*Simple Linear Regression1. review of least squares procedure2. inference for least squares lines

*IntroductionWe will examine the relationship between quantitative variables x and y via a mathematical equation.The motivation for using the technique:Forecast the value of a dependent variable (y) from the value of independent variables (x1, x2,xk.).Analyze the specific relationships between the independent variables and the dependent variable.

*House sizeHouseCostMost lots sell for $25,000Building a house costs about $75 per square foot. House cost = 25000 + 75(Size)The ModelThe model has a deterministic and a probabilistic components

*House cost = 25000 + 75(Size)House sizeHouseCostMost lots sell for $25,000+ eHowever, house cost vary even among same size houses!The ModelSince cost behave unpredictably, we add a random component.

*The ModelThe first order linear model

y = dependent variablex = independent variableb0 = y-interceptb1 = slope of the linee = error variablexyb0RunRiseb1 = Rise/Runb0 and b1 are unknown population parameters, therefore are estimated from the data.

*Estimating the CoefficientsThe estimates are determined by drawing a sample from the population of interest,calculating sample statistics.producing a straight line that cuts into the data.wwwww w w www www wwQuestion: What should be considered a good line?xy

*The Least Squares (Regression) LineA good line is one that minimizes the sum of squared differences between the points and the line.

*The Least Squares (Regression) Line33wwww44(1,2)22(2,4)(3,1.5)Sum of squared differences =(2 - 1)2 +(4 - 2)2 +(1.5 - 3)2 +(4,3.2)(3.2 - 4)2 = 6.892.5Let us compare two linesThe second line is horizontalThe smaller the sum of squared differencesthe better the fit of the line to the data.

*The Estimated CoefficientsTo calculate the estimates of the slope and intercept of the least squares line , use the formulas: The regression equation that estimatesthe equation of the first order linear modelis: Alternate formula for the slope b1

* Example:A car dealer wants to find the relationship between the odometer reading and the selling price of used cars.A random sample of 100 cars is selected, and the data recorded.Find the regression line.Independent variable xDependent variable yThe Simple Linear Regression Line

Sheet1

CarOdometerPrice

13738814636

24475814122

34583314016

43086215590

53170515568

63401014718

...

...

...

402375401

323595595

435335330

327445806

344705805

377205317

413505316

244695870

357815504

486135333

241885705

387755150

455635249

286765775

382315327

366835192

325175544

390505054

452515115

343845410

383835529

321615507

265615873

335335303

418495237

366685383

374955286

256295827

400995483

310145440

422335215

374075105

343565685

305995788

424855208

384305168

404525128

260305750

462964965

348445238

273795763

478755162

356485486

425015257

438035228

434815135

342795267

413705290

349665387

414275091

302415667

472285146

244645765

212215911

355215532

280065499

380795204

423325271

492235007

333585337

378195066

359755605

380855637

352365286

209625820

458085187

361835435

343995499

443304787

320635707

346415457

310495365

386365160

364685551

257455568

391985122

215355854

371355297

425815148

330235416

316445733

359695478

290515690

381805286

314945745

313725525

362385424

342125283

331905259

391965356

363925133

&A

Page &P

*The Simple Linear Regression LineSolutionSolving by hand: Calculate a number of statisticswhere n = 100.

*Solution continuedUsing the computer1. Scatterplot2. Trend function3. Tools > Data Analysis > Regression The Simple Linear Regression Line

*The Simple Linear Regression Line

Sheet1

PriceOdometerSUMMARY OUTPUT

1463637388

1412244758Regression Statistics

1401645833Multiple R0.8063

1559030862R Square0.6501

1556831705Adjusted R Square0.6466

1471834010Standard Error303.1

1447045854Observations100

1569019057

1507240149ANOVA

1480240237dfSSMSFSignificance F

1519032359Regression11673411116734111182.110.0000

1466043533Residual98900545091892

1561232744Total9925739561

1561034470

1463437720CoefficientsStandard Errort StatP-value

1463241350Intercept17067169100.970.0000

1574024469Odometer-0.06230.0046-13.490.0000

1500835781

1466648613

1541024188

1430038775

1449845563

1555028676

1465438231

1438436683

1508832517

1410839050

1423045251

1482034384

1505838383

1501432161

1574626561

1460633533

1447441849

1476636668

1457237495

1565425629

1496640099

1488031014

1443042233

1421037407

1537034356

1557630599

1441642485

1433638430

1425640452

1550026030

1393046296

1447634844

1552627379

1432447875

1497235648

1451442501

1445643803

1427043481

1453434279

1458041370

1477434966

1418241427

1533430241

1429247228

1553024464

1582221221

1506435521

1499828006

1440838079

1454242332

1401449223

1467433358

1413237819

1521035975

1527438085

1457235236

1564020962

1437445808

1487036183

1499834399

1357444330

1541432063

1491434641

1473031049

1432038636

1510236468

1513625745

1424439198

1570821535

1459437135

1429642581

1483233023

1546631644

1495635969

1538029051

1457238180

1549031494

1505031372

1484836238

1456634212

1451833190

1471239196

1426636392

*This is the slope of the line.For each additional mile on the odometer,the price decreases by an average of $0.0623Interpreting the Linear Regression -EquationThe intercept is b0 = $17067.0No dataDo not interpret the intercept as the Price of cars that have not been driven17067

