correlation and simple linear regression (problems with solutions)

8/10/2019 Correlation and Simple Linear Regression (Problems With Solutions)

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Correlationandregressionproblems

1.

ThefollowingdatarepresenttheyearsofexperienceXandsalaryY(inthousanddollars)ofarandom

sampleofprofessionalengineers(n=27)

X 1 2 2 3 3 4 4 6 6 6 7 7 9 9 9 12 12

Y 25 28 37 31 40 36 43 40 43 51 42 55 41 51 60 50 64

13 14 18 18 20 21 22 22 25 25

52 66 53 70 70 60 61 71 58 74

a. Findthecoefficientofcorrelationr.

b.

Isitsufficientsampleevidencetoindicatethatthereislinearcorrelationbetweentheyearsofexperienceand

salary?

c. Testthehypothesisthat= 0.6

againstthealternative> 0.6

.Use= 0.01

2. Achemicalengineerisstudyingtheeffectoftemperatureontheyieldofacertainproductinchemicalprocess.

Theprocessisrun10timesandthefollowingdataisobservedforthetemperatureofeachprocessXand

correspondingyieldY.

TemperatureX(in C ) 95 110 118 124 145 140 185 190 205 222

Yield(inkgs) 108 126 102 121 118 155 158 178 159 184

Thefollowinginformationisavailable: = 1534x ,2 = 252, 684x , y = 1409 ,

2 = 206, 319y

= 226,463xy

a.

Findthecoefficientofcorrelationr.

b.

Isitsufficientsampleevidencetoindicatethatthereislinearcorrelationbetweentemperatureandyieldof

chemicalproduct?

c.

Testthehypothesisthat= 0.8 againstthealternative 0.8 .Use= 0.05


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3. Inthefollowingdata,Xrepresentsthenumberofyearsofformaleducationand Yrepresentsthesalaryin

thousandsofdollarsofarandomsampleofadultmales.

X 13 17 9 18 16 18 13 16

Y 21.6 25.8 15.9 48.3 38.2 56.4 28.4 43.3

a. Plotthescatterdiagram.

b. Computethecoefficientofcorrelationandtestthehypothesis 0 : 0H = against 1 : 0H .

c.

Findthecoefficientsoftheleastsquareslineandwritetheequationforestimatedregressionline.

d. Computemeansquareerror.

e. Finda95%confidenceinterval for1 and 0 .

f.

Testthehypothesisthat 0 0 = againstthealternative 0 0 .

g. Testthehypothesisthat1 0 = againstthealternative 1 0 .

h. Estimatethetruemeanresponse|Y x tox=16yearsofeducation.

i. PredictthesalaryYcorrespondingto16yearsofeducation.

j.

Findthecoefficientofdeterminationandexplainitsmeaning.

k. ConstructANOVAtableforregression.

4.

Thetablebelowdisplaysthemathematicsachievementtestscoresforarandomsampleofn=10students,

selectedfromthepopulationof12th

graders,togetherwiththeirfinalcalculusgrades.

Mathachievement

Test

score

39 43 21 64 57 47 28 75 34 52

Finalcalculusgrade 65 78 52 82 92 89 73 98 56 75

a. Findthecoefficientsoftheleastsquareslineandwritetheequationforestimatedregressionline.

b. B.Determinewhetherthereissignificantrelationshipbetweenthecalculusgradesandtestscores.

Use 0.05=

c. Finda95%confidenceintervalfortheslopeofregressionline.

d. Estimatetheaveragecalculusgradeforthestudentswhoseachievementscoreis50witha95%CI.

e.

Astudenttooktheachievementtest,buthasnotyettakenthecalculustest.Predictthecalculusgradefor

thisstudentwitha95%predictioninterval.

f. Finda95%confidenceintervalfortheinterceptofregressionline.

g. Calculatethecoefficientofdeterminationandexplainitsmeaning.

h.

ConstructANOVAtableforregression.


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5. Youaregivendatasetwith6pairsofx valuesandyvalues.Weassumethatxisindependentvariableandyis

dependentvariable.

x 2 1 0 1 2

y 1 1 3 5 5

a.

Findthe

least

squares

line

for

the

data.

b. Dothedatapresentsufficientevidencetoindicatethatyandxarelinearlyrelated?

c. ConstructtheeANOVAtableforlinearregressionandusetocalculateF=MSR

MSE.Verifythatthesquareoftest

statisticusedinpartb.coincideswithcriticalvalueforFat 0.05= .

d.

Find90%confidenceintervalfortheslopeoftheline.

e. Estimatetheaveragevalueofywhenx=1,using90%confidenceinterval.

f. Finda90%prediction intervalforfuturevalueofywhenx=1.

6.

Thedata,togetherwithaportionofMinitabprintout,isgivenbelow.CertaininformationinMinitabprintoutis

missed.

x 1 2 3 4 5 6

y 5.6 4.6 4.5 3.7 3.2 3.7

a. Findtheleastsquareslineforthedata.

b.

FillinthemissingentriesintheMINITABanalysisofvariancetable.

c. Dothedatapresentsufficientevidencetoindicatethatyandxarelinearlyrelated?

UsetheinformationintheMINITABprintouttoanswerthisquestionatthe1%levelofsignificance.

d. Findthecoefficientofdetermination.

e. Find90%confidenceintervalfortheslopeoftheline.

f.

Estimatethe

average

value

of

ywhen

x=2,

using

90%

confidence

interval.

g. Finda95%prediction intervalforthefuturevalueofywhenx=2.

Predictor Coef St.dev t P

Constant 6.0000 0.1759 34.10 0.000

X 0.55714 0.04518 12.33 0.000

S=0.1890 Rsq=97.4%

Analysisofvariance

Source DF SS MS

Regression

*

*

5.4221

Residualerror * 0.1429 *

Total * 5.5750


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correlation and simple linear regression (problems with solutions)

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