Statistics for Business and Economics 8 th Edition Chapter 12 Multiple Regression Ch. 12-1 Copyright...

Statistics for Business and Economics 8th EditionChapter 12

Multiple RegressionCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall

Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall

Chapter GoalsAfter completing this chapter, you should be able to: Apply multiple regression analysis to business decision-making situationsAnalyze and interpret the computer output for a multiple regression modelPerform a hypothesis test for all regression coefficients or for a subset of coefficientsFit and interpret nonlinear regression modelsIncorporate qualitative variables into the regression model by using dummy variablesDiscuss model specification and analyze residualsCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


The Multiple Regression ModelIdea: Examine the linear relationship between 1 dependent (Y) & 2 or more independent variables (Xi)Multiple Regression Model with K Independent Variables:Y-interceptPopulation slopesRandom ErrorCh. 12-*12.1Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Multiple Regression EquationThe coefficients of the multiple regression model are estimated using sample dataEstimated (or predicted) value of yEstimated slope coefficientsMultiple regression equation with K independent variables:EstimatedinterceptIn this chapter we will always use a computer to obtain the regression slope coefficients and other regression summary measures.Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Three Dimensional GraphingTwo variable modelyx1x2yi yi

Estimation of Coefficients1. The xji terms are fixed numbers, or they are realizations of random variables Xj that are independent of the error terms, i 2. The expected value of the random variable Y is a linear function of the independent Xj variables.3. The error terms are normally distributed random variables with mean 0 and a constant variance, 2.

(The constant variance property is called homoscedasticity)Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall12.2Standard Multiple Regression Assumptions


4. The random error terms, i , are not correlated with one another, so that

5. It is not possible to find a set of numbers, c0, c1, . . . , ck, such that

(This is the property of no linear relation for the Xjs)(continued)Ch. 12-*Standard Multiple Regression AssumptionsCopyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Example: 2 Independent VariablesA distributor of frozen desert pies wants to evaluate factors thought to influence demand

Dependent variable: Pie sales (units per week)Independent variables: Price (in $) Advertising ($100s)Data are collected for 15 weeksCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Pie Sales ExampleSales = b0 + b1 (Price) + b2 (Advertising)Multiple regression equation:Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall

WeekPie SalesPrice($)Advertising($100s)13505.503.324607.503.333508.003.044308.004.553506.803.063807.504.074304.503.084706.403.794507.003.5104905.004.0113407.203.5123007.903.2134405.904.0144505.003.5153007.002.7


Estimating a Multiple Linear Regression EquationExcel can be used to generate the coefficients and measures of goodness of fit for multiple regression

Data / Data Analysis / RegressionCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Multiple Regression OutputCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall

Regression StatisticsMultiple R0.72213R Square0.52148Adjusted R Square0.44172Standard Error47.46341Observations15

ANOVA dfSSMSFSignificance FRegression229460.02714730.0136.538610.01201Residual1227033.3062252.776Total1456493.333

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept306.52619114.253892.682850.0199357.58835555.46404Price-24.9750910.83213-2.305650.03979-48.57626-1.37392Advertising74.1309625.967322.854780.0144917.55303130.70888


The Multiple Regression Equationb1 = -24.975: sales will decrease, on average, by 24.975 pies per week for each $1 increase in selling price, net of the effects of changes due to advertisingb2 = 74.131: sales will increase, on average, by 74.131 pies per week for each $100 increase in advertising, net of the effects of changes due to pricewhere Sales is in number of pies per week Price is in $ Advertising is in $100s.Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Example 12.3Y profit marginesX1: net revenue for deposit dollarX2 nunber of officesY = 1.56 + 0.237X1 -0.000249X2Corrleations YX1X1 -0.704X2 -.0.8680.941


Y = 1.33 -0.169X1Y = 1.55 0.000120X2


Explanatory Power of a Multiple Regression EquationCoefficient of Determination, R2

Reports the proportion of total variation in y explained by all x variables taken together

This is the ratio of the explained variability to total sample variabilityCh. 12-*12.3Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Estimation of Error VarianceConsider the population regression model

The unbiased estimate of the variance of the errors is

where

The square root of the variance, se , is called the standard error of the estimateCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Adjusted Coefficient of Determination, R2 never decreases when a new X variable is added to the model, even if the new variable is not an important predictor variableThis can be a disadvantage when comparing modelsWhat is the net effect of adding a new variable?We lose a degree of freedom when a new X variable is addedDid the new X variable add enough explanatory power to offset the loss of one degree of freedom?Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Adjusted Coefficient of Determination, Used to correct for the fact that adding non-relevant independent variables will still reduce the error sum of squares

(where n = sample size, K = number of independent variables)

Adjusted R2 provides a better comparison between multiple regression models with different numbers of independent variables Penalize excessive use of unimportant independent variablesValue is less than R2(continued)Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


44.2% of the variation in pie sales is explained by the variation in price and advertising, taking into account the sample size and number of independent variablesCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall





Coefficient of Multiple CorrelationThe coefficient of multiple correlation is the correlation between the predicted value and the observed value of the dependent variable

Is the square root of the multiple coefficient of determination Used as another measure of the strength of the linear relationship between the dependent variable and the independent variablesComparable to the correlation between Y and X in simple regressionCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Conf. Intervals and Hypothesis Tests for Regression CoefficientsThe variance of a coefficient estimate is affected by:the sample sizethe spread of the X variables the correlations between the independent variables, and the model error term

We are typically more interested in the regression coefficients bj than in the constant or intercept b0Ch. 12-*12.4Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Confidence IntervalsConfidence interval limits for the population slope j Example: Form a 95% confidence interval for the effect of changes in price (x1) on pie sales:-24.975 (2.1788)(10.832)So the interval is -48.576 < 1 < -1.374where t has (n K 1) d.f.Here, t has (15 2 1) = 12 d.f.Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall

CoefficientsStandard ErrorIntercept306.52619114.25389Price-24.9750910.83213Advertising74.1309625.96732


Confidence IntervalsConfidence interval for the population slope i Example: Excel output also reports these interval endpoints:Weekly sales are estimated to be reduced by between 1.37 to 48.58 pies for each increase of $1 in the selling price(continued)Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall

CoefficientsStandard ErrorLower 95%Upper 95%Intercept306.52619114.2538957.58835555.46404Price-24.9750910.83213-48.57626-1.37392Advertising74.1309625.9673217.55303130.70888


Hypothesis TestsUse t-tests for individual coefficientsShows if a specific independent variable is conditionally importantHypotheses:H0: j = 0 (no linear relationship)H1: j 0 (linear relationship does exist between xj and y)Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Evaluating Individual Regression CoefficientsH0: j = 0 (no linear relationship)H1: j 0 (linear relationship does exist between xi and y)

Test Statistic:

(df = n k 1)(continued)Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Evaluating Individual Regression Coefficientst-value for Price is t = -2.306, with p-value .0398

t-value for Advertising is t = 2.855, with p-value .0145(continued)Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall





