- in rst0order autocorrelation GGt t depends on depends on GGt t 0101..

AA!S*+ Anal"tic AA!S*+ Anal"tic *looms+ Remember *looms+ Remember Diclt"+  Eas" Diclt"+  Eas" /earning 0b1ecti2e+ 3-4 T

/earning 0b1ecti2e+ 3-4 Test residals 5or est residals 5or 2iolations o5 regression assmptions.2iolations o5 regression assmptions. Topic+ Residal Tests Topic+ Residal Tests

+7.

+7. n n a sa simimple ple bivbivariariate ate rregregressessionion-- F F calccalc * * t t calccalc22..

TRUE TRUE

"his statement is true onl4

"his statement is true onl4 in ain a simplesimple regression (one predictor). regression (one predictor).

AA!S*+ Anal"tic AA!S*+ Anal"tic *looms+ Remember *looms+ Remember Diclt"+ 3 Medim Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. Topic+ Anal"sis o5 7ariance+ 02erall Fit Topic+ Anal"sis o5 7ariance+ 02erall Fit

(52)

,/.

,/. CorCorrerelatlation anion anal4al4sis prsis primaimarilril4 me4 measurasures thees the degreedegree of the linear of the linear relationship beteen

relationship beteen X X and and Y Y ..

TRUE TRUE

"he sign of

"he sign of r r indicates the indicates the directiondirection and its magnitude indicates the and its magnitude indicates the

degree

degree of of linearit4linearit4..

AA!S*+ Anal"tic AA!S*+ Anal"tic *looms+ Remember *looms+ Remember Diclt"+ 3 Medim Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4 !alclate and test

/earning 0b1ecti2e+ 3-4 !alclate and test a correlation coecient 5or a correlation coecient 5or signi6cance.signi6cance. Topic+ 7isal Displa"s and !orrelation Anal"sis Topic+ 7isal Displa"s and !orrelation Anal"sis

Multiple Choice Questions Multiple Choice Questions

,1.

,1. "he v"he variablariable usee used to d to prepredict dict anothanother ver variablariable is e is callecalled thed the

A.

A. response response variable.variable. B.

B. regression regression variable.variable.

C.

C. independent independent variable.variable. .

. dependent dependent variable.variable. 9

9e might also call the e might also call the independent variable aindependent variable a predictor predictor of of Y Y ..

AA!S*+ Anal"tic AA!S*+ Anal"tic *looms+ Remember *looms+ Remember Diclt"+  Eas" Diclt"+  Eas" /earning 0b1ecti2e

/earning 0b1ecti2e+ 3-43 Interpret t)e slope + 3-43 Interpret t)e slope and intercept o5 a rand intercept o5 a r egression e:ation.egression e:ation. Topic+ Simple Regression Topic+ Simple Regression

(53)

,2.

,2. "he "he stastandandard rd ererroror of r of the the reregrgressessionion

A.

A. is based on s!uared deviations fris based on s!uared deviations from the rom the regression line.egression line. B.

B. ma4 ma4 assume assume negative negative values values ifif bb11 D /. D /. C.

C. is is in in s!uared s!uared units units of of the the dependent dependent variable.variable. .

. ma4 be cut in half to get an ma4 be cut in half to get an appro8appro8imate 7+ percent prediimate 7+ percent predictionction interval.

interval.

n a simple regression- the standard error is the s!uare root of the n a simple regression- the standard error is the s!uare root of the sum of the s!uared residuals divided b4 (

sum of the s!uared residuals divided b4 (nn 0 2). 0 2).

AA!S*+ Anal"tic AA!S*+ Anal"tic *looms+ Appl" *looms+ Appl" Diclt"+ 3 Medim Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. Topic+ Tests 5or Signi6cance Topic+ Tests 5or Signi6cance

,%.

,%. A locaA local trul truckincking compag compan4 ttn4 tted a red a regregressioession to rn to relate elate the trthe travel tavel timeime (da4s) of its shipments as

(da4s) of its shipments as a function of a function of the distance traveled (miles).the distance traveled (miles). "he tted regr

"he tted regression isession is TimeTime * 03.12, 5 /./21 * 03.12, 5 /./21 DistanceDistance- based on a- based on a sample of 2/ shipments. "he estimated standard error of the slope is sample of 2/ shipments. "he estimated standard error of the slope is /.//+%. #ind the value of

/.//+%. #ind the value of t t calccalc to test for zero slope. to test for zero slope.

A. A. 2.,2., B. B. +./2+./2 C. C. ././ . . %.1+%.1+ t t calccalc * * * * (/./21)L(/.(/./21)L(/.//+%) //+%) * * ./%6../%6. AA!S*+ Anal"tic AA!S*+ Anal"tic *looms+ Appl" *looms+ Appl" Diclt"+ 3 Medim Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4C T

/earning 0b1ecti2e+ 3-4C Test )"pot)eses abot t)e est )"pot)eses abot t)e slope and intercept b" slope and intercept b" sing t tests.sing t tests. Topic+ Tests 5or Signi6cance Topic+ Tests 5or Signi6cance

(54)

,. A local trucking compan4 tted a regression to relate the travel time (da4s) of its shipments as a function of the distance traveled (miles). "he tted regression is Time * 03.12, 5 ./21 Distance- based on a

sample of 2/ shipments. "he estimated standard error of the slope is /.//+%. #ind the critical value for a right0tailed test to see if the slope is positive- using α * ./+.

