© 2003 Prentice-Hall, Inc.Chap 13-1 Basic Business Statistics (9 th Edition) Chapter 13 Simple...

Basic Business Statistics(9th Edition)Chapter 13Simple Linear Regression

Chapter TopicsTypes of Regression ModelsDetermining the Simple Linear Regression EquationMeasures of VariationAssumptions of Regression and CorrelationResidual Analysis Measuring AutocorrelationInferences about the Slope

Chapter TopicsCorrelation - Measuring the Strength of the AssociationEstimation of Mean Values and Prediction of Individual ValuesPitfalls in Regression and Ethical Issues(continued)

Purpose of Regression AnalysisRegression Analysis is Used Primarily to Model Causality and Provide PredictionPredict the values of a dependent (response) variable based on values of at least one independent (explanatory) variableExplain the effect of the independent variables on the dependent variable

Types of Regression ModelsPositive Linear RelationshipNegative Linear RelationshipRelationship NOT LinearNo Relationship

Simple Linear Regression ModelRelationship between Variables is Described by a Linear FunctionThe Change of One Variable Causes the Other Variable to ChangeA Dependency of One Variable on the Other

Simple Linear Regression ModelPopulation RegressionLine (Conditional Mean)Population regression line is a straight line that describes the dependence of the average value (conditional mean) of one variable on the other Population Y Intercept Population Slope Coefficient Random ErrorDependent (Response) VariableIndependent (Explanatory) Variable(continued)

Simple Linear Regression Model(continued) = Random ErrorYX(Observed Value of Y) =Observed Value of Y(Conditional Mean)

Linear Regression EquationSample regression line provides an estimate of the population regression line as well as a predicted value of YSample Y InterceptSample Slope CoefficientResidual Simple Regression Equation (Fitted Regression Line, Predicted Value)

Linear Regression Equation and are obtained by finding the values of and that minimize the sum of the squared residuals

provides an estimate of provides an estimate of (continued)

Linear Regression Equation(continued)YXObserved Value

Interpretation of the Slopeand Intercept is the average value of Y when the value of X is zero

measures the change in the average value of Y as a result of a one-unit change in X

Interpretation of the Slopeand Intercept is the estimated average value of Y when the value of X is zero

is the estimated change in the average value of Y as a result of a one-unit change in X

(continued)

Simple Linear Regression: ExampleYou wish to examine the linear dependency of the annual sales of produce stores on their sizes in square footage. Sample data for 7 stores were obtained. Find the equation of the straight line that fits the data best.Annual Store Square Sales Feet($1000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760

Scatter Diagram: ExampleExcel Output

Chart1

3681

3395

6653

9543

3318

5563

3760

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet5

SUMMARY OUTPUT

Regression Statistics

Multiple R0.9705572038

R Square0.9419812858

Adjusted R Square0.930377543

Standard Error611.7515173314

Observations7

ANOVA

dfSSMSFSignificance F

Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857

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

Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT

ObservationPredicted YResiduals

14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet1

StoreSquare FeetAnnual Sales ($000)

117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet1

0

0

0

0

0

0

0

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet2

Sheet3

Simple Linear Regression Equation: ExampleFrom Excel Printout:

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet1


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet1

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet2

Sheet3

Graph of the Simple Linear Regression Equation: ExampleYi = 1636.415 +1.487Xi

Chart2

3681

3395

6653

9543

3318

5563

3760

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet1


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet1

0

0

0

0

0

0

0

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet2

Sheet3

Interpretation of Results: ExampleThe slope of 1.487 means that for each increase of one unit in X, we predict the average of Y to increase by an estimated 1.487 units.The equation estimates that for each increase of 1 square foot in the size of the store, the expected annual sales are predicted to increase by $1487.