H0: j = 0H1: j 0d.f. = 15-2-1 = 12 = .05t12, .025 = 2.1788The test statistic for each variable falls in the rejection region (p-values < .05)There is evidence that both Price and Advertising affect pie sales at = .05From Excel output: Reject H0 for each variableDecision:Conclusion:Reject H0Reject H0a/2=.025-t/2Do not reject H00t/2a/2=.025-2.17882.1788Example: Evaluating Individual Regression CoefficientsCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall

CoefficientsStandard Errort StatP-valuePrice-24.9750910.83213-2.305650.03979Advertising74.1309625.967322.854780.01449


Tests on Regression CoefficientsTests on All CoefficientsF-Test for Overall Significance of the ModelShows if there is a linear relationship between all of the X variables considered together and YUse F test statisticHypotheses: H0: 1 = 2 = = K = 0 (no linear relationship) H1: at least one i 0 (at least one independent variable affects Y) Ch. 12-*12.5Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


F-Test for Overall SignificanceTest statistic:

where F has K (numerator) and (n K 1) (denominator) degrees of freedom The decision rule isCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


F-Test for Overall SignificanceH0: 1 = 2 = 0H1: 1 and 2 not both zero = .05df1= 2 df2 = 12 Test Statistic:

Decision:

Conclusion:

Since F test statistic is in the rejection region (p-value < .05), reject H0There is evidence that at least one independent variable affects Y0 = .05F.05 = 3.885Reject H0Do not reject H0Critical Value: F = 3.885(continued)FCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Test on a Subset of Regression CoefficientsConsider a multiple regression model involving variables Xj and Zj , and the null hypothesis that the Z variable coefficients are all zero:

Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Tests on a Subset of Regression CoefficientsGoal: compare the error sum of squares for the complete model with the error sum of squares for the restricted modelFirst run a regression for the complete model and obtain SSENext run a restricted regression that excludes the Z variables (the number of variables excluded is R) and obtain the restricted error sum of squares SSE(R) Compute the F statistic and apply the decision rule for a significance level (continued)Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Newbold 12.41n=30 families Explain household milk consumptionY: milk consumption (quarts per week)X1: family income (dolars per week)X2: family sizeLS estimators :b0:-0.025, b1:0.052, b2:1.14SST: 162.1, SSE: 88.2: X3: number of preschool childeren ?SSE 83.7


Test the null hypothesis that Number of preschool childeren in a family dose not significantly ieffects weekly family milk consumption



12.41 Let

be the coefficient on the number of preschool children in the household

,

, F 1,26,.05 = 4.23

Therefore, do not reject

at the 5% level

_1081419590.unknown

_1081420368.unknown

_1383148208.unknown

_1065789789.unknown

PredictionGiven a population regression model

then given a new observation of a data point (x1,n+1, x 2,n+1, . . . , x K,n+1)the best linear unbiased forecast of yn+1 is

It is risky to forecast for new X values outside the range of the data used to estimate the model coefficients, because we do not have data to support that the linear model extends beyond the observed range.

^Ch. 12-*12.6Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Predictions from a Multiple Regression ModelPredict sales for a week in which the selling price is $5.50 and advertising is $350:Predicted sales is 428.62 piesNote that Advertising is in $100s, so $350 means that X2 = 3.5Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Transformations for Nonlinear Regression ModelsThe relationship between the dependent variable and an independent variable may not be linearCan review the scatter diagram to check for non-linear relationshipsExample: Quadratic model

The second independent variable is the square of the first variableCh. 12-*12.7Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Quadratic Model Transformations Let

And specify the model as

where:0 = Y intercept1 = regression coefficient for linear effect of X on Y2 = regression coefficient for quadratic effect on Yi = random error in Y for observation iQuadratic model form:Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Quadratic Regression ModelQuadratic models may be considered when the scatter diagram takes on one of the following shapes:X1YX1X1YYY1 < 01 > 01 < 01 > 01 = the coefficient of the linear term 2 = the coefficient of the squared termX12 > 02 > 02 < 02 < 0Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Testing for Significance: Quadratic EffectTesting the Quadratic EffectCompare the linear regression estimate

with quadratic regression estimate

Hypotheses (The quadratic term does not improve the model) (The quadratic term improves the model)H0: 2 = 0H1: 2 0Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Testing for Significance: Quadratic EffectTesting the Quadratic EffectHypotheses (The quadratic term does not improve the model) (The quadratic term improves the model)The test statistic isH0: 2 = 0H1: 2 0(continued)where: b2 = squared term slope coefficient 2 = hypothesized slope (zero) Sb = standard error of the slope2Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Testing for Significance: Quadratic EffectTesting the Quadratic Effect

Compare R2 from simple regression to R2 from the quadratic model

If R2 from the quadratic model is larger than R2 from the simple model, then the quadratic model is a better model

(continued)Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


ExmplePerformance by ageFirst increases and then falls again for older agesQuadartic relation



Sheet1

sigma 10.640.4096

sigma 20.560.3136

F1.306122449

nx20

ny40

sp0.028320.1682854717

mx2.71

my2.79

t-0.4753826885

Sheet2

0.8455172414

613

0.0065252855

0.0807792392

12.3794184031

1.03333333331.01653004553.458235214718.458235214711.5417647853

Sheet3

Y ( Performance)X ( age)X2 (age square)

3820400

1812144

52563136

27755625

61441936

58381444

11836889

42644096

51593481

3617289

Chart1

38

18

52

27

61

58

11

42

51

36

Sheet1

sigma 10.640.4096

sigma 20.560.3136

F1.306122449

nx20

ny40

sp0.028320.1682854717

mx2.71

my2.79

t-0.4753826885

Sheet2

0.8455172414

613

0.0065252855

0.0807792392

12.3794184031

1.03333333331.01653004553.458235214718.458235214711.5417647853

Sheet3

Y ( Performance)X ( age)X2 (age square)SUMMARY OUTPUT

3820400

1812144Regression Statistics

52563136Multiple R0.9876296727

27755625R Square0.9754123703

61441936Adjusted R Square0.9683873333

58381444Standard Error2.9895237316

11836889Observations10

42644096

51593481ANOVA

3617289dfSSMSFSignificance F

Regression22481.83923500881240.9196175044138.8480035970.0000023308

Residual762.56076499128.9372521416

Total92544.4

LaborCapitalOutput

20115410.1510.98560543312.5830901781425.6521424244CoefficientsStandard Errort StatP-value

25220568.5413.1326390222.9409289748579.3323792359Intercept-9.12559495563.782392223-2.41265168110.0465921946

18318499.1510.0975963093.165814209479.5067080849X Variable 13.01393969930.190829215215.79391130720.0000009884

40196911.1519.12704999582.873764756824.4996324708X Variable 2-0.03371967310.002036352-16.55886235380.0000007153

