Simple LReg

Chap 13-*

Introduction to Linear Regression and Correlation Analysis

Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.

Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Chapter GoalsAfter completing this chapter, you should be able to: Calculate and interpret the simple correlation between two variablesDetermine whether the correlation is significantCalculate and interpret the simple linear regression equation for a set of dataUnderstand the assumptions behind regression analysisDetermine whether a regression model is significant


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Chapter GoalsAfter completing this chapter, you should be able to:

Calculate and interpret confidence intervals for the regression coefficientsRecognize regression analysis applications for purposes of prediction and descriptionRecognize some potential problems if regression analysis is used incorrectlyRecognize nonlinear relationships between two variables

(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Scatter Plots and CorrelationA scatter plot (or scatter diagram) is used to show the relationship between two variablesCorrelation analysis is used to measure strength of the association (linear relationship) between two variablesOnly concerned with strength of the relationship No causal effect is implied


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Scatter Plot ExamplesyxyxyyxxLinear relationshipsCurvilinear relationships


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Scatter Plot ExamplesyxyxyyxxStrong relationshipsWeak relationships(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Scatter Plot ExamplesyxyxNo relationship(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Correlation CoefficientThe population correlation coefficient (rho) measures the strength of the association between the variables

The sample correlation coefficient r is an estimate of and is used to measure the strength of the linear relationship in the sample observations(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Features of 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


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*r = +.3r = +1Examples of Approximate r Valuesyxyxyxyxyxr = -1r = -.6r = 0


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Calculating the Correlation Coefficientwhere:r = Sample correlation coefficientn = Sample sizex = Value of the independent variabley = Value of the dependent variableSample correlation coefficient:or the algebraic equivalent:


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Calculation Example

Tree HeightTrunk Diameteryxxyy2x23582801225644994412401812771897294933619810893660137803600169217147441494511495202512151126122601144=321=73=3142=14111=713


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Trunk Diameter, xTreeHeight, yCalculation Example(continued)r = 0.886 relatively strong positive linear association between x and y


Chart1

35

49

27

33

60

21

45

51

Sheet1

Tree HeightTrunk Diameter

yx

358

499

277

336

6013

217

4511

5112

Sheet1

0

0

0

0

0

0

0

0

Sheet2

Sheet3

Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Excel OutputExcel Correlation OutputTools / data analysis / correlationCorrelation between Tree Height and Trunk Diameter


Sheet4

Tree HeightTrunk Diameter

Tree Height1

Trunk Diameter0.8862311

Sheet1

SalesYears with Midwestyxy2x2

4873$1,461$237,1699

4455$2,225$198,02525

2722$544$73,9844

6418$5,128$410,88164

1872$374$34,9694

4406$2,640$193,60036

3467$2,422$119,71649

2381$238$56,6441

3124$1,248$97,34416

2692$538$72,3614

6559$5,895$429,02581

5636$3,378$316,96936

Sheet2

Sheet3

Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Significance Test for CorrelationHypotheses H0: = 0 (no correlation) HA: 0 (correlation exists)

Test statistic

(with n 2 degrees of freedom)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Example: Produce StoresIs there evidence of a linear relationship between tree height and trunk diameter at the .05 level of significance?H0: = 0 (No correlation)H1: 0 (correlation exists) =.05 , df = 8 - 2 = 6


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Example: Test SolutionConclusion: There is evidence of a linear relationship at the 5% level of significanceDecision: Reject H0Reject H0Reject H0a/2=.025-t/2Do not reject H00t/2a/2=.025-2.44692.44694.68d.f. = 8-2 = 6


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Introduction to Regression AnalysisRegression analysis is used to:Predict the value of a dependent variable based on the value of at least one independent variableExplain the impact of changes in an independent variable on the dependent variableDependent variable: the variable we wish to explainIndependent variable: the variable used to explain the dependent variable


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Simple Linear Regression ModelOnly one independent variable, xRelationship between x and y is described by a linear functionChanges in y are assumed to be caused by changes in x


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Types of Regression ModelsPositive Linear RelationshipNegative Linear RelationshipRelationship NOT LinearNo Relationship


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Linear componentPopulation Linear RegressionThe population regression model:Population y intercept Population Slope Coefficient Random Error term, or residualDependent VariableIndependent VariableRandom Error component


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Linear Regression AssumptionsError values () are statistically independentError values are normally distributed for any given value of xThe probability distribution of the errors is normalThe probability distribution of the errors has constant varianceThe underlying relationship between the x variable and the y variable is linear


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Population Linear Regression(continued)Random Error for this x valueyxObserved Value of y for xiPredicted Value of y for xi xiSlope = 1Intercept = 0 i