Simple Linear Regressionin PHStatIn Excel, use PHStat | Regression | Simple Linear Regression Excel Spreadsheet of Regression Sales on Footage

Scatter

3681

3395

6653

9543

3318

5563

3760

Sales

X

Y

Scatter Diagram

DataCopy

FootageSales(X-XBar)^2

17263681389732.653061224

15423395653325.795918367

28166653216889.795918367

5555954310270193.6530612

129233181119968.65306122

2208556320245.2244897959

131337601075961.65306122

SLR

Regression Analysis



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%

Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536

Footage1.48663365650.16499921239.00994395940.00028120091.06249037151.9107769416

RESIDUAL OUTPUT

ObservationPredicted SalesResiduals

14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

SLR

1726

1542

2816

5555

1292

2208

1313

Footage

Residuals

Footage Residual Plot

Estimate

Confidence Interval Estimate

X Value2000

Confidence Level95%

Sample Size7

Degrees of Freedom5

t Value2.5705776352

Sample Mean2350.2857142857

Sum of Squared Difference13746317.43

Standard Error of the Estimate611.7515173314

h Statistic0.1517831758

Average Predicted Y (YHat)4609.6820391647

For Average Predicted Y (YHat)

Interval Half Width612.6572787957

Confidence Interval Lower Limit3997.024760369

Confidence Interval Upper Limit5222.3393179604

For Individual Response Y


Prediction Interval Lower Limit2921.9979923185

Prediction Interval Upper Limit6297.3660860109

This sheet is generated by:(1) PHStat | Regression | Simple Linear Regression (2) checking the "ANOVA and Coefficients Table", and the "Residual Plot" boxes

&A

Page &P

This sheet is generated by:(1) PHStat | Regression | Simple Linear Regression (2) checking the "Confidence and Prediction Intervals for X=" box

data

StoreFootageSales

11,7263,681

21,5423,395

32,8166,653

45,5559,543

51,2923,318

62,2085,563

71,3133,760

Measures of Variation: The Sum of SquaresSST = SSR + SSETotal Sample Variability=Explained Variability+Unexplained Variability

Measures of Variation: The Sum of SquaresSST = Total Sum of Squares Measures the variation of the Yi values around their mean, SSR = Regression Sum of Squares Explained variation attributable to the relationship between X and YSSE = Error Sum of Squares Variation attributable to factors other than the relationship between X and Y(continued)

Measures of Variation: The Sum of Squares(continued)XiYXYSST = (Yi - Y)2SSE =(Yi - Yi )2 SSR = (Yi - Y)2 ___

Venn Diagrams and Explanatory Power of RegressionSalesSizesVariations in Sales explained by Sizes or variations in Sizes used in explaining variation in SalesVariations in Sales explained by the error term or unexplained by SizesVariations in store Sizes not used in explaining variation in Sales

The ANOVA Table in Excel

ANOVAdfSSMSFSignificance FRegressionkSSRMSR=SSR/kMSR/MSEP-value of the F TestResidualsn-k-1SSEMSE=SSE/(n-k-1)Totaln-1SST

Measures of VariationThe Sum of Squares: ExampleExcel Output for Produce StoresSSRSSERegression (explained) dfDegrees of freedomError (residual) dfTotal dfSST

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet1


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet1

0

0

0

0

0

0

0

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet2

Sheet3

The Coefficient of Determination

Measures the proportion of variation in Y that is explained by the independent variable X in the regression model

Venn Diagrams and Explanatory Power of RegressionSalesSizes

Coefficients of Determination (r 2) and Correlation (r)r2 = 1,r2 = 1,r2 = .81,r2 = 0,YYi = b0 + b1XiX^YYi = b0 + b1XiX^YYi = b0 + b1XiX^YYi = b0 + b1XiX^r = +1r = -1r = +0.9r = 0

Standard Error of Estimate

Measures the standard deviation (variation) of the Y values around the regression equation

Measures of Variation: Produce Store ExampleExcel Output for Produce Stores r2 = .94 94% of the variation in annual sales can be explained by the variability in the size of the store as measured by square footage.Syxn

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet1


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet1

0

0

0

0

0

0

0

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet2

Sheet3

Linear Regression AssumptionsNormalityY values are normally distributed for each XProbability distribution of error is normalHomoscedasticity (Constant Variance)Independence of Errors

Consequences of Violationof the AssumptionsViolation of the AssumptionsNon-normality (error not normally distributed)Heteroscedasticity (variance not constant)Usually happens in cross-sectional dataAutocorrelation (errors are not independent)Usually happens in time-series dataConsequences of Any Violation of the AssumptionsPredictions and estimations obtained from the sample regression line will not be accurateHypothesis testing results will not be reliableIt is Important to Verify the Assumptions

Variation of Errors Aroundthe Regression Line Y values are normally distributed around the regression line. For each X value, the spread or variance around the regression line is the same.X1X2XY f(e)Sample Regression Line