55185958.1524.67687445492.84075870181051.5156876149

28217749.9514.37892521932.9328641487632.5715140807

26256682.8513.55122881163.031433133616.1946601897

51175904.4823.23037073982.8093613917978.9376000743

611931148.5526.80796624622.86491315631152.0374278968

49201928.5622.4986709482.8882794581974.7367369982

LaborCapitalOutputLn(Y/x2)ln(X1/X2)

20115410.151.2715908181-1.7491998548SUMMARY OUTPUT

25220568.540.949444125-2.1747517215

18318499.150.450855269-2.8716796249Regression Statistics

40196911.151.5365928787-1.5892352051Multiple R0.9837101462

55185958.151.6446485168-1.2130226398R Square0.9676856518

28217749.951.2401091841-2.0476928434Adjusted R Square0.9636463583

26256682.850.9810977716-2.2870809065Standard Error0.0785864247

51175904.481.642574219-1.2329603412Observations10

611931148.551.7835653673-1.1518163247

49201928.561.530330091-1.4114846099ANOVA

dfSSMSFSignificance F

Regression11.47953059571.4795305957239.56804481250.0000003021

Residual80.04940660920.0061758261

13.157606073b0Total91.5289372049

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95,0%Upper 95,0%

Intercept2.57726181120.08599138429.97116329950.00000000172.37896532432.7755582982.37896532432.775558298

X Variable 10.71870181290.046433807415.47798581250.00000030210.6116252610.82577836480.6116252610.8257783648

Sheet3


Sheet1

sigma 10.640.4096

sigma 20.560.3136

F1.306122449

nx20

ny40

sp0.028320.1682854717

mx2.71

my2.79

t-0.4753826885

Sheet2

0.8455172414

613

0.0065252855

0.0807792392

12.3794184031

1.03333333331.01653004553.458235214718.458235214711.5417647853

Sheet3

Y ( Performance)X ( age)X2 (age square)SUMMARY OUTPUT

3820400

1812144Regression Statistics

52563136Multiple R0.9876296727

27755625R Square0.9754123703

61441936Adjusted R Square0.9683873333

58381444Standard Error2.9895237316

11836889Observations10

42644096

51593481ANOVA

3617289dfSSMSFSignificance F

Regression22481.83923500881240.9196175044138.8480035970.0000023308

Residual762.56076499128.9372521416

Total92544.4

LaborCapitalOutput

20115410.1510.98560543312.5830901781425.6521424244CoefficientsStandard Errort StatP-value

25220568.5413.1326390222.9409289748579.3323792359Intercept-9.12559495563.782392223-2.41265168110.0465921946

18318499.1510.0975963093.165814209479.5067080849X Variable 13.01393969930.190829215215.79391130720.0000009884

40196911.1519.12704999582.873764756824.4996324708X Variable 2-0.03371967310.002036352-16.55886235380.0000007153

55185958.1524.67687445492.84075870181051.5156876149

28217749.9514.37892521932.9328641487632.5715140807

26256682.8513.55122881163.031433133616.1946601897

51175904.4823.23037073982.8093613917978.9376000743

611931148.5526.80796624622.86491315631152.0374278968

49201928.5622.4986709482.8882794581974.7367369982


20115410.151.2715908181-1.7491998548SUMMARY OUTPUT

25220568.540.949444125-2.1747517215


40196911.151.5365928787-1.5892352051Multiple R0.9837101462

55185958.151.6446485168-1.2130226398R Square0.9676856518


26256682.850.9810977716-2.2870809065Standard Error0.0785864247

51175904.481.642574219-1.2329603412Observations10

611931148.551.7835653673-1.1518163247

49201928.561.530330091-1.4114846099ANOVA


Regression11.47953059571.4795305957239.56804481250.0000003021

Residual80.04940660920.0061758261

13.157606073b0Total91.5289372049


Intercept2.57726181120.08599138429.97116329950.00000000172.37896532432.7755582982.37896532432.775558298

X Variable 10.71870181290.046433807415.47798581250.00000030210.6116252610.82577836480.6116252610.8257783648

Example: Quadratic ModelPurity increases as filter time increases:Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall

PurityFilterTime31728315522733840105412671370147815851587169917


Chart6

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7

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Purity

Time

Purity

Purity vs. Time

Scatter

5

7

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30

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47

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70

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Purity

X

Y

Scatter Diagram

DataCopy

TimePurity

15

27

38

512

722

830

1035

1247

1367

1470

1578

1585

1687

1799

SLR

Regression Analysis

Regression Statistics

Multiple R0.9730725797

R Square0.9468702454

Adjusted R Square0.9424427659

Standard Error8.0957938003

Observations14

ANOVA


Regression114016.926044352814016.9260443528213.86213869850.0000000052

Residual12786.502527075865.5418772563

Total1314803.4285714286

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%

Intercept-12.09458483754.5579211896-2.65353092660.0210413723-22.0254418323-2.1637278428

Time5.95162454870.406975788814.62402607690.00000000525.06490049386.8383486037

RESIDUAL OUTPUT

ObservationPredicted PurityResiduals

1-6.142960288811.1429602888

2-0.19133574017.1913357401

35.76028880872.2397111913

417.6635379061-5.6635379061

529.5667870036-7.5667870036

635.5184115523-5.5184115523

747.4216606498-12.4216606498

859.3249097473-12.3249097473

965.2765342961.723465704

1071.2281588448-1.2281588448

1177.17978339350.8202166065

1277.17978339357.8202166065

1383.13140794223.8685920578

1489.0830324919.916967509

SLR

1

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13

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Time

Residuals

Time Residual Plot

MR

Regression Analysis



R Square0.9906914106



Observations14

ANOVA


Regression214665.62953260117332.8147663005585.35214117330

Residual11137.799038827512.527185348

Total1314803.4285714286


Intercept4.62966632133.06137873211.51228146740.1586479489-2.108386245311.3677188878

Time0.18585157010.82075477540.22643982790.8250121623-1.62061842421.9923215643

Time-squared0.31978197880.04443831887.19608633340.00001761370.22197384910.4175901086

RESIDUAL OUTPUT


15.1352998702-0.1352998702

26.28049737670.7195026233

38.0652588409-0.0652588409

413.5534736424-1.5534736424

521.59994427460.4000557254

626.58252552723.4174744728

738.4663799053-3.4663799053

852.9084901142-5.9084901142

961.08889115515.9111088449

1069.90885615370.0911438463

1179.36838511-1.36838511

1279.368385115.63161489

1389.467478024-2.467478024

14100.2061348956-1.2061348956

MR

1

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Residuals

Time Residual Plot

MR2

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49

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169

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Time-squared

Residuals

Time-squared Residual Plot

MR3

Regression Analysis



R Square0.9905215027



Observations14

ANOVA


Regression214635.16745644087317.5837282204574.76075272810

Residual11140.046829273512.731529934

Total1314775.2142857143


Intercept3.72339133823.08624647521.20644652590.2529517479-3.069394789610.5161774659