Chart2

1463614736.9149985422

1412214277.6499296334

1401614210.6607913597

1559015143.5858044715

1556815091.0538569694

1471814947.4166814152

1447014209.352166333

1569015879.2200159326

1507214564.8619652644

1480214559.3782032476

1519015050.299534708

1466014353.9863895266

1561215026.3080758844

1561014918.7515617818

1463414716.2262600242

1463214490.0210768303

1574015541.968651898

1500814837.0559708266

1466614037.4237640101

1541015559.4793010654

1430014650.4834312998

1449814227.485970275

1555015279.8074382075

1465414684.3830510401

1438414780.8474101542

1508815040.4536892687

1410814633.3466749972

1423014246.9283992437

1482014924.1106928437

1505814674.9110984656

1501415062.6379992459

1574615411.6046730436

1460614977.1411641654

1447414458.9256535758

1476614781.7821423162

1457214730.2472424535

1565415469.6826980399

1496614567.9777391376

1488015134.1138518969

1443014434.9965102297

1421014735.7310044704

1537014925.8555262127

1557615159.9747750445

1441614419.2930099088

1433614671.9822710248

1425614545.9803755928

1550015444.6941915769

1393014181.8087252939

1447614895.4455732103

1552615360.6306124781

1432414083.4125863785

1497214845.3439293293

1451414418.2959622694

1445614337.1612106114

1427014357.2267943547

1453414930.6538179774

1458014488.774767281

1477414887.8430849597

1418214485.2227850656

1533415182.2837159765

1429214123.7307002976

1553015542.2802292854

1582215744.3693227007

1506414853.2579949672

1499815321.5588081083

1440814693.8550036146

1454214428.8272779607

1401413999.4113227572

1467414988.0463727216

1413214710.0570277552

1521014824.9667681986

1527414693.4811107498

1457214871.0179060444

1564015760.5090313639

1437414212.2186782963

1487014812.0051488861

1499814923.1759606817

1357414304.3209539879

1541415068.7449160373

1491414908.0956151355

1473015131.9328101857

1432014659.1452826672

1510214794.2452378089

1513615462.4541026541

1424414624.1239843325

1570815724.8022627771

1459414752.6808143405

1429614413.3107240722

1483215008.922057672

1546615094.8551010947

1495614825.3406610634

1538015256.4391341585

1457214687.5611403908

1549015104.2024227143

1505015111.8049109649

1484814808.5777976256

1456614934.8289549675

1451814998.5153729355

1471214624.2486152875

1426614798.9812140962

Price

Predicted Price

Odometer

Price

Odometer Line Fit Plot

Sheet1

PriceOdometerSUMMARY OUTPUT

1463637388

1412244758Regression Statistics

1401645833Multiple R0.8063076039

1559030862R Square0.6501319521

1556831705Adjusted R Square0.64656187

1471834010Standard Error303.1375029266

1447045854Observations100

1569019057

1507240149ANOVA

1480240237dfSSMSFSignificance F

1519032359Regression116734110.883303616734110.8833036182.10560149890

1466043533Residual989005449.8766964191892.3456805756

1561232744Total9925739560.76

1561034470

1463437720CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

1463241350Intercept17066.7660699617169.0246439804100.9720574944016731.342171302517402.18996862116731.342171302517402.189968621

1574024469Odometer-0.06231547750.004617791-13.49465084760-0.0714793333-0.0531516217-0.0714793333-0.0531516217