Residual AnalysisPurposesExamine linearity Evaluate violations of assumptionsGraphical Analysis of ResidualsPlot residuals vs. X and time

Residual Analysis for LinearityNot LinearLinearX e eXYXYX

Residual Analysis for Homoscedasticity HeteroscedasticityHomoscedasticitySRXSRXYXXY

Residual Analysis: Excel Output for Produce Stores ExampleExcel Output

Chart3

-521.3444172716

-533.8038244674

830.224897095

-351.6646881801

-239.1454103313

644.098160274

171.6352828813

Square Feet

Residual Plot

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet6

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT

ObservationPredicted YResidualsStandard Residuals

14202.3444172716-521.3444172716-0.9335558294

23928.8038244674-533.8038244674-0.9558665166

35822.775102905830.2248970951.486658851

49894.6646881801-351.6646881801-0.6297154218

53557.1454103313-239.1454103313-0.4282305219

64918.901839726644.0981602741.1533672795

73588.3647171187171.63528288130.3073421591

Sheet6

0

0

0

0

0

0

0

Square Feet

Residuals

Residual Plot

Sheet1

36810

33950

66530

95430

33180

55630

37600

Y

Predicted Y

X Variable 1

Y

X Variable 1 Line Fit Plot

Sheet2


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet2

0

0

0

0

0

0

0

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet3

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet1


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet1

0

0

0

0

0

0

0

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet2

Sheet3

Residual Analysis for IndependenceThe Durbin-Watson StatisticUsed when data is collected over time to detect autocorrelation (residuals in one time period are related to residuals in another period)Measures violation of independence assumptionShould be close to 2. If not, examine the model for autocorrelation.

Durbin-Watson Statisticin PHStatPHStat | Regression | Simple Linear Regression Check the box for Durbin-Watson Statistic

Obtaining the Critical Values of Durbin-Watson StatisticTable 13.4 Finding Critical Values of Durbin-Watson Statistic

k=1

k=2

n

dL

dU

dL

dU

15

1.08

1.36

.95

1.54

16

1.10

1.37

.98

1.54

_1013327115.unknown

_1044645607.unknown

_1013326860.unknown

Using the Durbin-Watson StatisticAccept H0 (no autocorrelation) : No autocorrelation (error terms are independent) : There is autocorrelation (error terms are not independent)042dL4-dLdU4-dUReject H0 (positive autocorrelation)InconclusiveReject H0 (negative autocorrelation)

Residual Analysis for IndependenceNot IndependentIndependenteeTimeTimeResidual is Plotted Against Time to Detect Any AutocorrelationNo Particular PatternCyclical PatternGraphical Approach

Inference about the Slope: t Testt Test for a Population SlopeIs there a linear dependency of Y on X ?Null and Alternative HypothesesH0: 1 = 0(no linear dependency)H1: 1 0(linear dependency)Test Statistic

Example: Produce StoreData for 7 Stores:Estimated Regression Equation:Annual Store Square Sales Feet($000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760The slope of this model is 1.487. Does square footage affect annual sales?

Inferences about the Slope: t Test ExampleH0: 1 = 0H1: 1 0 .05df 7 - 2 = 5Critical Value(s):Test Statistic: Decision:

Conclusion:

There is evidence that square footage affects annual sales.t02.5706-2.5706.025RejectReject.025From Excel Printout Reject H0.p-value

SLR

Regression Analysis



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147451.49533.62440.01515475.81092619832797.0185259536

Footage1.48660.16509.00990.000281.06249037151.9107769416

Sheet1

FootageAnnual Sales

11,7263,681

21,5423,395

32,8166,653

45,5559,543

51,2923,318

62,2085,563

71,3133,760

Sheet2

Sheet3

Inferences about the Slope: Confidence Interval ExampleConfidence Interval Estimate of the Slope: Excel Printout for Produce StoresAt 95% level of confidence, the confidence interval for the slope is (1.062, 1.911). Does not include 0.Conclusion: There is a significant linear dependency of annual sales on the size of the store.