Time0.50387209740.82742181030.60896641970.5549146396-1.31727194922.325016144

Time-squared0.30258438030.04479929376.75422212060.00003139640.20398174990.4011870107

RESIDUAL OUTPUT


14.5298478159-1.5298478159

25.94147305421.0585269458

37.95826705320.0417329468

413.80736133291.1926386671

522.0771306552-0.0771306552

627.11976845722.8802315428

739.0205503432-4.0205503432

853.3420072716-6.3420072716

961.41048887685.5895111232

1070.0841392425-0.0841392425

1179.3629583689-1.3629583689

1279.36295836895.6370416311

1389.2469462559-2.2469462559

1499.7361029035-0.7361029035

MR3

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Residuals

Time Residual Plot

MR4

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Time-squared

Residuals


SLR2

Regression Analysis



R Square0.9931003854



Observations14

ANOVA


Regression214494.72573919767247.3628695988791.64597260530

Residual11100.7028322319.1548029301

Total1314595.4285714286


Intercept2.12818417982.61706680670.81319444130.4333554776-3.6319439387.8883122976

Time1.31518326560.70163487341.87445538350.0876529987-0.22910545952.8594719907

Time-squared0.25780759160.0379887826.7864137320.00003007680.17419480390.3414203792

RESIDUAL OUTPUT


13.701175037-0.701175037

25.78978107741.2102189226

38.394002301-0.394002301

415.1492902976-0.1492902976

523.9670390269-1.9670390269

629.14933616633.8506638337

741.0607759947-1.0607759947

855.0346765558-5.0346765558

962.79504961114.2049503889

1071.0710378496-1.0710378496

1179.8626412712-1.8626412712

1279.86264127125.1373587288

1389.1698598761-2.1698598761

1498.99269366410.0073063359

SLR2

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Time Residual Plot

Sheet1

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Time-squared

Residuals


Sheet2

Regression Analysis



R Square0.9949366297



Observations14

ANOVA


Regression214556.77569461997278.38784730991080.73300705430

Residual1174.08144823736.7346771125

Total1314630.8571428571


Intercept1.5386984382.24465034050.68549582550.5072203925-3.40174614966.4791430257

Time1.56496427930.60179012372.60051505930.02467132580.24043247762.889496081

Time-squared0.24515555310.03258286427.5240639260.00001164640.17344111630.31686999

RESIDUAL OUTPUT


13.3488182705-0.3488182705

25.64924920921.3507507908

38.4399912542-0.4399912542

415.4924086631-0.4924086631

524.5060704972-2.5060704972

629.74836807373.2516319263

741.7038965456-1.7038965456

855.6206694426-1.6206694426

963.31452255063.6854774494

1071.4986867648-1.4986867648

1180.1731620854-2.1731620854

1280.17316208544.8268379146

1389.3379485123-2.3379485123

1498.99304604540.0069539546

Sheet2

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Time Residual Plot

Sheet3

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Time-squared

Residuals


Regression Analysis



R Square0.9688779767



Observations14

ANOVA


Regression114175.515265600814175.5152656008373.5790439750.0000000002

Residual12455.341877256337.945156438

Total1314630.8571428571


Intercept-11.28267148013.4680515734-3.25331709790.0069137748-18.838906613-3.7264363473

Time5.9851985560.309661568519.32819298270.00000000025.31050396916.6598931428

RESIDUAL OUTPUT


1-5.29747292428.2974729242

20.68772563186.3122743682

36.67292418771.3270758123

418.6433212996-3.6433212996

530.6137184116-8.6137184116

636.5989169675-3.5989169675

748.5693140794-8.5693140794

860.5397111913-6.5397111913

966.52490974730.4750902527

1072.5101083032-2.5101083032

1178.4953068592-0.4953068592

1278.49530685926.5046931408

1384.48050541522.5194945848

1490.46570397118.5342960289

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Time Residual Plot

Filter

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9917289

Purity

Time

Purity

Purity vs. Time

Example: Quadratic ModelSimple regression results:y = -11.283 + 5.985 Time(continued)^t statistic, F statistic, and R2 are all high, but the residuals are not random:Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall

Regression StatisticsR Square0.96888Adjusted R Square0.96628Standard Error6.15997

CoefficientsStandard Errort StatP-valueIntercept-11.282673.46805-3.253320.00691Time5.985200.3096619.328192.078E-10

FSignificance F373.579042.0778E-10


Chart5

8.2974729242

6.3122743682

1.3270758123

-3.6433212996

-8.6137184116

-3.5989169675

-8.5693140794

-6.5397111913

0.4750902527

-2.5101083032

-0.4953068592

6.5046931408

2.5194945848

8.5342960289

Time

Residuals

Time Residual Plot

Scatter

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Scatter Diagram

DataCopy

TimePurity

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1687

1799

SLR

Regression Analysis



R Square0.9468702454



Observations14

ANOVA


Regression114016.926044352814016.9260443528213.86213869850.0000000052

Residual12786.502527075865.5418772563

Total1314803.4285714286


Intercept-12.09458483754.5579211896-2.65353092660.0210413723-22.0254418323-2.1637278428

Time5.95162454870.406975788814.62402607690.00000000525.06490049386.8383486037

RESIDUAL OUTPUT


1-6.142960288811.1429602888

2-0.19133574017.1913357401

35.76028880872.2397111913

417.6635379061-5.6635379061

529.5667870036-7.5667870036

635.5184115523-5.5184115523

747.4216606498-12.4216606498

859.3249097473-12.3249097473

965.2765342961.723465704

1071.2281588448-1.2281588448

1177.17978339350.8202166065

1277.17978339357.8202166065

1383.13140794223.8685920578

1489.0830324919.916967509

SLR

0

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Time

Residuals

Time Residual Plot

MR

Regression Analysis



R Square0.9906914106



Observations14

ANOVA


Regression214665.62953260117332.8147663005585.35214117330

Residual11137.799038827512.527185348

Total1314803.4285714286


Intercept4.62966632133.06137873211.51228146740.1586479489-2.108386245311.3677188878

Time0.18585157010.82075477540.22643982790.8250121623-1.62061842421.9923215643

Time-squared0.31978197880.04443831887.19608633340.00001761370.22197384910.4175901086

RESIDUAL OUTPUT


15.1352998702-0.1352998702

26.28049737670.7195026233

38.0652588409-0.0652588409

413.5534736424-1.5534736424

521.59994427460.4000557254

626.58252552723.4174744728

738.4663799053-3.4663799053

852.9084901142-5.9084901142

961.08889115515.9111088449

1069.90885615370.0911438463

1179.36838511-1.36838511

1279.368385115.63161489

1389.467478024-2.467478024

14100.2061348956-1.2061348956

MR

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Time Residual Plot

MR2

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Time-squared

Residuals


MR3

Regression Analysis



R Square0.9905215027



Observations14

ANOVA


Regression214635.16745644087317.5837282204574.76075272810

Residual11140.046829273512.731529934

Total1314775.2142857143


Intercept3.72339133823.08624647521.20644652590.2529517479-3.069394789610.5161774659

