Statistical Methods for the Social Sciences

Alan Agresti, Barbara Finlay

Chapter 9 Linear Regression and Correlation - all with Video Answers

Educators

Chapter Questions

Problem 1

For the following variables in a regression analysis, which variable mote naturally plays the role of $X$ (cxplanalory variable) and which plays the role of $Y$ (response variable)?
a) College grade point average (GPA) and high school GPA
b) Number of children and mother's education level.
c) Annual income and number of years of education.
d) Annual income and assessed value of hume

Nick Johnson

Numerade Educator

03:02

Problem 2

Sketch plots of the following lincs. for values of $X$ between 0 and 10:
a) $Y=7 \div .5 X$
b) $Y=7+X$
c) $Y=7-X$
d) $Y=7-.5 X$
e) $Y=7$
f) $Y=X$

Lucas Finney

Numerade Educator

01:57

Problem 3

For the data on the 50 stales in Table 91 on $Y=$ violent crime rate and $X=$ poverty rate. the prediction equation is $\hat{Y}=209.9+25.5 X$.
a) Shetch a plot of the prediction equation for $X$ berween 0 and 100 .
b) Interpret the $Y$-intercept and the slope.
c) Find the predicted violeat crime rate for Massachusetls, which has $X=10.7$ and $Y=805$.
d) Find the residual for the Massaclusetts prediction. Interpret
e) Two states differ by 10.0 in their povcrty rates. Find the difference in their precicied violent crime rates.
f) The stace poverty rates range from 8.0 (for Hawaii) to 24.7 (for Mississippi). Over this range. find the rauge of piedicted values for violent crime rate.
g) What is the sign of the Pearson correlation between these variables? Why?

Tyler Moulton

Numerade Educator

07:25

Problem 4

A collcge adnissions ofticer claims that the piediction equarion $\hat{Y}=.5+7.0 X$ approximates the relarionship betwecn $Y=$ college GPA and $X=$ high school GPA (both mcasured on an four-point scale) for sludents af that college.
a) Is this equation realistic? Why or why not?
b) Suppose that the prediction equation is actually $\hat{Y}=.5+.7 X$ Interpret the slope.
c) Using the prediction equation in (b), find the predicted GPA for a student having a lugh school GPA of (i) 3.0 . (ii) 40
d) Suppose the prediction equation is $\hat{Y}=X$. Identify the $Y$-intercept and slope, and interpret theil values

Oluwadamilola Ameobi

Numerade Educator

Problem 5

A recent study of mail survey response rale paterus of the elderly found a predicLion equation relating $X=$ age and $Y=$ percentage of subjects responding of $\hat{Y}=90.2-.6 X$, for ages between about 60 and 90 (D. Kaldenberg et al . Public Optnion Quarterly, Vol 58; 1994. p. 68).
a) Interpict the slope.
b) Find the predicted response rate for a (i) 60 year old, (ii) 90 , ear old.
c) Find the difference in predicted response rates for two age groups that are ten years apart.

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03:06

Problem 6

Refer to Problem 9.24. For those countics, Table 9.9 shows part of the printout for the regression analysis relating $Y=$ median income (thousand of dollars) to $X=$ percent of residents with at least a high school education.
a) Report the predicton equation, and interpret the slope
b) Is the $Y$-intercept meaningful? Explain
c) Find the predicted median incomc for a county with $70 \%$ of the residents having at least a high school education.
d) County A has 10\% more of its revidents than county $B$ with at least a high school education Find thcir difference in predicted median incomes.
e) Find the Pearson correlation Interpret using (i) the sign (ii) the magnitude. (iii) the standardized slope.
f) Find the coefficient ol delermination Fxplain its PRE inlerpuetation.

Sheryl Ezze

Numerade Educator

06:46

Problem 7

A high school student analyzes whether a relationship exists between $X=$ number of books read for pleasure in the previons year and $Y=$ daily average number of hours spent watching television. For ber three best friends. the observations are as shown in Table 9.10 .
a) Construct a scatter diagram. Fioun inspection of the diagram, state the prodiction equation, and interpret. (Note: You can do this without using the least squeres formulas.)
b) Report the sample corrclation between $X$ and $Y$, and interpret.

