BB ABS 10th Univariate and Bivariate Corr SR MR 81Pages Final

8/6/2019 BB ABS 10th Univariate and Bivariate Corr SR MR 81Pages Final

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Business Statistics

ABS-Bangalore


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Dr. R. Venkatamuni ReddyAssociate Professor

Contact: 09632326277, 080-30938181

[email protected]

[email protected]
mailto:[email protected]:[email protected]:[email protected]:[email protected]


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Correlation, Simple andMultiple Regression


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Learning Objectives

In this chapter

How to use regression analysis to predict the value of a

dependent variable based on an independent variable

The meaning of the regression coefficients b0 and b1

How to evaluate the assumptions of regression analysis and

know what to do if the assumptions are violated

To make inferences about the slope and correlation coefficient

To estimate mean values and predict individual values


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Correlation vs. Regression

A scatter diagram can be used to show therelationship between two variables

Correlation analysis is used to measure strengthof the association (linear relationship)between two variables

Correlation is only concerned with strength of therelationship

No causal effect is implied with correlation


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Introduction toRegression Analysis

Regression analysis is used to:

Predict the value of a dependent variable based on the value

of at least one independent variable

Explain the impact of changes in an independent variable on

the dependent variable

Dependent variable: the variable we wish to predict

or explainIndependent variable: the variable used to explain

the dependent variable


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Simple Linear RegressionModel

Only one independent variable, X

Relationship between X and Y is described by

a linear function

Changes in Y are assumed to be caused by

changes in X


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Types of Relationships

YY

XX

YY

XX

YY

YY

XX

XX

Linear relationshipsLinear relationships Curvilinear relationshipsCurvilinear relationships


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YY

XX

YY

XX

YY

YY

XX

XX

Strong relationshipsStrong relationships Weak relationshipsWeak relationships

(continued)(continued)


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YY

XX

YY

XX

No relationshipNo relationship



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ii10i XY ++=Linear componentLinear component


PopulationPopulation

Y interceptY intercept

PopulationPopulation

SlopeSlope

CoefficientCoefficient

RandomRandom

ErrorError

termtermDependentDependent

VariableVariable

IndependentIndependent

VariableVariable

Random ErrorRandom Error

componentcomponent


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Random Error forRandom Error for

this Xthis Xii valuevalue

YY

XX

Observed ValueObserved Value

of Y for Xof Y for Xii

Predicted ValuePredicted Value

of Y for Xof Y for Xii

ii10i XY ++=

XXii

Slope =Slope = 11

Intercept =Intercept = 00

ii



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i10i XbbY +=

The simple linear regression equation provides anThe simple linear regression equation provides an estimateestimate ofof

the population regression linethe population regression line

Simple Linear RegressionEquation (Prediction Line)

Estimate of theEstimate of theregressionregression

interceptintercept

Estimate of theEstimate of theregression sloperegression slope

Estimated (orEstimated (or

predicted) Ypredicted) Yvalue forvalue for

observation iobservation i

Value of X forValue of X for

observation iobservation i

The individual random error terms eThe individual random error terms eii have a mean of zerohave a mean of zero


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Least Squares Method

b0 and b1 are obtained by finding the values of b0 and b1

that minimize the sum of the squared differences

between Y and :

2

i10i

2

ii ))Xb(b(Ymin)Y(Ymin +=

Y


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Finding the Least SquaresEquation

The coefficients b0

and b1, and other regression

results will be found using Excel or SPSS or

STATA


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b0is the estimated average value of Y

when the value of X is zero

b1is the estimated change in the average

value of Y as a result of a one-unit changein X

Interpretation of theSlope and the Intercept


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Simple Linear RegressionExample

A 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 selected

Dependent variable (Y) = house price in $1000s

Independent variable (X) = square feet


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Sample Data for HousePrice Model

House Price in $1000s(Y)

Square Feet(X)

245 1400

312 1600

279 1700308 1875

199 1100

219 1550

405 2350

324 2450

319 1425

255 1700


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0

50

100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500 3000

Square Fee

HousePrice($1000s)

Graphical Presentation

House price model: scatter plot


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Regression Using ExcelTools / Data Analysis / Regression