Time0.50387209740.82742181030.60896641970.5549146396-1.31727194922.325016144

Time-squared0.30258438030.04479929376.75422212060.00003139640.20398174990.4011870107

RESIDUAL OUTPUT


14.5298478159-1.5298478159

25.94147305421.0585269458

37.95826705320.0417329468

413.80736133291.1926386671

522.0771306552-0.0771306552

627.11976845722.8802315428

739.0205503432-4.0205503432

853.3420072716-6.3420072716

961.41048887685.5895111232

1070.0841392425-0.0841392425

1179.3629583689-1.3629583689

1279.36295836895.6370416311

1389.2469462559-2.2469462559

1499.7361029035-0.7361029035

MR3

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Time

Residuals

Time Residual Plot

MR4

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Time-squared

Residuals


SLR2

Regression Analysis



R Square0.9931003854



Observations14

ANOVA


Regression214494.72573919767247.3628695988791.64597260530

Residual11100.7028322319.1548029301

Total1314595.4285714286


Intercept2.12818417982.61706680670.81319444130.4333554776-3.6319439387.8883122976

Time1.31518326560.70163487341.87445538350.0876529987-0.22910545952.8594719907

Time-squared0.25780759160.0379887826.7864137320.00003007680.17419480390.3414203792

RESIDUAL OUTPUT


13.701175037-0.701175037

25.78978107741.2102189226

38.394002301-0.394002301

415.1492902976-0.1492902976

523.9670390269-1.9670390269

629.14933616633.8506638337

741.0607759947-1.0607759947

855.0346765558-5.0346765558

962.79504961114.2049503889

1071.0710378496-1.0710378496

1179.8626412712-1.8626412712

1279.86264127125.1373587288

1389.1698598761-2.1698598761

1498.99269366410.0073063359

SLR2

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Time Residual Plot

Sheet1

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Time-squared

Residuals


Sheet2

Regression Analysis



R Square0.9949366297



Observations14

ANOVA


Regression214556.77569461997278.38784730991080.73300705430

Residual1174.08144823736.7346771125

Total1314630.8571428571


Intercept1.5386984382.24465034050.68549582550.5072203925-3.40174614966.4791430257

Time1.56496427930.60179012372.60051505930.02467132580.24043247762.889496081

Time-squared0.24515555310.03258286427.5240639260.00001164640.17344111630.31686999

RESIDUAL OUTPUT


13.3488182705-0.3488182705

25.64924920921.3507507908

38.4399912542-0.4399912542

415.4924086631-0.4924086631

524.5060704972-2.5060704972

629.74836807373.2516319263

741.7038965456-1.7038965456

855.6206694426-1.6206694426

963.31452255063.6854774494

1071.4986867648-1.4986867648

1180.1731620854-2.1731620854

1280.17316208544.8268379146

1389.3379485123-2.3379485123

1498.99304604540.0069539546

Sheet2

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Time Residual Plot

Sheet3

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Time-squared

Residuals


Regression Analysis



R Square0.9688779767



Observations14

ANOVA


Regression114175.515265600814175.5152656008373.5790439750.0000000002

Residual12455.341877256337.945156438

Total1314630.8571428571


Intercept-11.28267148013.4680515734-3.25331709790.0069137748-18.838906613-3.7264363473

Time5.9851985560.309661568519.32819298270.00000000025.31050396916.6598931428

RESIDUAL OUTPUT


1-5.29747292428.2974729242

20.68772563186.3122743682

36.67292418771.3270758123

418.6433212996-3.6433212996

530.6137184116-8.6137184116

636.5989169675-3.5989169675

748.5693140794-8.5693140794

860.5397111913-6.5397111913

966.52490974730.4750902527

1072.5101083032-2.5101083032

1178.4953068592-0.4953068592

1278.49530685926.5046931408

1384.48050541522.5194945848

1490.46570397118.5342960289

0

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Time Residual Plot

Filter


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Purity

Time

Purity

Purity vs. Time

Example: Quadratic ModelQuadratic regression results:y = 1.539 + 1.565 Time + 0.245 (Time)2^(continued)The quadratic term is significant and improves the model: R2 is higher and se is lower, residuals are now randomCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall

CoefficientsStandard Errort StatP-valueIntercept1.538702.244650.685500.50722Time1.564960.601792.600520.02467Time-squared0.245160.032587.524061.165E-05

Regression StatisticsR Square0.99494Adjusted R Square0.99402Standard Error2.59513

FSignificance F1080.73302.368E-13


Chart3

-0.3488182705

1.3507507908

-0.4399912542

-0.4924086631

-2.5060704972

3.2516319263

-1.7038965456

-1.6206694426

3.6854774494

-1.4986867648

-2.1731620854

4.8268379146

-2.3379485123

0.0069539546

Time

Residuals

Time Residual Plot

Scatter

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DataCopy

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SLR

Regression Analysis



R Square0.9468702454



Observations14

ANOVA


Regression114016.926044352814016.9260443528213.86213869850.0000000052

Residual12786.502527075865.5418772563

Total1314803.4285714286


Intercept-12.09458483754.5579211896-2.65353092660.0210413723-22.0254418323-2.1637278428

Time5.95162454870.406975788814.62402607690.00000000525.06490049386.8383486037

RESIDUAL OUTPUT


1-6.142960288811.1429602888

2-0.19133574017.1913357401

35.76028880872.2397111913

417.6635379061-5.6635379061

529.5667870036-7.5667870036

635.5184115523-5.5184115523

747.4216606498-12.4216606498

859.3249097473-12.3249097473

965.2765342961.723465704

1071.2281588448-1.2281588448

1177.17978339350.8202166065

1277.17978339357.8202166065

1383.13140794223.8685920578

1489.0830324919.916967509

SLR

0

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Time

Residuals

Time Residual Plot

MR

Regression Analysis



R Square0.9906914106



Observations14

ANOVA


Regression214665.62953260117332.8147663005585.35214117330

Residual11137.799038827512.527185348

Total1314803.4285714286


Intercept4.62966632133.06137873211.51228146740.1586479489-2.108386245311.3677188878

Time0.18585157010.82075477540.22643982790.8250121623-1.62061842421.9923215643

Time-squared0.31978197880.04443831887.19608633340.00001761370.22197384910.4175901086

RESIDUAL OUTPUT


15.1352998702-0.1352998702

26.28049737670.7195026233

38.0652588409-0.0652588409

413.5534736424-1.5534736424

521.59994427460.4000557254

626.58252552723.4174744728

738.4663799053-3.4663799053

852.9084901142-5.9084901142

961.08889115515.9111088449

1069.90885615370.0911438463

1179.36838511-1.36838511

1279.368385115.63161489

1389.467478024-2.467478024

14100.2061348956-1.2061348956

MR

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Time

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Time Residual Plot

MR2

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Time-squared

Residuals


MR3

Regression Analysis



R Square0.9905215027



Observations14

ANOVA


Regression214635.16745644087317.5837282204574.76075272810

Residual11140.046829273512.731529934

Total1314775.2142857143


Intercept3.72339133823.08624647521.20644652590.2529517479-3.069394789610.5161774659