Srikar Katta

Numerade Educator

Problem 8

For the WWW data sel described in Pioblem 1.7, the sample correlation berween $Y=$ political ideolugy (xcored 1 to 7) and $X=$ mumber of timcs a week reiding a ncw'spaper is $r=-.066$
a) Inverpret the sign of the correlation
b) Interpret the square of the correlation. Would you conclude that the sample association is strong, or weak?
c) When $Y$ is predicted using $X=$ religiosity (how often attend religious services. scored $0,1,2,3$ ), the sample corrclation is $1=.580$. Which of thesc two explanator) variables seems to have a stronger linear rclationship with $Y$ ? Explain.

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01:54

Problem 9

The prediction equation for a sample of 100 people relating $X=$ years of education and $Y=$ annual incoine (in dollarsi is $\hat{Y}=-6000+3000 X$, and the Pearson correlation equals . 50 .
a) Suppose instead that $Y$ refers to annual income, in thousands of dollars. State the prediction equation and the correlation.
b) Suppose that $Y$ is treated as the explanatory sanable and $X$ is treated as the response varable. Will the correlation coefficient or the slope changc in value? Explain

Tyler Moulton

Numerade Educator

11:32

Problem 10

Fon the house sales data in Table 9.4. Table 9.11 shows a computer printout for the regression analy sis rclating selling pnce (thousands of dollars) to number of bedrooms
a) Report the prediction equation, and interpret the slope. Is the relationship positive. or begative?
b) Find the piedicted selling price for a bome with (i) two. (ii) three. (iii) four bedrooms.
c) The first observation in the data sct has thrce bedrooms and a selling price of 48.5 thousand dollars. Find the residual, and interpret.
d) Using the sample slope and the standard deviations, find the Pearson corrclation. Interpret its value.
c) Report the coelficient of determination. and interpret its value
f) Report the standerd crior of the sample slope. Inerjret
g) Find the test statistic and $P$-value for 1esting $H_0 \cdot \beta=0$ against $H_n: \beta \neq 0$, and interpret.
h) Consinucl a $95 \%$ confidence interval for $\beta$, and interpref.
i) Use the result of the previous part to forin a $95 \%$ confidence interval lor the differcnce in the mean housing prices for homes witb $X=4$ bedrooms and with $X=2$ bedrooms. Interpret.
j) Interpret the value labeled "Root MSE."

Omer Ceyhan

Numerade Educator

02:36

Problem 11

Refer to Table 9.1. Table 9.12 shows an SPSS printout for the relationship for all 51 observations betwecn $Y=$ inurder rate and $X=$ percentage white
a) Report the prediction equation. Interpret the $Y$-intercept and slope
b) Report the coefficient of determination, and interpret.
c) Find the correlation. and interpret.
d) Report and interprel the estimated conditional standard deviation of murder rate
e) When the D.C. observation is delcted, the estimated slope changes to -216 . and $r^2$ changes to 359 Explain how a single observation can lave such a large effect.

Tyler Moulton

Numerade Educator

Problem 12

Refer to Table 9.1 For all 51 observations. use software to analyıe the relationship between murder rate and violent crime rale, treating murder ratc as the response varable.
a) Construct a scatter diagram. Does there seem to be a positive, on a negative, relationship?
b) Find the prediction equation. and interpret the coefficients.
c) Find the predicted murder rate and the residual for D.C. Interpret.
d) Find the predicted murder rate at the mean of 612.8 for violent crinte rale.
e) Using the slope and the standard deviations 4411 for violent crime rate and 107 for murder rate, find the correlation. Interpret.
f) Repor TSS and SSE from your printout, and use them to find the coufficient of determination and the correlation.
g) Now. treating violent cnme rate as the response rariable, find the predietion equation Interprel the coefficients.
h) Find the picdicted violent crime rale at the mean of 8.73 for murder 1ate (Note from this and from (d) thar the predicted value at the mcan of the explanatory variable is simply the mean of the response vanable.)
1) Using the standard deviations and the slope of this sccond prediction cquation. find the correlation. Compare to (e). How would you characterize thic association, strong or weak?
j) Based on box plos for the individual vanables or the scatter diagram. would you regard D.C. as an outlier? Retit the model in (b) without it. and note the effect on the slope and corrclation.