A. 2.1/1

B. 2.++2

C. 1.7,/

D. 1.3%

#or d.5. * n 0 2 * 2/ 0 2 * 16- Appendi8  gives t ./+ * 1.3%.

AA!S*+ Anal"tic *looms+ Appl" Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4C Test )"pot)eses abot t)e slope and intercept b" sing t tests. Topic+ Tests 5or Signi6cance

,+. f the attendance at a baseball game is to be predicted b4 the

e!uation Attendance * 1,-+// 0 3+ Temperatre- hat ould be the predicted attendance if Temperatre is 7/ degreesE

A. ,-3+/

B. 7-3+/

C. 12-2+/

. 1/- /2/

"he predicted Attendance is 1,-+// 0 3+(7/) * 7-3+/.

AA!S*+ Anal"tic *looms+ Appl" Diclt"+  Eas" /earning 0b1ecti2e+ 3-43 Interpret t)e slope and intercept o5 a r egression e:ation.

(55)

,,.

,,. A h4pA h4pothesothesis teis test is st is conducconducted ated at the t the + per+ percent lcent level evel of sigof signicanicancence to test hether the population correlation is zero. f the sample

to test hether the population correlation is zero. f the sample consists of 2+ observations and

consists of 2+ observations and the correlatithe correlation coe$cient is /.,/-on coe$cient is /.,/-then the computed test statistic ould be

then the computed test statistic ould be

A. A. 2./31.2./31. B. B. 1.7,/.1.7,/. C. C. %.+73.%.+73. . . 1.,+.1.,+. t t calccalc * * r r N(N(nn 0 2)L(1 0 0 2)L(1 0 r r 22)O)O1L21L2 * (.,/)N(2+ 0 2)L(1 0 .,/ * (.,/)N(2+ 0 2)L(1 0 .,/22)O)O1L21L2 * %.+73. * %.+73. Comment Re!uir

Comment Re!uires formula handout es formula handout or memorizing the or memorizing the formula.formula.

AA!S*+ Anal"tic AA!S*+ Anal"tic *looms+ Appl" *looms+ Appl" Diclt"+ 3 Medim Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4 !alclate and test

/earning 0b1ecti2e+ 3-4 !alclate and test a correlation coecient 5or a correlation coecient 5or signi6cance.signi6cance. Topic+ 7isal Displa"s and !orrelation Anal"sis Topic+ 7isal Displa"s and !orrelation Anal"sis

,3

,3.. 9h9hicich of th of the fhe fololloloiing ing iss not not a characteristic of the a characteristic of the F-F-test in a simpletest in a simple regressionE

regressionE

A.

A. t t is is a a test test for for overall overall t t of of the the model.model. B.

B. "he "he test test statistic statistic can can never never be be negative.negative. C.

C. t t re!uirre!uires a table ith es a table ith numerator and denominator degrees ofnumerator and denominator degrees of freedom.

freedom.

D.

D. "he "he F F 0test gives a dierent0test gives a dierent p p0value than the0value than the t t 0test.0test.

F

F calccalc is the ratio is the ratio of to variances (mean s!uares) that measuresof to variances (mean s!uares) that measures overall t. "he test statistic cannot be

overall t. "he test statistic cannot be negative because thenegative because the variances are non0negative. n a

variances are non0negative. n a simple regrsimple regression- theession- the F F 0test ala4s0test ala4s agrees ith the

agrees ith the t t 0test.0test.

AA!S*+ Anal"tic AA!S*+ Anal"tic *looms+ Remember *looms+ Remember Diclt"+ 3 Medim Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. /earning 0b1ecti2e+ 3-4< Interpret t)e standard error= R3= A>07A table= and F test. Topic+ Anal"sis o5 7ariance+ 02erall Fit Topic+ Anal"sis o5 7ariance+ 02erall Fit

(56)

,6.

,6. A rA researesearchercher's F8's F8cel cel reresults sults are are shon shon belo belo usingusingFemlabFemlab (labor (labor force participation rate among females) to tr4 to predict

force participation rate among females) to tr4 to predict !ancer !ancer

(death rate per 1//-/// population due to cancer) in

(death rate per 1//-/// population due to cancer) in the +/ ;.S.the +/ ;.S. states.

states.

9hich of the folloing statements is

9hich of the folloing statements is not not trueE trueE

A.

A."he standard error is too high for this "he standard error is too high for this model to be of model to be of an4an4 predictive use.

predictive use. B.

B. "he 7+ percent condence interval for the "he 7+ percent condence interval for the coe$cient ofcoe$cient of FemlabFemlab is is 0.27 to 0/.26.

0.27 to 0/.26. C.

C. Signicant correSignicant correlation e8ists beteenlation e8ists beteen FemlabFemlab and and !ancer !ancer at at αα * . * . /+.

/+. .

. "he "he to0tailedto0tailed p p0value for0value for FemlabFemlab ill be less than ./+. ill be less than ./+. "he magnitude of

"he magnitude of ssee depends on depends on Y Y (and- in this case- the (and- in this case- the t t calccalc indicates signicance). indicates signicance). AA!S*+ Anal"tic AA!S*+ Anal"tic *looms+ Appl" *looms+ Appl" Diclt"+ 3 Medim Diclt"+ 3 Medim /earning 0b1ecti2e+ 3-4C T

/earning 0b1ecti2e+ 3-4C Test )"pot)eses abot t)e est )"pot)eses abot t)e slope and intercept b" slope and intercept b" sing t tests.sing t tests. Topic+ Tests 5or Signi6cance Topic+ Tests 5or Signi6cance