Time0.50387209740.82742181030.60896641970.5549146396-1.31727194922.325016144

Time-squared0.30258438030.04479929376.75422212060.00003139640.20398174990.4011870107

RESIDUAL OUTPUT


14.5298478159-1.5298478159

25.94147305421.0585269458

37.95826705320.0417329468

413.80736133291.1926386671

522.0771306552-0.0771306552

627.11976845722.8802315428

739.0205503432-4.0205503432

853.3420072716-6.3420072716

961.41048887685.5895111232

1070.0841392425-0.0841392425

1179.3629583689-1.3629583689

1279.36295836895.6370416311

1389.2469462559-2.2469462559

1499.7361029035-0.7361029035

MR3

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Time

Residuals

Time Residual Plot

MR4

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Time-squared

Residuals


Sheet1

Regression Analysis



R Square0.9931003854



Observations14

ANOVA


Regression214494.72573919767247.3628695988791.64597260530

Residual11100.7028322319.1548029301

Total1314595.4285714286


Intercept2.12818417982.61706680670.81319444130.4333554776-3.6319439387.8883122976

Time1.31518326560.70163487341.87445538350.0876529987-0.22910545952.8594719907

Time-squared0.25780759160.0379887826.7864137320.00003007680.17419480390.3414203792

RESIDUAL OUTPUT


13.701175037-0.701175037

25.78978107741.2102189226

38.394002301-0.394002301

415.1492902976-0.1492902976

523.9670390269-1.9670390269

629.14933616633.8506638337

741.0607759947-1.0607759947

855.0346765558-5.0346765558

962.79504961114.2049503889

1071.0710378496-1.0710378496

1179.8626412712-1.8626412712

1279.86264127125.1373587288

1389.1698598761-2.1698598761

1498.99269366410.0073063359

Sheet1

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Sheet2

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Time-squared

Residuals


Sheet3

Regression Analysis



R Square0.9949366297



Observations14

ANOVA


Regression214556.77569461997278.38784730991080.73300705430

Residual1174.08144823736.7346771125

Total1314630.8571428571


Intercept1.5386984382.24465034050.68549582550.5072203925-3.40174614966.4791430257

Time1.56496427930.60179012372.60051505930.02467132580.24043247762.889496081

Time-squared0.24515555310.03258286427.5240639260.00001164640.17344111630.31686999

RESIDUAL OUTPUT


13.3488182705-0.3488182705

25.64924920921.3507507908

38.4399912542-0.4399912542

415.4924086631-0.4924086631

524.5060704972-2.5060704972

629.74836807373.2516319263

741.7038965456-1.7038965456

855.6206694426-1.6206694426

963.31452255063.6854774494

1071.4986867648-1.4986867648

1180.1731620854-2.1731620854

1280.17316208544.8268379146

1389.3379485123-2.3379485123

1498.99304604540.0069539546

Sheet3

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Time

Residuals

Time Residual Plot

0

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Time-squared

Residuals


Filter


311

724

839

15525

22749

33864

4010100

5412144

6713169

7014196

7815225

8515225

8716256

9917289

0

0

0

0

0

0

0

0

0

0

0

0

0

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Purity

Time

Purity

Purity vs. Time

Chart4

-0.3488182705

1.3507507908

-0.4399912542

-0.4924086631

-2.5060704972

3.2516319263

-1.7038965456

-1.6206694426

3.6854774494

-1.4986867648

-2.1731620854

4.8268379146

-2.3379485123

0.0069539546

Time-squared

Residuals


Scatter

5

7

8

12

22

30

35

47

67

70

78

85

87

99

Purity

X

Y

Scatter Diagram

DataCopy

TimePurity

15

27

38

512

722

830

1035

1247

1367

1470

1578

1585

1687

1799

SLR

Regression Analysis



R Square0.9468702454



Observations14

ANOVA


Regression114016.926044352814016.9260443528213.86213869850.0000000052

Residual12786.502527075865.5418772563

Total1314803.4285714286


Intercept-12.09458483754.5579211896-2.65353092660.0210413723-22.0254418323-2.1637278428

Time5.95162454870.406975788814.62402607690.00000000525.06490049386.8383486037

RESIDUAL OUTPUT


1-6.142960288811.1429602888

2-0.19133574017.1913357401

35.76028880872.2397111913

417.6635379061-5.6635379061

529.5667870036-7.5667870036

635.5184115523-5.5184115523

747.4216606498-12.4216606498

859.3249097473-12.3249097473

965.2765342961.723465704

1071.2281588448-1.2281588448

1177.17978339350.8202166065

1277.17978339357.8202166065

1383.13140794223.8685920578

1489.0830324919.916967509

SLR

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Time

Residuals

Time Residual Plot

MR

Regression Analysis



R Square0.9906914106



Observations14

ANOVA


Regression214665.62953260117332.8147663005585.35214117330

Residual11137.799038827512.527185348

Total1314803.4285714286


Intercept4.62966632133.06137873211.51228146740.1586479489-2.108386245311.3677188878

Time0.18585157010.82075477540.22643982790.8250121623-1.62061842421.9923215643

Time-squared0.31978197880.04443831887.19608633340.00001761370.22197384910.4175901086

RESIDUAL OUTPUT


15.1352998702-0.1352998702

26.28049737670.7195026233

38.0652588409-0.0652588409

413.5534736424-1.5534736424

521.59994427460.4000557254

626.58252552723.4174744728

738.4663799053-3.4663799053

852.9084901142-5.9084901142

961.08889115515.9111088449

1069.90885615370.0911438463

1179.36838511-1.36838511

1279.368385115.63161489

1389.467478024-2.467478024

14100.2061348956-1.2061348956

MR

0

0

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0

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Time

Residuals

Time Residual Plot

MR2

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0

0

0

0

Time-squared

Residuals


MR3

Regression Analysis



R Square0.9905215027



Observations14

ANOVA


Regression214635.16745644087317.5837282204574.76075272810

Residual11140.046829273512.731529934

Total1314775.2142857143


Intercept3.72339133823.08624647521.20644652590.2529517479-3.069394789610.5161774659

Time0.50387209740.82742181030.60896641970.5549146396-1.31727194922.325016144

Time-squared0.30258438030.04479929376.75422212060.00003139640.20398174990.4011870107