1500835781

1466648613

1541024188

1430038775RESIDUAL OUTPUT

1449845563

1555028676ObservationPredicted PriceResiduals

1465438231114736.9149985422-100.9149985422

1438436683214277.6499296334-155.6499296334

1508832517314210.6607913597-194.6607913597

1410839050415143.5858044715446.4141955285

1423045251515091.0538569694476.9461430306

1482034384614947.4166814152-229.4166814152

1505838383714209.352166333260.647833667

1501432161815879.2200159326-189.2200159326

1574626561914564.8619652644507.1380347356

14606335331014559.3782032476242.6217967524

14474418491115050.299534708139.700465292

14766366681214353.9863895266306.0136104734

14572374951315026.3080758844585.6919241156

15654256291414918.7515617818691.2484382182

14966400991514716.2262600242-82.2262600242

14880310141614490.0210768303141.9789231697

14430422331715541.968651898198.031348102

14210374071814837.0559708266170.9440291734

15370343561914037.4237640101628.5762359899

15576305992015559.4793010654-149.4793010654

14416424852114650.4834312998-350.4834312998

14336384302214227.485970275270.514029725

14256404522315279.8074382075270.1925617925

15500260302414684.3830510401-30.3830510401

13930462962514780.8474101542-396.8474101542

14476348442615040.453689268747.5463107313

15526273792714633.3466749972-525.3466749972

14324478752814246.9283992437-16.9283992437

14972356482914924.1106928437-104.1106928437

14514425013014674.9110984656383.0889015344

14456438033115062.6379992459-48.6379992459

14270434813215411.6046730436334.3953269564

14534342793314977.1411641654-371.1411641654

14580413703414458.925653575815.0743464242

14774349663514781.7821423162-15.7821423162

14182414273614730.2472424535-158.2472424535

15334302413715469.6826980399184.3173019601

14292472283814567.9777391376398.0222608624

15530244643915134.1138518969-254.1138518969

15822212214014434.9965102297-4.9965102297

15064355214114735.7310044704-525.7310044704

14998280064214925.8555262127444.1444737873

14408380794315159.9747750445416.0252249555

14542423324414419.2930099088-3.2930099088

14014492234514671.9822710248-335.9822710248

14674333584614545.9803755928-289.9803755928

14132378194715444.694191576955.3058084231

15210359754814181.8087252939-251.8087252939

15274380854914895.4455732103-419.4455732103

14572352365015360.6306124781165.3693875219

15640209625114083.4125863785240.5874136215

14374458085214845.3439293293126.6560706707

14870361835314418.295962269495.7040377306

14998343995414337.1612106114118.8387893886

13574443305514357.2267943547-87.2267943547

15414320635614930.6538179774-396.6538179774

14914346415714488.77476728191.225232719

14730310495814887.8430849597-113.8430849597

14320386365914485.2227850656-303.2227850656

15102364686015182.2837159765151.7162840235

15136257456114123.7307002976168.2692997024

14244391986215542.2802292854-12.2802292854

15708215356315744.369322700777.6306772993

14594371356414853.2579949672210.7420050328

14296425816515321.5588081083-323.5588081083

14832330236614693.8550036146-285.8550036146

15466316446714428.8272779607113.1727220393

14956359696813999.411322757214.5886772428

15380290516914988.0463727216-314.0463727216

14572381807014710.0570277552-578.0570277552

15490314947114824.9667681986385.0332318014

15050313727214693.4811107498580.5188892502

14848362387314871.0179060444-299.0179060444

14566342127415760.5090313639-120.5090313639

14518331907514212.2186782963161.7813217037

14712391967614812.005148886157.9948511139

14266363927714923.175960681774.8240393183

7814304.3209539879-730.3209539879

7915068.7449160373345.2550839627

8014908.09561513555.9043848645

8115131.9328101857-401.9328101857

8214659.1452826672-339.1452826672

8314794.2452378089307.7547621911

8415462.4541026541-326.4541026541

8514624.1239843325-380.1239843325

8615724.8022627771-16.8022627771

8714752.6808143405-158.6808143405

8814413.3107240722-117.3107240722

8915008.922057672-176.922057672

9015094.8551010947371.1448989053

9114825.3406610634130.6593389366

9215256.4391341585123.5608658415

9314687.5611403908-115.5611403908

9415104.2024227143385.7975772857

9515111.8049109649-61.8049109649

9614808.577797625639.4222023744

9714934.8289549675-368.8289549675

9814998.5153729355-480.5153729355

9914624.248615287587.7513847125

10014798.9812140962-532.9812140962

Sheet1

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Price

Predicted Price

Odometer

Price

Odometer Line Fit Plot

*Error Variable: Required ConditionsThe error e is a critical part of the regression model.Four requirements involving the distribution of e must be satisfied.The probability distribution of e is normal.The mean of e is zero: E(e) = 0.The standard deviation of e is se for all values of x.The set of errors associated with different values of y are all independent.

*The Normality of eFrom the first three assumptions we have:y is normally distributed with meanE(y) = b0 + b1x, and a constant standard deviation sem3x1x2x3m1m2The standard deviation remains constant,but the mean value changes with x

*Assessing the ModelThe least squares method will produces a regression line whether or not there is a linear relationship between x and y.Consequently, it is important to assess how well the linear model fits the data.Several methods are used to assess the model. All are based on the sum of squares for errors, SSE.

*This is the sum of differences between the points and the regression line.It can serve as a measure of how well the line fits the data. SSE is defined by Sum of Squares for Errors

*The mean error is equal to zero.If se is small the errors tend to be close to zero (close to the mean error). Then, the model fits the data well.Therefore, we can, use se as a measure of the suitability of using a linear model.An estimator of se is given by se Standard Error of Estimate

*Example:Calculate the standard error of estimate for the previous example and describe what it tells you about the model fit.Solution Standard Error of Estimate,Example

* Testing the slopeWhen no linear relationship exists between two variables, the regression line should be horizontal.qqDifferent inputs (x) yielddifferent outputs (y).No linear relationship.Different inputs (x) yieldthe same output (y).The slope is not equal to zeroThe slope is equal to zeroLinear relationship.Linear relationship.Linear relationship.Linear relationship.

*We can draw inference about b1 from b1 by testingH0: b1 = 0H1: b1 = 0 (or < 0,or > 0)The test statistic is

If the error variable is normally distributed, the statistic is Student t distribution with d.f. = n-2.where Testing the Slope

*ExampleTest to determine whether there is enough evidence to infer that there is a linear relationship between the car auction price and the odometer reading for all three-year-old Tauruses in the previous example . Use a = 5%. Testing the Slope,Example

*Solving by handTo compute t we need the values of b1 and sb1. The rejection region is t > t.025 or t < -t.025 with n = n-2 = 98. Approximately, t.025 = 1.984 Testing the Slope,Example

*Using the computerThere is overwhelming evidence to inferthat the odometer reading affects the auction selling price. Testing the Slope,Example

Sheet1

PriceOdometerSUMMARY OUTPUT

1463637388

1412244758Regression Statistics

1401645833Multiple R0.8063

1559030862R Square0.6501

1556831705Adjusted R Square0.6466

1471834010Standard Error303.1

1447045854Observations100

1569019057

1507240149ANOVA

1480240237dfSSMSFSignificance F

1519032359Regression11673411116734111182.110.0000

1466043533Residual98900545091892

1561232744Total9925739561

1561034470

1463437720CoefficientsStandard Errort StatP-value

1463241350Intercept17067169100.970.0000

1574024469Odometer-0.06230.0046-13.490.0000

1500835781

1466648613

1541024188

1430038775

1449845563

1555028676

1465438231

1438436683

1508832517

1410839050

1423045251

1482034384

1505838383

1501432161

1574626561

1460633533

1447441849

1476636668

1457237495

1565425629

1496640099

1488031014

1443042233

1421037407

1537034356

1557630599

1441642485

1433638430

1425640452

1550026030

1393046296

1447634844

1552627379

1432447875

1497235648

1451442501

1445643803

1427043481

1453434279

1458041370

1477434966

1418241427

1533430241

1429247228

1553024464

1582221221

1506435521

1499828006

1440838079

1454242332

1401449223

1467433358

1413237819

1521035975

1527438085

1457235236

1564020962

1437445808

1487036183

1499834399

1357444330

1541432063

1491434641

1473031049

1432038636

1510236468

1513625745

1424439198

1570821535

1459437135

1429642581

1483233023

1546631644

1495635969

1538029051

1457238180

1549031494

1505031372

1484836238

1456634212

1451833190

1471239196

1426636392

*To measure the strength of the linear relationship we use the coefficient of determination.Coefficient of determinationNote that the coefficient of determination is r2