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*The sample regression line provides an estimate of the population regression lineEstimated Regression ModelEstimate of the regression interceptEstimate of the regression slope Estimated (or predicted) y valueIndependent variableThe individual random error terms ei have a mean of zero


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Least Squares Criterionb0 and b1 are obtained by finding the values of b0 and b1 that minimize the sum of the squared residuals


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*The Least Squares EquationThe formulas for b1 and b0 are:algebraic equivalent:and


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*b0 is the estimated average value of y when the value of x is zero

b1 is the estimated change in the average value of y as a result of a one-unit change in xInterpretation of the Slope and the Intercept


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Finding the Least Squares EquationThe coefficients b0 and b1 will usually be found using computer software, such as Excel or Minitab

Other regression measures will also be computed as part of computer-based regression analysis


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Simple Linear Regression ExampleA real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet)

A random sample of 10 houses is selectedDependent variable (y) = house price in $1000sIndependent variable (x) = square feet


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Sample Data for House Price Model

House Price in $1000s(y)Square Feet (x)2451400312160027917003081875199110021915504052350324245031914252551700


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Regression Using ExcelTools / Data Analysis / Regression


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Excel OutputThe regression equation is:

Regression StatisticsMultiple R0.76211R Square0.58082Adjusted R Square0.52842Standard Error41.33032Observations10

ANOVAdfSSMSFSignificance FRegression118934.934818934.934811.08480.01039Residual813665.56521708.1957Total932600.5000

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Intercept98.2483358.033481.692960.12892-35.57720232.07386Square Feet0.109770.032973.329380.010390.033740.18580


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Graphical PresentationHouse price model: scatter plot and regression lineSlope = 0.10977Intercept = 98.248


Chart2

245

312

279

308

199

219

405

324

319

255

House Price

Square Feet

House Price ($1000s)

Sheet4

SUMMARY OUTPUT

Regression Statistics

Multiple R0.76211

R Square0.58082

Adjusted R Square0.52842

Standard Error41.33032

Observations10

ANOVA

dfSSMSFSignificance F

Regression118934.934818934.934811.084760.01039

Residual813665.56521708.1957

Total932600.5000

CoefficientsStandard Errort StatP-valueLower 95%Upper 95%

Intercept98.2483358.033481.692960.12892-35.57720232.07386

Square Feet0.109770.032973.329380.010390.033740.18580

RESIDUAL OUTPUT

ObservationPredicted House PriceResiduals

1251.9231625835-6.9231625835

2273.876710149538.1232898505

3284.8534839325-5.8534839325

4304.06283805283.9371619472

5218.9928412345-19.9928412345

6268.388323258-49.388323258

7356.202513522148.7974864779

8367.1792873051-43.1792873051

9254.667356029364.3326439707

10284.8534839325-29.8534839325

Sheet4

2450

3120

2790

3080

1990

2190

4050

3240

3190

2550

House Price

Predicted House Price

Square Feet

House Price

Square Feet Line Fit Plot

Sheet1

House PriceSquare Feet

2451400

3121600

2791700

3081875

1991100

2191550

4052350

3242450

3191425

2551700

Sheet1

0

0

0

0

0

0

0

0

0

0

House Price

Square Feet

House Price

Sheet2

Sheet3

Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Interpretation of the Intercept, b0b0 is the estimated average value of Y when the value of X is zero (if x = 0 is in the range of observed x values)Here, no houses had 0 square feet, so b0 = 98.24833 just indicates that, for houses within the range of sizes observed, $98,248.33 is the portion of the house price not explained by square feet


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Interpretation of the Slope Coefficient, b1b1 measures the estimated change in the average value of Y as a result of a one-unit change in XHere, b1 = .10977 tells us that the average value of a house increases by .10977($1000) = $109.77, on average, for each additional one square foot of size


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Least Squares Regression PropertiesThe sum of the residuals from the least squares regression line is 0 ( )The sum of the squared residuals is a minimum (minimized )The simple regression line always passes through the mean of the y variable and the mean of the x variableThe least squares coefficients are unbiased estimates of 0 and 1


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Explained and Unexplained VariationTotal variation is made up of two parts:Total sum of SquaresSum of Squares RegressionSum of Squares Errorwhere: = Average value of the dependent variabley = Observed values of the dependent variable = Estimated value of y for the given x value


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*SST = total sum of squares Measures the variation of the yi values around their mean ySSE = error sum of squares Variation attributable to factors other than the relationship between x and ySSR = regression sum of squares Explained variation attributable to the relationship between x and y(continued)Explained and Unexplained Variation


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*(continued)XiyxyiSST = (yi - y)2SSE = (yi - yi )2 SSR = (yi - y)2 ___Explained and Unexplained Variationyyy_y