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Excel OutputRegression Statistics

Multiple R 0.76211

R Square 0.58082

Adjusted R Square 0.52842

Standard Error 41.33032

Observations 10

ANOVA

df SS MS F Significance F

Regression 1 18934.9348 18934.9348 11.0848 0.01039

Residual 8 13665.5652 1708.1957

Total 9 32600.5000

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386

Square Feet 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580

The regression equation is:The regression equation is:

feet)(square0.1097798.24833pricehouse +=


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0

50100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500 3000

Square Fee

HousePrice($1000s)

Graphical PresentationHouse price model: scatter plot andregression line


SlopeSlope

= 0.10977= 0.10977

InterceptIntercept= 98.248= 98.248


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Interpretation of theIntercept, b0

b0

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 = 98.24833just 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



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317.85

0)0.1098(20098.25

(sq.ft.)0.109898.25pricehouse

=

+=

+=

Predict the price for a housePredict the price for a house

with 2000 square feet:with 2000 square feet:

The predicted price for a house with 2000The predicted price for a house with 2000

square feet is 317.85($1,000s) = $317,850square feet is 317.85($1,000s) = $317,850

Predictions usingRegression Analysis


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0

50

100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500 3000

HousePrice($1

000s)

Interpolation vs. Extrapolation

When using a regression model for prediction,only predict within the relevant range of data

Relevant range forRelevant range for

interpolationinterpolation

Do not try toDo not try to

extrapolate beyondextrapolate beyond

the range ofthe range of

observed Xsobserved Xs


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Measures of Variation

Total variation is made up of two parts:

SSESSRSST +=Total Sum ofTotal Sum of

SquaresSquaresRegression Sum ofRegression Sum of

SquaresSquaresError Sum ofError Sum of

SquaresSquares

=

2

i )YY(SST =

2

ii )Y

Y(SSE=

2

i )YY

(SSRwhere:where:

= Average value of the dependent variable= Average value of the dependent variable

YYii = Observed values of the dependent variable= Observed values of the dependent variable

ii = Predicted value of Y for the given X= Predicted value of Y for the given Xii valuevalueY

Y


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SST = Total sum of squares

Measures the variation of the Yi values around their mean Y

SSR = regression sum of squares

Explained variation attributable to the relationship between X

and Y

SSE = error sum of squares

Variation attributable to factors other than the relationship

between X and Y


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XXii

YY

XX

YYii

SSTSST ==(Y(Yii-- YY))22SSESSE == (Y(Yii-- YYii ))22

SSR =SSR = ((YYii-- YY))22

__

__

__

YY

YY

YY

__YY



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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 r2

Coefficient of Determination,r2

1r0 2 note:note:

squaresofsumtotalsquaresofsumregression

SSTSSRr2 ==


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rr22 = 1= 1

Examples of Approximater2 Values

YY

XX

YY

XX

rr22 = 1= 1

rr22 = 1= 1

Perfect linear relationship betweenPerfect linear relationship betweenX and Y:X and Y:

100% of the variation in Y is100% of the variation in Y is

explained by variation in Xexplained by variation in X


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YY

XX

YY

XX

0 < r0 < r22 < 1< 1

Weaker linear relationshipsWeaker linear relationshipsbetween X and Y:between X and Y:

Some but not all of the variationSome but not all of the variation

in Y is explained by variation inin Y is explained by variation in

XX


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rr22 = 0= 0

No linear relationship between XNo linear relationship between Xand Y:and Y:

The value of Y does not dependThe value of Y does not depend

on X. (None of the variation in Yon X. (None of the variation in Y

is explained by variation in X)is explained by variation in X)