*Coefficient of determinationTo understand the significance of this coefficient note:Overall variability in yThe regression modelThe error

*Coefficient of determinationx1x2y1y2Two data points (x1,y1) and (x2,y2) of a certain sample are shown.Total variation in y =Variation explained by the regression line+ Unexplained variation (error)Variation in y = SSR + SSE

*Coefficient of determinationR2 measures the proportion of the variation in y that is explained by the variation in x.R2 takes on any value between zero and one.R2 = 1: Perfect match between the line and the data points.R2 = 0: There are no linear relationship between x and y.

*ExampleFind the coefficient of determination for the used car price odometer example.what does this statistic tell you about the model?SolutionSolving by hand;Coefficient of determination,Example

* Using the computer From the regression output we have

65% of the variation in the auctionselling price is explained by the variation in odometer reading. Therest (35%) remains unexplained bythis model.Coefficient of determination

Sheet1

PriceOdometerSUMMARY OUTPUT

1463637388

1412244758Regression Statistics

1401645833Multiple R0.8063

1559030862R Square0.6501

1556831705Adjusted R Square0.6466

1471834010Standard Error303.1

1447045854Observations100

1569019057

1507240149ANOVA

1480240237dfSSMSFSignificance F

1519032359Regression11673411116734111182.110.0000

1466043533Residual98900545091892

1561232744Total9925739561

1561034470

1463437720CoefficientsStandard Errort StatP-value

1463241350Intercept17067169100.970.0000

1574024469Odometer-0.06230.0046-13.490.0000

1500835781

1466648613

1541024188

1430038775

1449845563

1555028676

1465438231

1438436683

1508832517

1410839050

1423045251

1482034384

1505838383

1501432161

1574626561

1460633533

1447441849

1476636668

1457237495

1565425629

1496640099

1488031014

1443042233

1421037407

1537034356

1557630599

1441642485

1433638430

1425640452

1550026030

1393046296

1447634844

1552627379

1432447875

1497235648

1451442501

1445643803

1427043481

1453434279

1458041370

1477434966

1418241427

1533430241

1429247228

1553024464

1582221221

1506435521

1499828006

1440838079

1454242332

1401449223

1467433358

1413237819

1521035975

1527438085

1457235236

1564020962

1437445808

1487036183

1499834399

1357444330

1541432063

1491434641

1473031049

1432038636

1510236468

1513625745

1424439198

1570821535

1459437135

1429642581

1483233023

1546631644

1495635969

1538029051

1457238180

1549031494

1505031372

1484836238

1456634212

1451833190

1471239196

1426636392

*If we are satisfied with how well the model fits the data, we can use it to predict the values of y.To make a prediction we usePoint prediction, andInterval predictionUsing the Regression EquationBefore using the regression model, we need to assess how well it fits the data.

*Point PredictionExamplePredict the selling price of a three-year-old Taurus with 40,000 miles on the odometer. It is predicted that a 40,000 miles car would sell for $14,575.How close is this prediction to the real price?

*Interval EstimatesTwo intervals can be used to discover how closely the predicted value will match the true value of y.Prediction interval predicts y for a given value of x,Confidence interval estimates the average y for a given x.

*Interval Estimates,ExampleExample - continued Provide an interval estimate for the bidding price on a Ford Taurus with 40,000 miles on the odometer.Two types of predictions are required:A prediction for a specific carAn estimate for the average price per car

*Interval Estimates,Example SolutionA prediction interval provides the price estimate for a single car: t.025,98Approximately

*Solution continuedA confidence interval provides the estimate of the mean price per car for a Ford Taurus with 40,000 miles reading on the odometer.

The confidence interval (95%) =Interval Estimates,Example

*As xg moves away from x the interval becomes longer. That is, the shortest interval is found at x.The effect of the given xg on the length of the interval

*As xg moves away from x the interval becomes longer. That is, the shortest interval is found at x.The effect of the given xg on the length of the interval

*As xg moves away from x the interval becomes longer. That is, the shortest interval is found at x. The effect of the given xg on the length of the interval

*Regression Diagnostics - IThe three conditions required for the validity of the regression analysis are:the error variable is normally distributed.the error variance is constant for all values of x.The errors are independent of each other.How can we diagnose violations of these conditions?

* Residual AnalysisExamining the residuals (or standardized residuals), help detect violations of the required conditions.Example continued:Nonnormality. Use Excel to obtain the standardized residual histogram.Examine the histogram and look for a bell shaped. diagram with a mean close to zero.

*For each residual we calculate the standard deviation as follows:

A Partial list ofStandard residuals Residual Analysis

Sheet1

PriceOdometerSUMMARY OUTPUT

1463637388

1412244758Regression Statistics

1401645833Multiple R0.8063076039

1559030862R Square0.6501319521

1556831705Adjusted R Square0.64656187

1471834010Standard Error303.1375029266

1447045854Observations100

1569019057

1507240149ANOVA

1480240237dfSSMSFSignificance F

1519032359Regression116734110.883303616734110.8833036182.10560149890

1466043533Residual989005449.8766964191892.3456805756

1561232744Total9925739560.76

1561034470

1463437720CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

1463241350Intercept17066.7660699617169.0246439804100.9720574944016731.342171302517402.18996862116731.342171302517402.189968621

1574024469Odometer-0.06231547750.004617791-13.49465084760-0.0714793333-0.0531516217-0.0714793333-0.0531516217