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01:57

Problem 13

Refer to Problem 9 24. For those data, use software to analyze $Y=$ crme rate and $X=$ percentage living in an urban environment.
a) Construct a stem and Icaf plot and a box plot to $Y$ Interprel.
b) Show that $\hat{Y}=245+.56 X$. Interpret the $Y$-intercept and slope.
c) Find the predicted crome rate and the residual for Alachua County Interpiet.
d) Using the slope. find the difference in predicted crime rales between counties that are $100 \%$ urban and counties that are $0 \%$ urban. Interpiet.
e) Report and interpret the Pearson correlation. Show the connection between it and the slope and the stundard deviations of 28.3 lor crime rate and 34.0 for percentage urban.
f) Find TSS and SSE on your printout, and use then to verify the coefficient of determination. InterpreL.
g) Does it make scnse to conduct statistical inference, such as a test ol independence, for these data? Explain.

Tyler Moulton

Numerade Educator

Problem 14

Using software. plor the relationship between percentage single-parent fanilies and jlercentage white. for the data in Table 9.1 .
a) Based on your ploc. identify the two observations that secm quite different froun the others
b) Find the prediction equation and the comelation (i) tor the entire data set. (ii) deleting the first of the two outlying observations, (iii) delcting the second of the two outlying abservations, fiv) deleting both outlying obscrvations. Discuss the influence of these points

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01:32

Problem 15

Refer to the housing data in Table 9 4. U'se software of fit the inodel with numbel of bathruorns as the predictor for proce.
a) Construce a scatta diagran. Note the effect of using such a highly discocte predictor. having only threc values.
b) Find the prediction equation. Interpret the slope.
c) Find the predicted selling price for hones with onc, two, and three balhroorns.
d) Find the Pearson corrclation and interpret its value.
e) Find the coefficient of determination and interpet its value
f) Test the null liypothesis that mean selling price is independent of nuinber of bathrooms. dnd report the $P$-value. Why is this inference not especiall? informative fol these variables?
g) Find a $95 \%$ confidence inter al for the difference borweed the mean selling price for honles with two bathrooms and homes with one bathroom. Interpret.
h) Find a $95 \%$ coufidence inter $\$ al for the difference betwecn the mean selling pnce for homes with three bathrooms and homes with one bathroom. Interprct.

Shu Naito

Numerade Educator

07:16

Problem 16

A study was conducled using 49 Catholic lemale undeagraduates at Texas A \& M University. The variables measured refer to the parents of these students. The response 1 ariable is the number of children that the pasents have One of the explanatory variables is the mother 8 educational level. measuied as the number of years of formal education For these datd. $\bar{X}=9.88.\lrcorner X=3.77, \dot{Y}=3.35,5 Y=219$. the prediction equation is $\dot{Y}=5.40-.207 X$. the standard error of the slope estimate is 079 , and SSE $=201.95$
a) Interprel the $Y$-intercept and slope.
b) Find the predicted numbers of children for wornen with (i) 8. (ii) 12 . (iii) 16 years of education
c) Find the Pearson correlation and interpret its value.
d) Test the null liypothesis that mean number of children is independent of mother's educational level, and report and interpret the $P$-valuc.
e) Find a $95 \%$ confidencc interval for the slope of the regrcssion equation. Interpret.
f) Construct and iuterpret a $95 \%$ confidence interval for the difference berween the mean nunber of children fol two sets of mothers who are eight years apart in educational level. g) Sketch a potcntiat scatter diagram for these variables such that the analyses you conducted above would be inappropnate.

Heather Duong

Numerade Educator

Problem 17

Table 9.13 lists recent values for several nations on the crude birlh rate (number of births per 1000 population sizc) nomer's economic activity (female labor force as percentage of nale). percentage women using contraccption, female life expectancy, female adult literacy rate, a human development index (HDI, which has components referriog to life expectancy at birth, educational attainnent. and income per capita). gross national product (GNP, per capita, in thousands of dollars). daily newspaper circulation per 100 people, and number of televisions per 100 people This exercise uses burth rate as the response variable and wonen's econome activity as the explanatory variable. Table 914 shows part of a SPSS printout for a regression analysis.
a) Report the prediction equation and interpret the $\gamma$-intercept and slope.
b) Report $t$ and $r^2$, and interptet their values.
c) Find the predicted value and residual for Nigena. and interptet.