YY

XXrr22 = 0= 0


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Multiple R 0.76211

R Square 0.58082



Observations 10

ANOVA


Regression 1 18934.9348 18934.9348 11.0848 0.01039

Residual 8 13665.5652 1708.1957

Total 9 32600.5000


Intercept 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386

Square Feet 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580

58.08% of the variation in house58.08% of the variation in house

prices is explained by variation inprices is explained by variation in

square feetsquare feet

0.5808232600.500018934.9348

SSTSSRr2 ===


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Standard Error of EstimateThe standard deviation of the variation of

observations around the regression line is estimated

by

2n

)YY(

2n

SSES

n

1i

2

ii

YX

=

==

WhereWhere

SSE = error sum of squaresSSE = error sum of squares

n = sample sizen = sample size


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Multiple R 0.76211

R Square 0.58082



Observations 10

ANOVA


Regression 1 18934.9348 18934.9348 11.0848 0.01039

Residual 8 13665.5652 1708.1957

Total 9 32600.5000


Intercept 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386

Square Feet 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580

41.33032SYX =


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Comparing Standard Errors

YYYY

XX XXYXssmall YXslarge

SSYXYX is a measure of the variation of observed Y valuesis a measure of the variation of observed Y values

from the regression linefrom the regression line

The magnitude of SThe magnitude of SYXYX should always be judged relative to the size ofshould always be judged relative to the size of

the Y values in the sample datathe Y values in the sample data

i.e., Si.e., SYXYX = $41.33K is= $41.33K ismoderately small relative to house prices in themoderately small relative to house prices in the

$200 - $300K range$200 - $300K range


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Assumptions of RegressionUse the acronym LINE:

Linearity

The underlying relationship between X and Y is linear

Independence of Errors Error values are statistically independent

Normality of Error Error values () are normally distributed for any given value of X

Equal Variance (Homoscedasticity) The probability distribution of the errors has constant variance


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Residual Analysis

The residual for observation i, ei, is the difference between its

observed and predicted value

Check the assumptions of regression by examining the residuals Examine for linearity assumption

Evaluate independence assumption

Evaluate normal distribution assumption

Examine for constant variance for all levels of X (homoscedasticity)

Graphical Analysis of Residuals

Can plot residuals vs. X

iii YYe =

Resid al Anal sis for


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Residual Analysis forLinearity

Not LinearNot Linear LinearLinear

xx

res

id

ual

s

res

id

ual

s

xx

YY

xx

YY

xx

res

id

ual

s

res

id

ual

s


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Residual Analysis forResidual Analysis forIndependenceIndependence

Not IndependentNot Independent

IndependentIndependent

XX

XXres

id

ua

ls

res

id

ua

ls

res

id

ua

ls

res

id

ua

ls

XX

res

id

ua

ls

res

id

ua

ls

Residual Analysis for


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Residual Analysis forNormality

PercentPercent

ResidualResidual

A normal probability plot of the residuals can be used toA normal probability plot of the residuals can be used to

check for normality:check for normality:

-3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 3

00

100100

R id l A l i f


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Residual Analysis forEqual Variance

Non-constant varianceNon-constant variance Constant varianceConstant variance

xx xx

YY

xx xx

YY

res

id

ual

s

res

id

ua

ls

res

id

ual

s

res

id

ua

ls


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House Price Model Residual Pl

-60

-40

-20

0

20

40

60

80

0 1000 2000 300

Square Fee

Residuals

Excel Residual Output

RESIDUAL OUTPUT

Predicted

House Price

Residuals

1 251.92316 -6.923162

2 273.87671 38.123293 284.85348 -5.853484

4 304.06284 3.937162

5 218.99284 -19.99284

6 268.38832 -49.38832

7 356.20251 48.797498 367.17929 -43.17929

9 254.6674 64.33264

10 284.85348 -29.85348

Does not appear to violateDoes not appear to violate

any regression assumptionsany regression assumptions

easur ng u ocorre a on:


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Used when data are collected over time

to detect if autocorrelation is present

Autocorrelation exists if residuals in one

time period are related to residuals in

another period

easur ng u ocorre a on:The Durbin-Watson

Statistic


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AutocorrelationAutocorrelation is correlation of the errors (residuals) over time

Violates the regression assumption that residuals areViolates the regression assumption that residuals are

random and independentrandom and independent

Time (t) Residual Plot

-15

-10

-5

0

5

10

15

0 2 4 6 8

Time (t)