1500835781

1466648613

1541024188

1430038775RESIDUAL OUTPUT

1449845563

1555028676ObservationPredicted PriceResidualsStandard Residuals

1465438231114736.91-100.91-0.33

1438436683214277.65-155.65-0.52

1508832517314210.66-194.66-0.65

1410839050415143.59446.411.48

1423045251515091.05476.951.58

1482034384614947.4166814152-229.4166814152-0.7606587824

1505838383714209.352166333260.6478336670.8642094488

1501432161815879.2200159326-189.2200159326-0.6273818714

1574626561914564.8619652644507.13803473561.6814775527

14606335331014559.3782032476242.62179675240.804441941

14474418491115050.299534708139.7004652920.463193806

14766366681214353.9863895266306.01361047341.0146251742

14572374951315026.3080758844585.69192411561.9419324833

15654256291414918.7515617818691.24843821822.2919178854

14966400991514716.2262600242-82.2262600242-0.2726311201

14880310141614490.0210768303141.97892316970.470748309

14430422331715541.968651898198.0313481020.65659691

14210374071814837.0559708266170.94402917340.5667856247

15370343561914037.4237640101628.57623598992.0841206113

15576305992015559.4793010654-149.4793010654-0.4956167199

14416424852114650.4834312998-350.4834312998-1.162070249

14336384302214227.485970275270.5140297250.8969220163

14256404522315279.8074382075270.19256179250.8958561505

15500260302414684.3830510401-30.3830510401-0.1007386842

13930462962514780.8474101542-396.8474101542-1.3157956341

14476348442615040.453689268747.54631073130.1576455496

15526273792714633.3466749972-525.3466749972-1.7418505039

14324478752814246.9283992437-16.9283992437-0.0561281572

14972356482914924.1106928437-104.1106928437-0.3451916067

14514425013014674.9110984656383.08890153441.2701776331

14456438033115062.6379992459-48.6379992459-0.161265175

14270434813215411.6046730436334.39532695641.1087281913

14534342793314977.1411641654-371.1411641654-1.2305634633

14580413703414458.925653575815.07434642420.0499808206

14774349663514781.7821423162-15.7821423162-0.0523276036

14182414273614730.2472424535-158.2472424535-0.5246878911

15334302413715469.6826980399184.31730196010.6111263297

14292472283814567.9777391376398.02226086241.3196909939

15530244643915134.1138518969-254.1138518969-0.8425452412

15822212214014434.9965102297-4.9965102297-0.0165665346

15064355214114735.7310044704-525.7310044704-1.7431247948

14998280064214925.8555262127444.14447378731.4726147748

14408380794315159.9747750445416.02522495551.3793820009

14542423324414419.2930099088-3.2930099088-0.010918373

14014492234514671.9822710248-335.9822710248-1.1139898965

14674333584614545.9803755928-289.9803755928-0.9614650428

14132378194715444.694191576955.30580842310.1833731036

15210359754814181.8087252939-251.8087252939-0.8349023148

15274380854914895.4455732103-419.4455732103-1.3907225797

14572352365015360.6306124781165.36938752190.5483022254

15640209625114083.4125863785240.58741362150.7976966974

14374458085214845.3439293293126.65607067070.4199435364

14870361835314418.295962269495.70403773060.3173183239

14998343995414337.1612106114118.83878938860.3940243939

13574443305514357.2267943547-87.2267943547-0.2892109971

15414320635614930.6538179774-396.6538179774-1.3151537558

14914346415714488.77476728191.2252327190.3024683036

14730310495814887.8430849597-113.8430849597-0.377460531

14320386365914485.2227850656-303.2227850656-1.0053718546

15102364686015182.2837159765151.71628402350.5030337078

15136257456114123.7307002976168.26929970240.5579172353

14244391986215542.2802292854-12.2802292854-0.0407165869

15708215356315744.369322700777.63067729930.2573939092

14594371356414853.2579949672210.74200503280.6987406319

14296425816515321.5588081083-323.5588081083-1.0727984011

14832330236614693.8550036146-285.8550036146-0.9477868726

15466316446714428.8272779607113.17272203930.3752378616

14956359696813999.411322757214.58867724280.0483705256

15380290516914988.0463727216-314.0463727216-1.0412587699

14572381807014710.0570277552-578.0570277552-1.9166180601

15490314947114824.9667681986385.03323180141.2766242955

15050313727214693.4811107498580.51888925021.9247806599

14848362387314871.0179060444-299.0179060444-0.9914300692

14566342127415760.5090313639-120.5090313639-0.3995622833

14518331907514212.2186782963161.78132170370.5364055587

14712391967614812.005148886157.99485111390.1922889502

14266363927714923.175960681774.82403931830.2480881612

7814304.3209539879-730.3209539879-2.4214675419

7915068.7449160373345.25508396271.1447350304

8014908.09561513555.90438486450.0195767029

8115131.9328101857-401.9328101857-1.3326568936

8214659.1452826672-339.1452826672-1.1244772445

8314794.2452378089307.75476219111.0203981735

8415462.4541026541-326.4541026541-1.0823981007

8514624.1239843325-380.1239843325-1.2603470911

8615724.8022627771-16.8022627771-0.0557099365

8714752.6808143405-158.6808143405-0.5261254512

8814413.3107240722-117.3107240722-0.3889579083

8915008.922057672-176.922057672-0.5866065019

9015094.8551010947371.14489890531.2305758463

9114825.3406610634130.65933893660.433216857

9215256.4391341585123.56086584150.40968101

9314687.5611403908-115.5611403908-0.3831569518

9415104.2024227143385.79757728571.2791585755

9515111.8049109649-61.8049109649-0.2049216649

9614808.577797625639.42220237440.1307090848

9714934.8289549675-368.8289549675-1.2228970538

9814998.5153729355-480.5153729355-1.5932068943

9914624.248615287587.75138471250.29095034

10014798.9812140962-532.9812140962-1.7671637426

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Odometer

Residuals

Odometer Residual Plot

*It seems the residual are normally distributed with mean zero Residual Analysis

Chart1

1

19

30

32

16

2

Frequency

Standardized residuals

Sheet1

PriceOdometerSUMMARY OUTPUT

1463637388

1412244758Regression Statistics

1401645833Multiple R0.8063076039

1559030862R Square0.6501319521

1556831705Adjusted R Square0.64656187

1471834010Standard Error303.1375029266

1447045854Observations100

1569019057

1507240149ANOVA

1480240237dfSSMSFSignificance F

1519032359Regression116734110.883303616734110.8833036182.10560149890

1466043533Residual989005449.8766964191892.3456805756

1561232744Total9925739560.76

1561034470

1463437720CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

1463241350Intercept17066.7660699617169.0246439804100.9720574944016731.342171302517402.18996862116731.342171302517402.189968621

1574024469Odometer-0.06231547750.004617791-13.49465084760-0.0714793333-0.0531516217-0.0714793333-0.0531516217