RESIDUAL OUTPUT


14.5298478159-1.5298478159

25.94147305421.0585269458

37.95826705320.0417329468

413.80736133291.1926386671

522.0771306552-0.0771306552

627.11976845722.8802315428

739.0205503432-4.0205503432

853.3420072716-6.3420072716

961.41048887685.5895111232

1070.0841392425-0.0841392425

1179.3629583689-1.3629583689

1279.36295836895.6370416311

1389.2469462559-2.2469462559

1499.7361029035-0.7361029035

MR3

0

0

0

0

0

0

0

0

0

0

0

0

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0

Time

Residuals

Time Residual Plot

MR4

0

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0

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0

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Time-squared

Residuals


Sheet1

Regression Analysis



R Square0.9931003854



Observations14

ANOVA


Regression214494.72573919767247.3628695988791.64597260530

Residual11100.7028322319.1548029301

Total1314595.4285714286


Intercept2.12818417982.61706680670.81319444130.4333554776-3.6319439387.8883122976

Time1.31518326560.70163487341.87445538350.0876529987-0.22910545952.8594719907

Time-squared0.25780759160.0379887826.7864137320.00003007680.17419480390.3414203792

RESIDUAL OUTPUT


13.701175037-0.701175037

25.78978107741.2102189226

38.394002301-0.394002301

415.1492902976-0.1492902976

523.9670390269-1.9670390269

629.14933616633.8506638337

741.0607759947-1.0607759947

855.0346765558-5.0346765558

962.79504961114.2049503889

1071.0710378496-1.0710378496

1179.8626412712-1.8626412712

1279.86264127125.1373587288

1389.1698598761-2.1698598761

1498.99269366410.0073063359

Sheet1

0

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Time

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Time Residual Plot

Sheet2

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Time-squared

Residuals


Sheet3

Regression Analysis



R Square0.9949366297



Observations14

ANOVA


Regression214556.77569461997278.38784730991080.73300705430

Residual1174.08144823736.7346771125

Total1314630.8571428571


Intercept1.5386984382.24465034050.68549582550.5072203925-3.40174614966.4791430257

Time1.56496427930.60179012372.60051505930.02467132580.24043247762.889496081

Time-squared0.24515555310.03258286427.5240639260.00001164640.17344111630.31686999

RESIDUAL OUTPUT


13.3488182705-0.3488182705

25.64924920921.3507507908

38.4399912542-0.4399912542

415.4924086631-0.4924086631

524.5060704972-2.5060704972

629.74836807373.2516319263

741.7038965456-1.7038965456

855.6206694426-1.6206694426

963.31452255063.6854774494

1071.4986867648-1.4986867648

1180.1731620854-2.1731620854

1280.17316208544.8268379146

1389.3379485123-2.3379485123

1498.99304604540.0069539546

Sheet3

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Time Residual Plot

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Filter


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Purity

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Purity

Purity vs. Time

Logarithmic TransformationsOriginal exponential model

Transformed logarithmic modelThe Exponential Model:Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Interpretation of coefficientsFor the logarithmic model:

When both dependent and independent variables are logged:The estimated coefficient bk of the independent variable Xk can be interpreted as

a 1 percent change in Xk leads to an estimated bk percentage change in the average value of Y

bk is the elasticity of Y with respect to a change in XkCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Example Cobb-Douglas Production FunctionProduction function output as a function of labor and capital


Sheet1

sigma 10.640.4096

sigma 20.560.3136

F1.306122449

nx20

ny40

sp0.028320.1682854717

mx2.71

my2.79

t-0.4753826885

Sheet2

0.8455172414

613

0.0065252855

0.0807792392

12.3794184031

1.03333333331.01653004553.458235214718.458235214711.5417647853

Sheet3


3820400

1812144

52563136

27755625

61441936

58381444

11836889

42644096

51593481

3617289

LaborCapitalOutput

20115410.1510.98560543312.5830901781425.6521424244

25220568.5413.1326390222.9409289748579.3323792359

18318499.1510.0975963093.165814209479.5067080849

40196911.1519.12704999582.873764756824.4996324708

55185958.1524.67687445492.84075870181051.5156876149

28217749.9514.37892521932.9328641487632.5715140807

26256682.8513.55122881163.031433133616.1946601897

51175904.4823.23037073982.8093613917978.9376000743

611931148.5526.80796624622.86491315631152.0374278968

49201928.5622.4986709482.8882794581974.7367369982

Original and new VariablesOriginal data and new variables


Sheet1

sigma 10.640.4096

sigma 20.560.3136

F1.306122449

nx20

ny40

sp0.028320.1682854717

mx2.71

my2.79

t-0.4753826885

Sheet2

0.8455172414

613

0.0065252855

0.0807792392

12.3794184031

1.03333333331.01653004553.458235214718.458235214711.5417647853

Sheet3


3820400

1812144

52563136

27755625

61441936

58381444

11836889

42644096

51593481

3617289

LaborCapitalOutput

20115410.1510.98560543312.5830901781425.6521424244

25220568.5413.1326390222.9409289748579.3323792359

18318499.1510.0975963093.165814209479.5067080849

40196911.1519.12704999582.873764756824.4996324708

55185958.1524.67687445492.84075870181051.5156876149

28217749.9514.37892521932.9328641487632.5715140807

26256682.8513.55122881163.031433133616.1946601897

51175904.4823.23037073982.8093613917978.9376000743

611931148.5526.80796624622.86491315631152.0374278968

49201928.5622.4986709482.8882794581974.7367369982


20115410.151.2715908181-1.7491998548

25220568.540.949444125-2.1747517215

18318499.150.450855269-2.8716796249

40196911.151.5365928787-1.5892352051

55185958.151.6446485168-1.2130226398

28217749.951.2401091841-2.0476928434

26256682.850.9810977716-2.2870809065

51175904.481.642574219-1.2329603412

611931148.551.7835653673-1.1518163247

49201928.561.530330091-1.4114846099

Regression Output


Sheet1

sigma 10.640.4096

sigma 20.560.3136

F1.306122449

nx20

ny40

sp0.028320.1682854717

mx2.71

my2.79

t-0.4753826885

Sheet2

0.8455172414

613

0.0065252855

0.0807792392

12.3794184031

1.03333333331.01653004553.458235214718.458235214711.5417647853

Sheet3


3820400

1812144

52563136

27755625

61441936

58381444

11836889

42644096

51593481

3617289

LaborCapitalOutput

20115410.1510.98560543312.5830901781425.6521424244

25220568.5413.1326390222.9409289748579.3323792359

18318499.1510.0975963093.165814209479.5067080849

40196911.1519.12704999582.873764756824.4996324708

55185958.1524.67687445492.84075870181051.5156876149

28217749.9514.37892521932.9328641487632.5715140807

26256682.8513.55122881163.031433133616.1946601897

51175904.4823.23037073982.8093613917978.9376000743

611931148.5526.80796624622.86491315631152.0374278968

49201928.5622.4986709482.8882794581974.7367369982


20115410.151.2715908181-1.7491998548SUMMARY OUTPUT

25220568.540.949444125-2.1747517215


40196911.151.5365928787-1.5892352051Multiple R0.9837101462

55185958.151.6446485168-1.2130226398R Square0.9676856518


26256682.850.9810977716-2.2870809065Standard Error0.0785864247

51175904.481.642574219-1.2329603412Observations10

611931148.551.7835653673-1.1518163247

49201928.561.530330091-1.4114846099ANOVA


Regression11.47953059571.4795305957239.56804481250.0000003021

Residual80.04940660920.0061758261

Total91.5289372049


Intercept2.57726181120.08599138429.97116329950.00000000172.37896532432.7755582982.37896532432.775558298