Residuals

Here, residuals show aHere, residuals show a

cyclic pattern, notcyclic pattern, not

random. Cyclical patternsrandom. Cyclical patterns

are a sign of positiveare a sign of positive

autocorrelationautocorrelation

Th D bi W t


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The Durbin-WatsonStatistic

=

=

= n

1i

2

i

n

2i

2

1ii

e

)ee(

D

The possible range is 0The possible range is 0 D 4 D 4

D should be close to 2 if HD should be close to 2 if H00 is trueis true

D less than 2 may signal positiveD less than 2 may signal positive

autocorrelation, D greater than 2 mayautocorrelation, D greater than 2 may

signal negative autocorrelationsignal negative autocorrelation

The Durbin-Watson statistic is used to test for

autocorrelation

HH00: residuals are not correlated: residuals are not correlated

HH11: positive: positive autocorrelation is presentautocorrelation is present

T ti f P iti


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Testing for PositiveAutocorrelation

Calculate the Durbin-Watson test statistic = DCalculate the Durbin-Watson test statistic = D

(The Durbin-Watson Statistic can be found using Excel or Minitab)(The Durbin-Watson Statistic can be found using Excel or Minitab)

Decision rule: reject HDecision rule: reject H00

if D < dif D < dLL

HH00: positive autocorrelation does not exist: positive autocorrelation does not exist

HH11::positive autocorrelation is presentpositive autocorrelation is present

00 ddUU 22ddLL

Reject HReject H00 Do not reject HDo not reject H00

Find the values dFind the values dLL and dand dUU from the Durbin-Watson tablefrom the Durbin-Watson table

(for sample size(for sample size nn and number of independent variablesand number of independent variables kk))

InconclusiveInconclusive

i f i i


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Suppose we have the following time series data:

Is there autocorrelation?

y = 30.65 + 4.7038

R2

= 0.8976

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30

Time

Sales

Testing for PositiveTesting for PositiveAutocorrelationAutocorrelation


T ti f P iti


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Example with n = 25:

Durbin-Watson Calculations

Sum of Squared

Difference of Residuals

3296.18

Sum of SquaredResiduals

3279.98

Durbin-Watson Statistic 1.00494

y = 30.65 + 4.7038

R2 = 0.8976

0

20

40

60

80

100

120

140

160

0 5 10 15 20 25 30

Time

Sales



Excel/PHStat output:Excel/PHStat output:

1.004943279.98

3296.18

e

)e(e

Dn

1i

2

i

n

2i

2

1ii

==

=

=

=

T ti f P iti


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Here, n = 25 and there is k = 1 one independent variable

Using the Durbin-Watson table, dL= 1.29 and d

U= 1.45

D = 1.00494 < dL= 1.29, soreject H

0and conclude that

significant positive autocorrelation existsTherefore the linear model is not the appropriate model to

forecast sales



Decision:Decision: reject Hreject H00 sincesince

D = 1.00494 < dD = 1.00494 < dLL

00 ddUU=1.45=1.45 22ddLL=1.29=1.29Reject HReject H00 Do not reject HDo not reject H00InconclusiveInconclusive


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Inferences About the Slope

The standard error of the regression slope

coefficient (b1) is estimated by

==

2

i

YXYXb

)X(X

S

SSX

SS1

where:where:

= Estimate of the standard error of the least squares slope= Estimate of the standard error of the least squares slope

= Standard error of the estimate= Standard error of the estimate

1bS

2n

SSESYX

=


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Multiple R 0.76211

R Square 0.58082



Observations 10

ANOVA


Regression 1 18934.9348 18934.9348 11.0848 0.01039

Residual 8 13665.5652 1708.1957

Total 9 32600.5000


Intercept 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386

Square Feet 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580

0.03297S1b=

C i St d d E


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Comparing Standard Errorsof the Slope

YY

XX

YY

XX1bSsmall 1bSlarge

is a measure of the variation in the slope of regression lines fromis a measure of the variation in the slope of regression lines from

different possible samplesdifferent possible samples1b

S

Inference about the Slope:


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Inference about the Slope:t Test

t test for a population slopeIs there a linear relationship between X and Y?