1500835781

1466648613

1541024188

1430038775RESIDUAL OUTPUT

1449845563

1555028676ObservationPredicted PriceResidualsStandard ResidualsBean

1465438231114736.91-100.91-0.33-2.00

1438436683214277.65-155.65-0.52-1.00

1508832517314210.66-194.66-0.650.00

1410839050415143.59446.411.481.00

1423045251515091.05476.951.582.00

1482034384614947.4166814152-229.4166814152-0.7606587824

1505838383714209.352166333260.6478336670.8642094488

1501432161815879.2200159326-189.2200159326-0.6273818714BeanFrequency

1574626561914564.8619652644507.13803473561.6814775527-21

14606335331014559.3782032476242.62179675240.804441941-119

14474418491115050.299534708139.7004652920.463193806030

14766366681214353.9863895266306.01361047341.0146251742132

14572374951315026.3080758844585.69192411561.9419324833216

15654256291414918.7515617818691.24843821822.2919178854More2

14966400991514716.2262600242-82.2262600242-0.2726311201

14880310141614490.0210768303141.97892316970.470748309

14430422331715541.968651898198.0313481020.65659691

14210374071814837.0559708266170.94402917340.5667856247

15370343561914037.4237640101628.57623598992.0841206113

15576305992015559.4793010654-149.4793010654-0.4956167199

14416424852114650.4834312998-350.4834312998-1.162070249

14336384302214227.485970275270.5140297250.8969220163

14256404522315279.8074382075270.19256179250.8958561505

15500260302414684.3830510401-30.3830510401-0.1007386842

13930462962514780.8474101542-396.8474101542-1.3157956341

14476348442615040.453689268747.54631073130.1576455496

15526273792714633.3466749972-525.3466749972-1.7418505039

14324478752814246.9283992437-16.9283992437-0.0561281572

14972356482914924.1106928437-104.1106928437-0.3451916067

14514425013014674.9110984656383.08890153441.2701776331

14456438033115062.6379992459-48.6379992459-0.161265175

14270434813215411.6046730436334.39532695641.1087281913

14534342793314977.1411641654-371.1411641654-1.2305634633

14580413703414458.925653575815.07434642420.0499808206

14774349663514781.7821423162-15.7821423162-0.0523276036

14182414273614730.2472424535-158.2472424535-0.5246878911

15334302413715469.6826980399184.31730196010.6111263297

14292472283814567.9777391376398.02226086241.3196909939

15530244643915134.1138518969-254.1138518969-0.8425452412

15822212214014434.9965102297-4.9965102297-0.0165665346

15064355214114735.7310044704-525.7310044704-1.7431247948

14998280064214925.8555262127444.14447378731.4726147748

14408380794315159.9747750445416.02522495551.3793820009

14542423324414419.2930099088-3.2930099088-0.010918373

14014492234514671.9822710248-335.9822710248-1.1139898965

14674333584614545.9803755928-289.9803755928-0.9614650428

14132378194715444.694191576955.30580842310.1833731036

15210359754814181.8087252939-251.8087252939-0.8349023148

15274380854914895.4455732103-419.4455732103-1.3907225797

14572352365015360.6306124781165.36938752190.5483022254

15640209625114083.4125863785240.58741362150.7976966974

14374458085214845.3439293293126.65607067070.4199435364

14870361835314418.295962269495.70403773060.3173183239

14998343995414337.1612106114118.83878938860.3940243939

13574443305514357.2267943547-87.2267943547-0.2892109971

15414320635614930.6538179774-396.6538179774-1.3151537558

14914346415714488.77476728191.2252327190.3024683036

14730310495814887.8430849597-113.8430849597-0.377460531

14320386365914485.2227850656-303.2227850656-1.0053718546

15102364686015182.2837159765151.71628402350.5030337078

15136257456114123.7307002976168.26929970240.5579172353

14244391986215542.2802292854-12.2802292854-0.0407165869

15708215356315744.369322700777.63067729930.2573939092

14594371356414853.2579949672210.74200503280.6987406319

14296425816515321.5588081083-323.5588081083-1.0727984011

14832330236614693.8550036146-285.8550036146-0.9477868726

15466316446714428.8272779607113.17272203930.3752378616

14956359696813999.411322757214.58867724280.0483705256

15380290516914988.0463727216-314.0463727216-1.0412587699

14572381807014710.0570277552-578.0570277552-1.9166180601

15490314947114824.9667681986385.03323180141.2766242955

15050313727214693.4811107498580.51888925021.9247806599

14848362387314871.0179060444-299.0179060444-0.9914300692

14566342127415760.5090313639-120.5090313639-0.3995622833

14518331907514212.2186782963161.78132170370.5364055587

14712391967614812.005148886157.99485111390.1922889502

14266363927714923.175960681774.82403931830.2480881612

7814304.3209539879-730.3209539879-2.4214675419

7915068.7449160373345.25508396271.1447350304

8014908.09561513555.90438486450.0195767029

8115131.9328101857-401.9328101857-1.3326568936

8214659.1452826672-339.1452826672-1.1244772445

8314794.2452378089307.75476219111.0203981735

8415462.4541026541-326.4541026541-1.0823981007