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08:42

Problem 18

Refer to the pievious exercise. Now use GNP as the explanatory variable for predicting birth rate,
a) Construct a stem and leaf plot or box plot for burch rate, and describe its distribution.
b) Construct a scatter dingram. and indicate whether a lnear nodel seems appropriatc.
c) Fit the model, and interpict the parameter estimates
d) Canl you compare the slopes of the prediction equations with the two predictors to determine which has the stronger elfect? Explain.
e) Which \ariable, GNP or women's economic activity, seems to have the stronger association with birth rate?

James Kiss

Numerade Educator

Problem 19

Refer to the previous two cxercises. Using softwase, obtain the correlation matrix for these data. Which pairs of variables are highly correlatcd? Describe the nature of those correlations, and explain how your software handled the missing values. (For a particular analysis. most software deletes observations for which data are inissing on at least one variable used in the auljsis. Betrer strategies exist. see. for instance, R. Little and D Rubin. Sociological Methods and Research. Vol. I8. 1989. pp. 292-326)

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01:49

Problem 20

For a random sample of U . counties, data are obtained on $X=$ percentage of the population aged over 50 and $Y=$ per student expenditure on education. Table 9.15 is part of the computer printout for the analysis.
a) What was the sample size for this stody?
b) Fill in the blanks in Table 9.15 .

James Kiss

Numerade Educator

Problem 21

For Table 9.1 , use software to analyze the data on violent crime rate and percent single parent fannilies.
a) Construct a scatter diagram. What does it show'?
b) One point is quite far ıemoved froin the others. having a much highes value on both variables than the 1 est of the sample. but it fits in well with the lineas trend exhibited by the rest of the points Show that the correlation changes from .839 to .649 when you delete this obser vation. Why does it drop so dramatically?

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Problem 22

Refer to the WWW data sel (Problein 1.7) Using softu are, conduct regression analyses relating (i) $Y=$ political ideology and $X=$ religiosity, (ii) $Y=$ high school GPA and $X=$ hours of TV watching. Prepare a report.
a) Using graphical ways of portraying the individual variables and their relationship
b) Interpreting descnptive statistics for summarizing the individual varaables and theit relationship.
c) Summarizing and interpreting results of infcrential analy ses.
d) Checkang effects of possihly influential oulliers

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Problem 23

Refer to the data filc you creared in P1oblem 17. For variables chosen by your instructor, conduct a regression and correlation analysis. Report both descnptive and infcrential statistical analyses, interpreting and summarzing your findings.

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Problem 24

Table 9.16 shows recent data from dll 67 Florida countics on crine rate (number of crines per 1000 residents). median income (in thousands of dellars). percentage of residents with at least a high school education (of thosc aged at least 25 ). and the percentage of the county's residents living in an urban environunent. Using crime late as the response vari-able and percent urban as the predictor. analyze these data. In your report, provide intcrpretations of all the analyses

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Problem 25

Refer 10 the previous exercise. Using income as the response vanable and percentagre of high school graduarcs as the explanatory variable, analyze these data Piepare a report. and explain carcfully the interpretations of all your analyses

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04:09

Problem 26

Refer to Table 9.1. Analyze the relationship betueen violent cnme rate and peicentage having at least a hgh school education Write a report showing your analyses, ploviding interpretations, and summarizing your findings.

Kelsey Dondelinger

Numerade Educator

Problem 27

Reler to Table 9.1. Analyze the relationship between violent crime rate and percentage of single-parent families. Wrute a 1 eport shouing you analyses, providing interpretations. and summarizing your findings.

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Problem 28

Repeat the previous exercise. using murder rate as the response variable.

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Problem 29

Reler to Table 9.13. Analize the relationship between neu spaper circulation and gross national pioduct Tell uhy you conducted each parl of the analysis and explain hou 10 interpret the results

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Problem 30

Recently the General Social Survey has asked subjects to rale various groups using the "feeling thermometer." Ratings between 50 and 100 mean you feel favorablc and u arm toward the group, whereas ralings between 0 and 50 mean that you don i fecl favorable. It also asks subjects to rate themselves on political ideology, using scores $1=$ extremely liberal, $2=$ liberal. $3=$ slightlj ljbcral. $4=$ inoderatc, $5=$ slightly conservative, $6=$ conservaure, 7 = extiemely conservative, and to describc their religious attendance, using the categolies (never. less than once a year, once or twice a year, several times a year. about once a mond. 2-3 times a month. nearly every' weeh. evely week, several times a week). Table 9.17 shows data for ten of the sibjects in a recent survey, where the foclings the1momcter refers to feelings abou liberals and using religion scores that are the category numbers.
a) Analyze Table 917 Tell why y ou conducted each analy sis, and explain how to interpret the results
d) Suppose the feelings response for the first subject had incorrectly been recol dod as 90 instead of 10 Hou would this have allected results of your analy ses?