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet6

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

Footage1.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT

ObservationPredicted YResidualsStandard Residuals

14202.3444172716-521.3444172716-0.9335558294

23928.8038244674-533.8038244674-0.9558665166

35822.775102905830.2248970951.486658851

49894.6646881801-351.6646881801-0.6297154218

53557.1454103313-239.1454103313-0.4282305219

64918.901839726644.0981602741.1533672795

73588.3647171187171.63528288130.3073421591

Sheet6

Residual Plot

Sheet1

36811726

33951542

66532816

95435555

33181292

55632208

37601313

Y

Predicted Y

X Variable 1

Y

X Variable 1 Line Fit Plot

Sheet2


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet2

0

0

0

0

0

0

0

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet3

Inferences about the Slope: F TestF Test for a Population SlopeIs there a linear dependency of Y on X ?Null and Alternative HypothesesH0: 1 = 0(no linear dependency)H1: 1 0(linear dependency)Test Statistic

Numerator d.f.=1, denominator d.f.=n-2

Relationship between a t Test and an F TestNull and Alternative HypothesesH0: 1 = 0(no linear dependency)H1: 1 0(linear dependency)

The p value of a t Test and the p value of an F Test are Exactly the SameThe Rejection Region of an F Test is Always in the Upper Tail

Inferences about the Slope: F Test ExampleTest Statistic: Decision:Conclusion:

H0: 1 = 0H1: 1 0 .05numerator df = 1denominator df 7 - 2 = 5

There is evidence that square footage affects annual sales.From Excel Printout Reject H0.06.61Reject = .05p-value

Scatter

3681

3395

6653

9543

3318

5563

3760

Sales

X

Y

Scatter Diagram

DataCopy


17263681389732.653061224

15423395653325.795918367

28166653216889.795918367

5555954310270193.6530612

129233181119968.65306122

2208556320245.2244897959

131337601075961.65306122

SLR

Regression Analysis



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.000281

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536

Footage1.48663365650.16499921239.00994395940.00028120091.06249037151.9107769416

RESIDUAL OUTPUT

ObservationPredicted SalesResiduals

14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

SLR

1726

1542

2816

5555

1292

2208

1313

Footage

Residuals

Footage Residual Plot

Estimate


X Value2000

Confidence Level95%

Sample Size7

Degrees of Freedom5

t Value2.5705776352

Sample Mean2350.2857142857













&A

Page &P

data

StoreFootageSales

11,7263,681

21,5423,395

32,8166,653

45,5559,543

51,2923,318

62,2085,563

71,3133,760

Purpose of Correlation Analysis Correlation Analysis is Used to Measure Strength of Association (Linear Relationship) Between 2 Numerical VariablesOnly strength of the relationship is concernedNo causal effect is implied

Purpose of Correlation AnalysisPopulation Correlation Coefficient (Rho) is Used to Measure the Strength between the Variables

(continued)

Purpose of Correlation AnalysisSample Correlation Coefficient r is an Estimate of and is Used to Measure the Strength of the Linear Relationship in the Sample Observations

(continued)

Sample Observations from Various r Valuesr = .6r = 1YXYXYXYXYXr = -1r = -.6r = 0

Features of r and rUnit FreeRange between -1 and 1The Closer to -1, the Stronger the Negative Linear RelationshipThe Closer to 1, the Stronger the Positive Linear RelationshipThe Closer to 0, the Weaker the Linear Relationship

t Test for CorrelationHypotheses H0: = 0 (no correlation) H1: 0 (correlation)Test Statistic

Example: Produce StoresFrom Excel Printout rIs there any evidence of linear relationship between annual sales of a store and its square footage at .05 level of significance?H0: = 0 (no association)H1: 0 (association) .05df 7 - 2 = 5

Sheet5

SUMMARY OUTPUT



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536475.81092619832797.0185259536

X Variable 11.48663365650.16499921239.00994395940.00028120091.06249037151.91077694161.06249037151.9107769416

RESIDUAL OUTPUT


14202.3444172716-521.3444172716

23928.8038244674-533.8038244674

35822.775102905830.224897095

49894.6646881801-351.6646881801

53557.1454103313-239.1454103313

64918.901839726644.098160274

73588.3647171187171.6352828813

Sheet1


117263681

215423395

328166653

455559543

512923318

622085563

713133760

Sheet1

0

0

0

0

0

0

0

Annual Sales ($000)

Square Feet

Annual Sales ($000)

Sheet2

Sheet3

Example: Produce Stores Solution02.5706-2.5706.025RejectReject.025Critical Value(s):Conclusion: There is evidence of a linear relationship at 5% level of significance.Decision: Reject H0.The value of the t statistic is exactly the same as the t statistic value for test on the slope coefficient.