8514624.1239843325-380.1239843325-1.2603470911

8615724.8022627771-16.8022627771-0.0557099365

8714752.6808143405-158.6808143405-0.5261254512

8814413.3107240722-117.3107240722-0.3889579083

8915008.922057672-176.922057672-0.5866065019

9015094.8551010947371.14489890531.2305758463

9114825.3406610634130.65933893660.433216857

9215256.4391341585123.56086584150.40968101

9314687.5611403908-115.5611403908-0.3831569518

9415104.2024227143385.79757728571.2791585755

9515111.8049109649-61.8049109649-0.2049216649

9614808.577797625639.42220237440.1307090848

9714934.8289549675-368.8289549675-1.2228970538

9814998.5153729355-480.5153729355-1.5932068943

9914624.248615287587.75138471250.29095034

10014798.9812140962-532.9812140962-1.7671637426

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55.3058084231

-251.8087252939

-419.4455732103

165.3693875219

240.5874136215

126.6560706707

95.7040377306

118.8387893886

-87.2267943547

-396.6538179774

91.225232719

-113.8430849597

-303.2227850656

151.7162840235

168.2692997024

-12.2802292854

77.6306772993

210.7420050328

-323.5588081083

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113.1727220393

14.5886772428

-314.0463727216

-578.0570277552

385.0332318014

580.5188892502

-299.0179060444

-120.5090313639

161.7813217037

57.9948511139

74.8240393183

-730.3209539879

345.2550839627

5.9043848645

-401.9328101857

-339.1452826672

307.7547621911

-326.4541026541

-380.1239843325

-16.8022627771

-158.6808143405

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-176.922057672

371.1448989053

130.6593389366

123.5608658415

-115.5611403908

385.7975772857

-61.8049109649

39.4222023744

-368.8289549675

-480.5153729355

87.7513847125

-532.9812140962

Odometer

Residuals

Odometer Residual Plot

Frequency

Standardized residuals

* HeteroscedasticityWhen the requirement of a constant variance is violated we have a condition of heteroscedasticity.Diagnose heteroscedasticity by plotting the residual against the predicted y.++++++++++++++++++++++++y^Residual

* HomoscedasticityWhen the requirement of a constant variance is not violated we have a condition of homoscedasticity.Example - continued

Chart2

-100.9149985422

-155.6499296334

-194.6607913597

446.4141955285

476.9461430306

-229.4166814152

260.647833667

-189.2200159326

507.1380347356

242.6217967524

139.700465292

306.0136104734

585.6919241156

691.2484382182

-82.2262600242

141.9789231697

198.031348102

170.9440291734

628.5762359899

-149.4793010654

-350.4834312998

270.514029725

270.1925617925

-30.3830510401

-396.8474101542

47.5463107313

-525.3466749972

-16.9283992437

-104.1106928437

383.0889015344

-48.6379992459

334.3953269564

-371.1411641654

15.0743464242

-15.7821423162

-158.2472424535

184.3173019601

398.0222608624

-254.1138518969

-4.9965102297

-525.7310044704

444.1444737873

416.0252249555

-3.2930099088

-335.9822710248

-289.9803755928

55.3058084231

-251.8087252939

-419.4455732103

165.3693875219

240.5874136215

126.6560706707

95.7040377306

118.8387893886

-87.2267943547

-396.6538179774

91.225232719

-113.8430849597

-303.2227850656

151.7162840235

168.2692997024

-12.2802292854

77.6306772993

210.7420050328

-323.5588081083

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113.1727220393

14.5886772428

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385.0332318014

580.5188892502

-299.0179060444

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161.7813217037

57.9948511139

74.8240393183

-730.3209539879

345.2550839627

5.9043848645

-401.9328101857

-339.1452826672

307.7547621911

-326.4541026541

-380.1239843325

-16.8022627771

-158.6808143405

-117.3107240722

-176.922057672

371.1448989053

130.6593389366

123.5608658415

-115.5611403908

385.7975772857

-61.8049109649

39.4222023744

-368.8289549675

-480.5153729355

87.7513847125

-532.9812140962

Residuals

Predicted Price

Residuals

Sheet1

PriceOdometerSUMMARY OUTPUT

1463637388

1412244758Regression Statistics

1401645833Multiple R0.8063076039

1559030862R Square0.6501319521

1556831705Adjusted R Square0.64656187

1471834010Standard Error303.1375029266

1447045854Observations100

1569019057

1507240149ANOVA

1480240237dfSSMSFSignificance F

1519032359Regression116734110.883303616734110.8833036182.10560149890

1466043533Residual989005449.8766964191892.3456805756

1561232744Total9925739560.76

1561034470

1463437720CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0%

1463241350Intercept17066.7660699617169.0246439804100.9720574944016731.342171302517402.18996862116731.342171302517402.189968621

1574024469Odometer-0.06231547750.004617791-13.49465084760-0.0714793333-0.0531516217-0.0714793333-0.0531516217