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02:06

Problem 31

In ans article in USA Todan (December 28, 1984), sociologisıs N. Glenn and B. A Shelton arc quoted as shou ing a strong link between residential mobilify and divorce rates In Table 9.18 , divorce rate is the annual number of div orces and annulments per 1000 population. and mobility rale is the percentage of people living in a diflerent house from fिve years ago. Analy ze these dald.

Charles Carter

Numerade Educator

Problem 32

Describe a situation in which it is inappropriate to use the Pearson coryclation to measure the association between two quantirative variables.

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Problem 33

Annual income. in dollars. is the response variable in a regression aualysis. For a British version of the report on the analysis, all responses are converted to British pounds sterling (1 pound cquals about 1.5 dollars. as of 1997).
a) Ilow, if at all. does the slope of the prediction equation clange?
b) How, if at all. does the correlation change?

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Problem 34

The । ariables $Y=$ aunual incone (thousands of dollars). $X_1=$ number of years of educarion, and $X_2=$ number of years experience in job are measured tor all the employees having city-funded jobs, in Knoxville. Tennessec. The following prediction equations and correlations apply
i. $\hat{Y}=10+1.0 X_i, \quad 1=.30$.
ii. $\hat{Y}=14+.4 \lambda_2 . \quad r=60$.
The correlation is -40 herween $X_1$ and $X_2$. Which of the following statements are true?
a) The strongest samplc association is between $Y$ and $X_2$.
b) The weakest sample association is between $X_1$ and $X_2$.
c) The prediction equatuon using $X_2$ to predict $X_1$ has negative slope.
d) A standald deviation incredse in education corresponds to a precicled increase of 3 standard deviations in income.
e) There is a $30 \%$ reduction in error in using education. instead of $\bar{Y}$. to predict income.
f) Each additional year on the job conesponds to a $\$ 400$ increase in predicted incomc.
g) When $X_1$ is the predictor of $Y$, the sum ol squared residuals (SSE) is larger than when $X_2$ is the predictor of $Y$.
h) The prodicted mean income for employees baving 20 years of expenence is $\$ 4000$ higher than the pledicted mean income for employees having 10 ycats of expenence.
i) If $\hat{\sigma}=8$ for the model using $X_1$ to predict $Y$, then it is not unusual to obscrve an income of $\$ 70.000$ for an employee whe has 10 years of educalion.
j) It is possible that $s_y=12.0$ and $s_{X_1}=36$.
k) It is possible that $\bar{Y}=20$ and $\bar{X}_1=13$.
Select the best response(s) in Problems 7.35-7.37.

Shu Naito

Numerade Educator

00:46

Problem 35

Oue can interpret $r=.3$ as follows.
a) A $30 \%$ reduction in error occurs in using $X$ to predict $Y$.
b) A $9 \%$ reduction in error occurs in using $X$ to predict $Y$ compared to using $\bar{Y}$ to predict $Y$.
c) $9 \%$ of the time $\hat{Y}=Y$
d) $Y$ changes .3 unil for every one-unit inciease in $X$.
e) When $X$ prodicts $Y$, the average residual is .3 .
f) $X$ changes 3 standard deviations when $Y$ changes one standard deviation.

Maxime Rossetti

Numerade Educator

01:29

Problem 36

The conclation is inappropriate as a measure of association between two quantilative variables.
a) When different pleople measure the variables using different anits.
b) When the relationship is hghly nonlinear.
c) When the data points fall exactly on a straight line
d) When the slope of the piediction equation is 0 using nearly all the datd. but a couple of outlicis are extrenely high on $Y$ at the high end of the $X$ scale.
e) When $Y$ tends to decrease as $X$ increases.
f) When we huse data for the entire population rathel than a sanmle.
g) When the sample has a much narrower range of $X$-talues than does the population.