Null and alternative hypotheses

H0: 1 = 0 (no linear relationship)

H1: 1 0 (linear relationship does exist)

Test statistic

1b

11

S

bt

=

2nd.f. =

where:where:

bb11 = regression slope= regression slopecoefficientcoefficient

11 = hypothesized slope= hypothesized slope

SSbb = standard= standard

error of the slopeerror of the slope11

Inference about the Slope:


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House Price in$1000s

(y)

Square Feet(x)

245 1400

312 1600

279 1700

308 1875

199 1100

219 1550

405 2350

324 2450

319 1425

255 1700

(sq.ft.)0.109898.25pricehouse +=

Simple Linear Regression Equation:Simple Linear Regression Equation:

The slope of this model is 0.1098The slope of this model is 0.1098

Does square footage of the house affectDoes square footage of the house affect

its sales price?its sales price?

Inference about the Slope:t Test


Inferences about the Slope:


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Inferences about the Slope:tTest Example

H0: 1 = 0

H1: 1 0

From Excel output:From Excel output:

Coefficients Standard Error t Stat P-value

Intercept 98.24833 58.03348 1.69296 0.12892

Square Feet 0.10977 0.03297 3.32938 0.01039

1bS

tt

bb11

32938.303297.0

010977.0

S

b

t1b

11

=

=

=


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Inferences about the Slope:


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Inferences about the Slope:tTest Example

H0: 1 = 0

H1: 1 0

P-value =P-value = 0.010390.01039

There is sufficient evidenceThere is sufficient evidence

that square footage affectsthat square footage affects

house pricehouse price

From Excel output:From Excel output:

Reject HReject H00

Coefficients Standard Error t Stat P-value

Intercept 98.24833 58.03348 1.69296 0.12892

Square Feet 0.10977 0.03297 3.32938 0.01039

P-valueP-value

Decision:Decision: P-value 3.329)+P(t < -3.329) =P(t > 3.329)+P(t < -3.329) =

0.010390.01039

(for 8 d.f.)(for 8 d.f.)


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Multiple R 0.76211

R Square 0.58082



Observations 10

ANOVA


Regression 1 18934.9348 18934.9348 11.0848 0.01039

Residual 8 13665.5652 1708.1957

Total 9 32600.5000


Intercept 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386

Square Feet 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580

11.08481708.1957

18934.9348

MSE

MSRF ===

With 1 and 8 degrees ofWith 1 and 8 degrees of

freedomfreedom P-value forP-value forthe F Testthe F Test


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H0: 1 = 0

H1: 1 0

= .05

df1= 1 df2 = 8

Test Statistic:Test Statistic:

Decision:Decision:

Conclusion:Conclusion:

Reject HReject H00 atat = 0.05= 0.05

There is sufficient evidence that houseThere is sufficient evidence that house

size affects selling pricesize affects selling price00

= .05= .05

FF.05.05 = 5.32= 5.32

Reject HReject H00Do notDo notreject Hreject H00

11.08MSE

MSRF ==

CriticalCritical

Value:Value:

FF = 5.32= 5.32

F Test for Significance(continued)(continued)

FF

Confidence Interval Estimate


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Confidence Interval Estimatefor the Slope

Confidence Interval Estimate of the Slope:Confidence Interval Estimate of the Slope:

Excel Printout for House Prices:Excel Printout for House Prices:

At 95% level of confidence, the confidence interval for the slope isAt 95% level of confidence, the confidence interval for the slope is

(0.0337, 0.1858)(0.0337, 0.1858)

1b2n1Stb


Intercept 98.24833 58.03348 1.69296 0.12892 -35.57720 232.07386

Square Feet 0.10977 0.03297 3.32938 0.01039 0.03374 0.18580

d.f. = n - 2d.f. = n - 2


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t Test for a Correlation


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t Test for a CorrelationCoefficient

Hypotheses

H0: = 0 (no correlation between X and Y)

HA: 0 (correlation exists)

Test statistic

(with n 2 degrees of freedom)

2n

r1-rt 2

=

0bifrr

0bifrr

where

1

2

1

2

+=


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Example: House PricesIs there evidence of a linear relationship betweenIs there evidence of a linear relationship between

square feet and house price at the .05 level ofsquare feet and house price at the .05 level of

significance?significance?