1500835781

1466648613

1541024188

1430038775RESIDUAL OUTPUT

1449845563

1555028676ObservationPredicted PriceResiduals

1465438231114736.9149985422-100.9149985422

1438436683214277.6499296334-155.6499296334

1508832517314210.6607913597-194.6607913597

1410839050415143.5858044715446.4141955285

1423045251515091.0538569694476.9461430306

1482034384614947.4166814152-229.4166814152

1505838383714209.352166333260.647833667

1501432161815879.2200159326-189.2200159326

1574626561914564.8619652644507.1380347356

14606335331014559.3782032476242.6217967524

14474418491115050.299534708139.700465292

14766366681214353.9863895266306.0136104734

14572374951315026.3080758844585.6919241156

15654256291414918.7515617818691.2484382182

14966400991514716.2262600242-82.2262600242

14880310141614490.0210768303141.9789231697

14430422331715541.968651898198.031348102

14210374071814837.0559708266170.9440291734

15370343561914037.4237640101628.5762359899

15576305992015559.4793010654-149.4793010654

14416424852114650.4834312998-350.4834312998

14336384302214227.485970275270.514029725

14256404522315279.8074382075270.1925617925

15500260302414684.3830510401-30.3830510401

13930462962514780.8474101542-396.8474101542

14476348442615040.453689268747.5463107313

15526273792714633.3466749972-525.3466749972

14324478752814246.9283992437-16.9283992437

14972356482914924.1106928437-104.1106928437

14514425013014674.9110984656383.0889015344

14456438033115062.6379992459-48.6379992459

14270434813215411.6046730436334.3953269564

14534342793314977.1411641654-371.1411641654

14580413703414458.925653575815.0743464242

14774349663514781.7821423162-15.7821423162

14182414273614730.2472424535-158.2472424535

15334302413715469.6826980399184.3173019601

14292472283814567.9777391376398.0222608624

15530244643915134.1138518969-254.1138518969

15822212214014434.9965102297-4.9965102297

15064355214114735.7310044704-525.7310044704

14998280064214925.8555262127444.1444737873

14408380794315159.9747750445416.0252249555

14542423324414419.2930099088-3.2930099088

14014492234514671.9822710248-335.9822710248

14674333584614545.9803755928-289.9803755928

14132378194715444.694191576955.3058084231

15210359754814181.8087252939-251.8087252939

15274380854914895.4455732103-419.4455732103

14572352365015360.6306124781165.3693875219

15640209625114083.4125863785240.5874136215

14374458085214845.3439293293126.6560706707

14870361835314418.295962269495.7040377306

14998343995414337.1612106114118.8387893886

13574443305514357.2267943547-87.2267943547

15414320635614930.6538179774-396.6538179774

14914346415714488.77476728191.225232719

14730310495814887.8430849597-113.8430849597

14320386365914485.2227850656-303.2227850656

15102364686015182.2837159765151.7162840235

15136257456114123.7307002976168.2692997024

14244391986215542.2802292854-12.2802292854

15708215356315744.369322700777.6306772993

14594371356414853.2579949672210.7420050328

14296425816515321.5588081083-323.5588081083

14832330236614693.8550036146-285.8550036146

15466316446714428.8272779607113.1727220393

14956359696813999.411322757214.5886772428

15380290516914988.0463727216-314.0463727216

14572381807014710.0570277552-578.0570277552

15490314947114824.9667681986385.0332318014

15050313727214693.4811107498580.5188892502

14848362387314871.0179060444-299.0179060444

14566342127415760.5090313639-120.5090313639

14518331907514212.2186782963161.7813217037

14712391967614812.005148886157.9948511139

14266363927714923.175960681774.8240393183

7814304.3209539879-730.3209539879

7915068.7449160373345.2550839627

8014908.09561513555.9043848645

8115131.9328101857-401.9328101857

8214659.1452826672-339.1452826672

8314794.2452378089307.7547621911

8415462.4541026541-326.4541026541

8514624.1239843325-380.1239843325

8615724.8022627771-16.8022627771

8714752.6808143405-158.6808143405

8814413.3107240722-117.3107240722

8915008.922057672-176.922057672

9015094.8551010947371.1448989053

9114825.3406610634130.6593389366

9215256.4391341585123.5608658415

9314687.5611403908-115.5611403908

9415104.2024227143385.7975772857

9515111.8049109649-61.8049109649

9614808.577797625639.4222023744

9714934.8289549675-368.8289549675

9814998.5153729355-480.5153729355

9914624.248615287587.7513847125

10014798.9812140962-532.9812140962

Sheet1

Residuals

Predicted Price

Residuals

* Non Independence of Error VariablesA time series is constituted if data were collected over time.Examining the residuals over time, no pattern should be observed if the errors are independent.When a pattern is detected, the errors are said to be autocorrelated.Autocorrelation can be detected by graphing the residuals against time.

*Patterns in the appearance of the residuals over time indicates that autocorrelation exists.+++++++++++++++++++++++++TimeResidualResidualTime+++Note the runs of positive residuals,replaced by runs of negative residualsNote the oscillating behavior of the residuals around zero. 00 Non Independence of Error Variables

* OutliersAn outlier is an observation that is unusually small or large.Several possibilities need to be investigated when an outlier is observed:There was an error in recording the value.The point does not belong in the sample.The observation is valid.Identify outliers from the scatter diagram.It is customary to suspect an observation is an outlier if its |standard residual| > 2

*++++++++++++++++The outlier causes a shift in the regression line but, some outliers may be very influentialAn outlierAn influential observation

* Procedure for Regression DiagnosticsDevelop a model that has a theoretical basis.Gather data for the two variables in the model.Draw the scatter diagram to determine whether a linear model appears to be appropriate.Determine the regression equation.Check the required conditions for the errors.Check the existence of outliers and influential observationsAssess the model fit.If the model fits the data, use the regression equation.