Nick Johnson

Numerade Educator

01:29

Problem 37

The slope of the least squares prediction equalion and the Pearson comclation coefficient are similar in the seuse that
a) They do not depend on the units of measurement
b) Thes botl musi fall between -1 and +1 .
c) They both have the saune sign.
d) They both equal I when there is the stongest association
e) Their squares both have PRE interprelaions.
f) They have the siune $t$ statustic value for lesting $H_0$ : Independence.
g) They both can be strongly affected by severe oudiers.

Nick Johnson

Numerade Educator

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Problem 38

Describe the assumptions (a) in using the regı cssion equation $E\left(Y_J=\alpha \rightarrow \beta X\right.$ to repa esent the relationship between two vandbles and (b) in ınaking inferences about that equation using the leasi squares prediction cquation. Which assunptions are most critical?

Shu Naito

Numerade Educator

05:21

Problem 39

Rcfel to the previons exercise In r'iew of these assumptions. indicate wiyy such a model would or would not be good in the following situations
a) $X=$ time, $\mathrm{I}^*=$ percentage unemployed workers in the Unitcd States.
b) $X=$ income, $Y=$ chantable contrbutions within the previous yeur.
c) $X=$ age, $Y=$ annual medical expenses.
d) $X=$ per capita income. $Y^{\prime}=$ life expectancy, for nations. (Hiut. The uncreasung trend eventually lev cis off.)

Rashmi Sinha

Numerade Educator

01:01

Problem 40

For a class of 100 students. the teacher takes the 10 swudents who perform poorest on the nuidterm exam and enrolls them in a special tutoring program. The overall class mean is 70 both ou the midte in and final. but the mean for the specially tutored students increases from 50 to 60 . Can we conclude that the tutoring program was successful? Explain

Nick Johnson

Numerade Educator

02:39

Problem 41

Refer to I'roblein 924 For these counties, the correlalion between high school education rate and iucome equals 79 . Suppose we also bave data at the individual level uts well as aggregated for a county: Sketch a scatler diagraın to show that at the individual level. the corrclation could be inuch wcake.

Ivan Kochetkov

Numerade Educator

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Problem 42

Explain why the correlatoon between $X=$ number of years of educatoon and $Y=$ annual income is likely to be smaller of we use a random samplc of adults who have a college degree than if we use a random sample of all adults.

Emily Himsel

Numerade Educator

01:07

Problem 43

Fxplain carefully the interpretations of the standard deviations (a! $s_Y$, (b) $s_Y$. (c) $\hat{\sigma}$. (d) $\dot{\sigma}_j$

Sonam Khatri

Numerade Educator

02:39

Problem 44

*A icport summarizing the results of a study on the relationshup belween scores for sludents on a verbal apritude test $X$ and a inathematics aptitude lest l' states that $\bar{X}=480$. $\bar{y}=500, s_\lambda=80, s_y=120$, and $r=.60$.
a) Using the formulas for the correlation and for the least squares estimates, find the prediction equation.
b) Find the prediction equation for predicting terbal test result using matb test result.

Sheryl Ezze

Numerade Educator

Problem 45

*Observatuons on both $X$ and $Y$ are standa dized, having estimated means of 0 and standald detiations of 1 (see Section 42 ) Show that the piediction equation has the form $\hat{Y}=r \chi$. where $r$ is the sample correlation between $X$ and $Y$, that is, for the standardized variables, the $Y$-intercept equals 0 and the slope is the same as the correlation.