X Variable 10.71870181290.046433807415.47798581250.00000030210.6116252610.82577836480.6116252610.8257783648

Parameter valeusb1 =0.71B2= 1- 0.71 = 0.29Ln(b0) = 2.577 so b0 = exp(2.577) = 13.15The estimated cobb-douglas Prodcution Function isOutput = 13.15*L0.71K0.29,


Dummy Variables for Regression ModelsA dummy variable is a categorical independent variable with two levels:yes or no, on or off, male or femalerecorded as 0 or 1Regression intercepts are different if the variable is significantAssumes equal slopes for other variablesIf more than two levels, the number of dummy variables needed is (number of levels - 1)Ch. 12-*12.8Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Dummy Variable Example Let:y = Pie Salesx1 = Pricex2 = Holiday (X2 = 1 if a holiday occurred during the week) (X2 = 0 if there was no holiday that week)Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Dummy Variable Example Same slope(continued)x1 (Price)y (sales)b0 + b2b0 HolidayNo HolidayDifferent interceptHoliday (x2 = 1)No Holiday (x2 = 0)If H0: 2 = 0 is rejected, thenHoliday has a significant effect on pie salesCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Interpreting the Dummy Variable CoefficientSales: number of pies sold per weekPrice: pie price in $

Holiday:Example:1 If a holiday occurred during the week0 If no holiday occurredb2 = 15: on average, sales were 15 pies greater in weeks with a holiday than in weeks without a holiday, given the same priceCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Differences in SlopeHypothesizes interaction between pairs of x variablesResponse to one x variable may vary at different levels of another x variable

Contains two-way cross product terms

Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Effect of InteractionGiven:

Without interaction term, effect of X1 on Y is measured by 1With interaction term, effect of X1 on Y is measured by 1 + 3 X2Effect changes as X2 changes Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Interaction Examplex2 = 1:y = 1 + 2x1 + 3(1) + 4x1(1) = 4 + 6x1 x2 = 0: y = 1 + 2x1 + 3(0) + 4x1(0) = 1 + 2x1 Slopes are different if the effect of x1 on y depends on x2 valuex148120010.51.5ySuppose x2 is a dummy variable and the estimated regression equation is ^^Ch. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Significance of Interaction TermThe coefficient b3 is an estimate of the difference in the coefficient of x1 when x2 = 1 compared to when x2 = 0The t statistic for b3 can be used to test the hypothesis

If we reject the null hypothesis we conclude that there is a difference in the slope coefficient for the two subgroupsCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Multiple Regression Analysis Application ProcedureAssumptions:The errors are normally distributedErrors have a constant varianceThe model errors are independentei = (yi yi)

Analysis of ResidualsThese residual plots are used in multiple regression:Residuals vs. yiResiduals vs. x1iResiduals vs. x2iResiduals vs. time (if time series data)

Chapter SummaryDeveloped the multiple regression modelTested the significance of the multiple regression modelDiscussed adjusted R2 ( R2 )Tested individual regression coefficientsTested portions of the regression modelUsed quadratic terms and log transformations in regression modelsExplained dummy variablesEvaluated interaction effectsDiscussed using residual plots to check model assumptionsCh. 12-*Copyright 2013 Pearson Education, Inc. Publishing as Prentice Hall


Rcrtv,dr 12.78n=93 rate scale of 1(low)-10(high)opinion of residence hall lifelevels of satisfactionsroom mates, floor, hall, resident advisorY. overall opinion X1, X4


Source df SS MSS F R-sqmodel 4 37.016 9.2540 9.958 0.312error 88 81.780 0.9293total 92 118.79param estimate t std errorinter. 3.950 5.84 0.676x1 0.106 1.69 0.063x2 0.122 1.70 0.072x3 0.0921.75 0.053x4 0.169 2.64 0.064



31.2% of the variation in the overall opinion of residence hall can be explained by the variation in the satisfaction with roommates, with floor, with hall, and with resident advisor. The F-test of the significance of the overall model shows that we reject

that all of the slope coefficients are jointly equal to zero in favor of that at least one slope coefficient is not equal to zero. The F-test yielded a p-value of .000.

All of the variables are significant at the .1 level of significance but not at the .05 level of significance except the variable satisfaction with resident advisor. The variable satisfaction with resident advisor is significant at all common levels of alpha.

_1382966321.unknown

_1382966322.unknown

Exercise 12.82developing countiry - 48 countriesY:per manifacturing growth x1 per agricultural growthx2 per exorts growthx3 per rate of inflation



Regression Analysis: y_manufgrowt versus x1_aggrowth, x2_exportgro, ...

The regression equation is

y_manufgrowth = 2.15 + 0.493 x1_aggrowth + 0.270 x2_exportgrowth

- 0.117 x3_inflation

Predictor Coef SE Coef T P VIF

Constant 2.1505 0.9695 2.22 0.032

x1_aggro 0.4934 0.2020 2.44 0.019 1.0

x2_expor 0.26991 0.06494 4.16 0.000 1.0

x3_infla -0.11709 0.05204 -2.25 0.030 1.0

S = 3.624 R-Sq = 39.3% R-Sq(adj) = 35.1%

Analysis of Variance

Source DF SS MS F P

Regression 3 373.98 124.66 9.49 0.000

Residual Error 44 577.97 13.14

Total 47 951.95


39.3% of the variation in the manufacturing growth can be explained by the variation in agricultural growth, exports growth, and rate of inflation.

The F-test of the significance of the overall model shows that we reject

that all of the slope coefficients are jointly equal to zero in favor of that at least one slope coefficient is not equal to zero. The F-test yielded a p-value of .000.

All the independent variables are significant at the .1 and .05 levels of significance. Exports growth is significant in explaining the manufacturing growth at all common levels of alpha.

All else being equal, the agricultural growth and the exports growth variables have the expected sign. This is because, as the percentage of agricultural growth and the exports growth increases, the percentage of manufacturing growth also increases. The rate of inflation variable is negative which indicates that higher the inflation rate, lower is the manufacturing growth.

_1383032877.unknown

_1383032878.unknown

Copyright 2013 Pearson Education, Inc. Publishing as Prentice HallCh. 12-*All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America.


Statistics for Business and Economics 8 th Edition Chapter 12 Multiple Regression Ch. 12-1 Copyright...

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Transcript of Statistics for Business and Economics 8 th Edition Chapter 12 Multiple Regression Ch. 12-1 Copyright...