Estimation of Mean ValuesConfidence Interval Estimate for :The Mean of Y Given a Particular Xit value from table with df=n-2Standard error of the estimateSize of interval varies according to distance away from mean,

Prediction of Individual ValuesPrediction Interval for Individual Response Yi at a Particular XiAddition of 1 increases width of interval from that for the mean of Y

Interval Estimates for Different Values of XYXPrediction Interval for a Individual Yia given XConfidence Interval for the Mean of YYi = b0 + b1Xi

Example: Produce StoresYi = 1636.415 +1.487XiData for 7 Stores:Regression Model Obtained:Annual Store Square Sales Feet($000) 1 1,726 3,681 2 1,542 3,395 3 2,816 6,653 4 5,555 9,543 5 1,292 3,318 6 2,208 5,563 7 1,313 3,760Consider a store with 2000 square feet.

Estimation of Mean Values: ExampleFind the 95% confidence interval for the average annual sales for stores of 2,000 square feet.Predicted Sales Yi = 1636.415 +1.487Xi = 4610.45 ($000)X = 2350.29SYX = 611.75 tn-2 = t5 = 2.5706Confidence Interval Estimate for

Prediction Interval for Y : ExampleFind the 95% prediction interval for annual sales of one particular store of 2,000 square feet.Predicted Sales Yi = 1636.415 +1.487Xi = 4610.45 ($000)X = 2350.29SYX = 611.75 tn-2 = t5 = 2.5706Prediction Interval for Individual

Estimation of Mean Values and Prediction of Individual Values in PHStatIn Excel, use PHStat | Regression | Simple Linear Regression Check the Confidence and Prediction Interval for X= boxExcel Spreadsheet of Regression Sales on Footage

DataCopy


17263681389732.653061224

15423395653325.795918367

28166653216889.795918367

5555954310270193.6530612

129233181119968.65306122

2208556320245.2244897959

131337601075961.65306122

SLR

Regression Analysis



R Square0.9419812858



Observations7

ANOVA


Regression130380456.119499630380456.119499681.1790901520.0002812009

Residual51871199.59478615374239.91895723

Total632251655.7142857


Intercept1636.4147260759451.49533084953.62443333130.0151488191475.81092619832797.0185259536

Footage1.48663365650.16499921239.00994395940.00028120091.06249037151.9107769416

Estimate


X Value2000

Confidence Level95%

Sample Size7

Degrees of Freedom5

t Value2.5705776352

Sample Mean2350.2857142857













&A

Page &P

data

StoreFootageSales

11,7263,681

21,5423,395

32,8166,653

45,5559,543

51,2923,318

62,2085,563

71,3133,760

Pitfalls of Regression AnalysisLacking an Awareness of the Assumptions Underlining Least-Squares RegressionNot Knowing How to Evaluate the AssumptionsNot Knowing What the Alternatives to Least-Squares Regression are if a Particular Assumption is ViolatedUsing a Regression Model Without Knowledge of the Subject Matter

Strategy for Avoiding the Pitfalls of RegressionStart with a scatter plot of X on Y to observe possible relationshipPerform residual analysis to check the assumptionsUse a histogram, stem-and-leaf display, box-and-whisker plot, or normal probability plot of the residuals to uncover possible non-normality

Strategy for Avoiding the Pitfalls of RegressionIf there is violation of any assumption, use alternative methods (e.g., least absolute deviation regression or least median of squares regression) to least-squares regression or alternative least-squares models (e.g., curvilinear or multiple regression)If there is no evidence of assumption violation, then test for the significance of the regression coefficients and construct confidence intervals and prediction intervals(continued)

Chapter SummaryIntroduced Types of Regression ModelsDiscussed Determining the Simple Linear Regression EquationDescribed Measures of VariationAddressed Assumptions of Regression and CorrelationDiscussed Residual Analysis Addressed Measuring Autocorrelation

Chapter SummaryDescribed Inference about the SlopeDiscussed Correlation - Measuring the Strength of the Association Addressed Estimation of Mean Values and Prediction of Individual ValuesDiscussed Pitfalls in Regression and Ethical Issues

(continued)

© 2003 Prentice-Hall, Inc.Chap 13-1 Basic Business Statistics (9 th Edition) Chapter 13 Simple...

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Transcript of © 2003 Prentice-Hall, Inc.Chap 13-1 Basic Business Statistics (9 th Edition) Chapter 13 Simple...