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04:35

Problem 46

A confidence interval for a pojulation correlation $\rho$ requires a nathematical transfolınation of $r$ for which the sampling distribution is approximalely normal This transformation is $T(r)=(1 / 2) \log [(1+r) /(1-r)]$, where $\log$ denotes the natural (base-e) loganthm. The transiormation of the population value $\rho$ is denoted by $T(\rho)$. The vartable $T(r)$ is approximately nomally distributed aboul $T(\rho)$ u ith standald enor $\sigma_T=$ $1 / \sqrt{n-3}$. A confidence interral for $T(\rho)$ is $T(r) \pm \approx \sigma_T$ Once we ger the endpoints of the interval for $T(\rho)$. we substitute each endpoint for $T$ in the inverse transformation $\rho=\left(e^{2 T}-1\right)^{\prime}\left(c^{2 T}+1\right)$. whele $e$ denotes the exponential function (the inverse of the natural $\log$ function) These two values form the endpoints of the confidence interval for $\rho$
a) For the cornelation of 899 for housing price and size in Table 94 . show that $T(1)=$ 1.47 Shou that the standard en or of $T(1)$ is . 1054 .
b) Show that a $95 \%$ confidence interval for $T(\rho)$ is ( 126,167 ).
c) Show that the corresponding confidence interval for $\rho$ is $(.85, .93$ ). (Unless $r=0$. the confidence interval for $\rho$ is not symmerric about the point estumate $r$. bccausc of the nonsymmetr; of the samplung distribution of $r$.)
d) A confidence interval for the population value $\rho^2$ of the coefficient of determundtion follows durectly by squaring the limits of the confidence interval for $\rho$ Find and inlerpret this confidence intertal.
e) lf the confidence interval tor $\rho$ includes 0 . explain $u$ by the lower endpoint of the confidence interval for $\rho^2$ is also 0 . and the upper endpoint is the larger of the squared endpoints of the conidence interval for $\rho$

Sarvesh Somasundaram

Numerade Educator

00:52

Problem 47

${ }^7$ Refer to the previous exercise and to Problein 916 Find and interpret $95 \%$ confidence inter als for lhe population Pearson correlation and the population coefficient of detennination

James Kiss

Numerade Educator

02:39

Problem 48

*Refer to P1oblein 9.46. Let $\rho_1$ and $\rho_2$ denote the population correlation Yalues betwcen two vanables for two sepasate populations. Let $r_1$ and $r_2$ denote sample valucs for inde-

Carolyn Behr-Jerome

Numerade Educator

Problem 49

Show thal substituling $X=X$ into dre prediction equation $Y$ ' $=a+b X$ yicids the predicted $Y$-value of $\hat{Y}=\bar{Y}$. (

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05:37

Problem 50

${ }^{+}$Alternative formulas for defining the pearson correlation use the data in formulas simulal to the one for $b$ :
$$
r=\frac{\sum(X-\bar{X})(Y-\bar{Y})}{\sqrt{\left[\sum(X-\bar{X})^2\right]\left[\sum(Y-\bar{Y})^2\right]}}=\frac{1}{n-1} \sum\left(\frac{X-\bar{X}}{s_X}\right)\left(\frac{Y-\bar{Y}}{s_1}\right)
$$

Roughly, the correlation is the average cioss-product of the $=$-scole for $X$ tines the $z-$ score for 1 Using this fonmula. explan why (a) the correlation has the same value when $X$ predicts $Y$ as when $Y$ predicis $X$. (o) the correlation does not depend on the units of measulement. (Note Fot the popolation, the correlarion is oflen defined as
Covariance of X and $Y$
where the covauiance between $X$ and $Y$ is the average of the cioss-products $\left(X-\mu_\lambda\right)$ $(Y-\mu r)$ about the pupulation means.)

Sheryl Ezze

Numerade Educator

02:31

Problem 51

The values of $Y$ are multiplied by a constant c. From their formulas. sbou that the standard dc viation $s y$ and the least squares slope $b$ are also then multiplied by $c$. Thus, show that $r=b s_x$ isy temains the samc. so that $r$ does not depend on the units of measurement.

Donald Albin

Numerade Educator

01:52

Problem 52

Suppose thrat the lineur regıession equation $E(Y)=\alpha+\beta X$ with nomality and constant standard deviation $\sigma$ is trul) appropnate for the relationship between $Y$ and $X$ Then, the interval of numbers

Narayan Hari

Numerade Educator

03:17

Problem 53

${ }^3$ Refer to Problem 9.16 and the previous exercise
a) Construct a $95 \%$ confidence inter al for the mean number of children for mothers having $X=16$.
b) Explain why the prediction interval is probably inappropriate.

James Kiss

Numerade Educator

08:04

Problem 55

*To implement least squares, one can find the formulas for the $a$ and $b$ estimates that minimize SSE $=\sum(Y-\hat{Y})^2=\sum[Y-(a+b X)]^2$ using calculus. b) taking the denvative of this function with respect to $a$, takung the derivaduve with respect to $b$. selling the two der ivatuves equal to 0 , and solving the tho linear equations simultaneously for $a$ and $b$. Take the derivative with respect to $a$ and solve for $a$. showing that $a=\bar{Y}-b \bar{X}$.

Stella Li

Numerade Educator