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable

The coefficient of determination is also called R-squared and is denoted as R2Coefficient of Determination, R2where


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Coefficient of determination Coefficient of Determination, R2(continued)Note: In the single independent variable case, the coefficient of determination is

where:R2 = Coefficient of determination r = Simple correlation coefficient


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*R2 = +1Examples of Approximate R2 ValuesyxyxR2 = 1R2 = 1Perfect linear relationship between x and y:

100% of the variation in y is explained by variation in x


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Examples of Approximate R2 Valuesyxyx0 < R2 < 1Weaker linear relationship between x and y:

Some but not all of the variation in y is explained by variation in x


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Examples of Approximate R2 ValuesR2 = 0No linear relationship between x and y:

The value of Y does not depend on x. (None of the variation in y is explained by variation in x)yxR2 = 0


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Excel Output58.08% of the variation in house prices is explained by variation in square feet





Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Standard Error of EstimateThe standard deviation of the variation of observations around the regression line is estimated byWhereSSE = Sum of squares error n = Sample size k = number of independent variables in the model


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*The Standard Deviation of the Regression SlopeThe standard error of the regression slope coefficient (b1) is estimated bywhere:= Estimate of the standard error of the least squares slope

= Sample standard error of the estimate


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Excel Output





Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Comparing Standard ErrorsyyyxxxyxVariation of observed y values from the regression lineVariation in the slope of regression lines from different possible samples


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Inference about the Slope: t Testt test for a population slopeIs there a linear relationship between x and y?Null and alternative hypothesesH0: 1 = 0(no linear relationship)H1: 1 0(linear relationship does exist)Test statistic

where: b1 = Sample regression slope coefficient 1 = Hypothesized slope sb1 = Estimator of the standard error of the slope


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Estimated Regression Equation:The slope of this model is 0.1098 Does square footage of the house affect its sales price?Inference about the Slope: t Test(continued)



Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Inferences about the Slope: t Test ExampleH0: 1 = 0HA: 1 0Test Statistic: t = 3.329There is sufficient evidence that square footage affects house priceFrom Excel output: Reject H0tb1Decision:Conclusion:Reject H0Reject H0a/2=.025-t/2Do not reject H00t/2a/2=.025-2.30602.30603.329d.f. = 10-2 = 8

CoefficientsStandard Errort StatP-valueIntercept98.2483358.033481.692960.12892Square Feet0.109770.032973.329380.01039


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Regression Analysis for DescriptionConfidence Interval Estimate of the Slope:Excel Printout for House Prices:At 95% level of confidence, the confidence interval for the slope is (0.0337, 0.1858)d.f. = n - 2



Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Regression Analysis for DescriptionSince the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.70 and $185.80 per square foot of house sizeThis 95% confidence interval does not include 0.Conclusion: There is a significant relationship between house price and square feet at the .05 level of significance



Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Confidence Interval for the Average y, Given xConfidence interval estimate for the mean of y given a particular xp

Size of interval varies according to distance away from mean, x


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Confidence Interval for an Individual y, Given xConfidence interval estimate for an Individual value of y given a particular xp

This extra term adds to the interval width to reflect the added uncertainty for an individual case


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Interval Estimates for Different Values of xyxPrediction Interval for an individual y, given xp xpy = b0 + b1xxConfidence Interval for the mean of y, given xp


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Estimated Regression Equation:Example: House PricesPredict the price for a house with 2000 square feet



Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Example: House PricesPredict the price for a house with 2000 square feet:The predicted price for a house with 2000 square feet is 317.85($1,000s) = $317,850(continued)


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Estimation of Mean Values: ExampleFind the 95% confidence interval for the average price of 2,000 square-foot housesPredicted Price Yi = 317.85 ($1,000s)Confidence Interval Estimate for E(y)|xpThe confidence interval endpoints are 280.66 -- 354.90, or from $280,660 -- $354,900


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Estimation of Individual Values: ExampleFind the 95% confidence interval for an individual house with 2,000 square feetPredicted Price Yi = 317.85 ($1,000s)Prediction Interval Estimate for y|xpThe prediction interval endpoints are 215.50 -- 420.07, or from $215,500 -- $420,070


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Finding Confidence and Prediction Intervals PHStatIn Excel, use PHStat | regression | simple linear regression

Check the confidence and prediction interval for X= box and enter the x-value and confidence level desired


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Input valuesFinding Confidence and Prediction Intervals PHStat(continued)Confidence Interval Estimate for E(y)|xpPrediction Interval Estimate for y|xp


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Residual AnalysisPurposesExamine for linearity assumptionExamine for constant variance for all levels of x Evaluate normal distribution assumptionGraphical Analysis of ResidualsCan plot residuals vs. xCan create histogram of residuals to check for normality


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Residual Analysis for LinearityNot LinearLinearxresidualsxyxyxresiduals


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Residual Analysis for Constant Variance Non-constant varianceConstant variancexxyxxyresidualsresiduals


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Excel Output

RESIDUAL OUTPUTPredicted House Price Residuals1251.92316-6.9231622273.8767138.123293284.85348-5.8534844304.062843.9371625218.99284-19.992846268.38832-49.388327356.2025148.797498367.17929-43.179299254.667464.3326410284.85348-29.85348


Chart1

-6.9231625835

38.1232898505

-5.8534839325

3.9371619472

-19.9928412345

-49.388323258

48.7974864779

-43.1792873051

64.3326439707

-29.8534839325

Square Feet

Residuals

House Price Model Residual Plot

Sheet4

SUMMARY OUTPUT


Multiple R0.76211

R Square0.58082



Observations10

ANOVA


Regression118934.934818934.934811.084760.01039

Residual813665.56521708.1957

Total932600.5000


Intercept98.2483358.033481.692960.12892-35.57720232.07386

Square Feet0.109770.032973.329380.010390.033740.18580

RESIDUAL OUTPUT


1251.9231625835-6.9231625835

2273.876710149538.1232898505

3284.8534839325-5.8534839325

4304.06283805283.9371619472

5218.9928412345-19.9928412345

6268.388323258-49.388323258

7356.202513522148.7974864779

8367.1792873051-43.1792873051

9254.667356029364.3326439707

10284.8534839325-29.8534839325

Sheet4

2451400

3121600

2791700

3081875

1991100

2191550

4052350

3242450

3191425

2551700

House Price

Predicted House Price

Square Feet

House Price

Square Feet Line Fit Plot

DataCopy

Square FeetHouse Price(X-XBar)^2

140024599225

160031213225

1700279225

187530825600

1100199378225

155021927225

2350405403225

2450324540225

142531984100

1700255225

Estimate

Confidence Interval Estimate

Data

X Value2000

Confidence Level95%

Intermediate Calculations

Sample Size10

Degrees of Freedom8

t Value2.3060056265

Sample Mean1715

Sum of Squared Difference1571500

Standard Error of the Estimate41.3303236503

h Statistic0.151686287

Average Predicted Y (YHat)317.7838052816

For Average Predicted Y (YHat)

Interval Half Width37.1195180764

Confidence Interval Lower Limit280.6642872052

Confidence Interval Upper Limit354.9033233579

For Individual Response Y

Interval Half Width102.2813064456

Prediction Interval Lower Limit215.502498836

Prediction Interval Upper Limit420.0651117272

SLR

Regression Analysis


Multiple R0.7621137132

R Square0.5808173119



Observations10

ANOVA


Regression118934.93477569218934.93477569211.08475761660.0103940164

Residual813665.5652243081708.1956530385

Total932600.5


Intercept98.248329621458.03347858471.69295951260.1289188121-35.5771985175232.0738577603

Square Feet0.10976773780.03296944333.32937796240.01039401640.03374001620.1857954595

SLR2

Regression Analysis


Multiple R0.7621137132

R Square0.5808173119



Observations10

ANOVA


Regression118934.93477569218934.93477569211.08475761660.0103940164

Residual813665.5652243081708.1956530385

Total932600.5


Intercept98.248329621458.03347858471.69295951260.1289188121-35.5771985175232.0738577603

Square Feet0.10976773780.03296944333.32937796240.01039401640.03374001620.1857954595

RESIDUAL OUTPUT


1251.9231625835-6.9231625835

2273.876710149538.1232898505

3284.8534839325-5.8534839325

4304.06283805283.9371619472

5218.9928412345-19.9928412345

6268.388323258-49.388323258

7356.202513522148.7974864779

8367.1792873051-43.1792873051

9254.667356029364.3326439707

10284.8534839325-29.8534839325

SLR2

1400

1600

1700

1875

1100

1550

2350

2450

1425

1700

Square Feet

Residuals

House Price Model Residual Plot

Sheet1

House PriceSquare Feet

2451400

3121600

2791700

3081875

1991100

2191550

4052350

3242450

3191425

2551700

Sheet1

House Price

Square Feet

House Price

Sheet2

Sheet3

Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Chapter SummaryIntroduced correlation analysisDiscussed correlation to measure the strength of a linear associationIntroduced simple linear regression analysisCalculated the coefficients for the simple linear regression equationDescribed measures of variation (R2 and s)Addressed assumptions of regression and correlation


Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.Chap 13-*Chapter SummaryDescribed inference about the slopeAddressed estimation of mean values and prediction of individual valuesDiscussed residual analysis

(continued)


Documents

Simple LReg