HH00:: = 0 (No correlation)= 0 (No correlation)HH11:: 0 (correlation exists)0 (correlation exists)

=.05 , df=.05 , df==10 - 2 = 810 - 2 = 8

3.329

210

.76210.762

2n

r1rt

22=

=

=


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Example: Test Solution

Conclusion:Conclusion:ThereThere is evidenceis evidence ofof

a linear associationa linear association

at the 5% level ofat the 5% level of

significancesignificance

Decision:Decision:

Reject HReject H00

Reject HReject H00Reject HReject H00

/2=.025/2=.025

-t-t/2/2Do not reject HDo not reject H00

00tt/2/2

/2=.025/2=.025

-2.3060-2.3060 2.30602.30603.3293.329

d.f. = 10-2 = 8d.f. = 10-2 = 8

3.329

210

.7621

0.762

2n

r1

rt

22=

=

=

Estimating Mean Values and


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Estimating Mean Values andPredicting Individual Values

YY

XX XXii

Y = bY = b00+b+b11XXii

ConfidenceConfidence

Interval forInterval for

thethe meanmean ofofY, given XY, given Xii

Prediction IntervalPrediction Interval

for anfor an individualindividual YY,,

given Xgiven Xii

Goal: Form intervals around Y to express uncertaintyGoal: Form intervals around Y to express uncertaintyabout the value of Y for a given Xabout the value of Y for a given Xii

YY

Confidence Interval for


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Confidence Interval forthe Average Y, Given X

Confidence interval estimate for theConfidence interval estimate for themean value of Ymean value of Y given a particular Xgiven a particular Xii

Size of interval varies according toSize of interval varies according to

distance away from mean,distance away from mean,XX

iYX2n

XX|Y

hStY

:forintervalConfidencei

=

+=

+=2

i

2

i

2

ii

)X(X

)X(X

n

1

SSX

)X(X

n

1h

Prediction Interval for


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Prediction Interval foran Individual Y, Given X

Confidence interval estimate for anConfidence interval estimate for anIndividual value of YIndividual value of Y given a particular Xgiven a particular Xii

This extra term adds to the interval width to reflect theThis extra term adds to the interval width to reflect the

added uncertainty for an individual caseadded uncertainty for an individual case

iYX2n

XX

h1StY

:YforintervalConfidencei

+

=

Estimation of Mean Values:


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Estimation of Mean Values:Example

Find the 95% confidence interval for the mean price ofFind the 95% confidence interval for the mean price of

2,000 square-foot houses2,000 square-foot houses

Predicted Price YPredicted Price Yii = 317.85 ($1,000s)= 317.85 ($1,000s)

Confidence Interval Estimate forConfidence Interval Estimate forY|X=XY|X=X

37.12317.85

)X(X

)X(X

n

1StY

2

i

2

iYX2-n =

+

The confidence interval endpoints are 280.66 and 354.90, or fromThe confidence interval endpoints are 280.66 and 354.90, or from

$280,660 to $354,900$280,660 to $354,900

ii

Estimation of Individual


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Estimation of IndividualValues: Example

Find the 95% prediction interval for an individual houseFind the 95% prediction interval for an individual house

with 2,000 square feetwith 2,000 square feet

Predicted Price YPredicted Price Yii = 317.85 ($1,000s)= 317.85 ($1,000s)

Prediction Interval Estimate for YPrediction Interval Estimate for YX=XX=X

102.28317.85

)X(X

)X(X

n

11StY

2

i

2

iYX1-n =

++

The prediction interval endpoints are 215.50 and 420.07, or fromThe prediction interval endpoints are 215.50 and 420.07, or from

$215,500 to $420,070$215,500 to $420,070

ii

Finding Confidence and


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Finding Confidence andPrediction Intervals in Excel

In Excel, use

PHStat | regression | simple linear regression

Check the

confidence and prediction interval for X=

box and enter the X-value and confidence leveldesired

Finding Confidence and


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Input values

Finding Confidence andPrediction Intervals in Excel


Confidence Interval Estimate forConfidence Interval Estimate forY|X=XiY|X=Xi

Prediction Interval Estimate for YPrediction Interval Estimate for YX=XiX=Xi

YY


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P itfalls of Regression

AnalysisLacking an awareness of the assumptionsunderlying least-squares regression

Not knowing how to evaluate the assumptions

Not knowing the alternatives to least-squares

regression if a particular assumption is violated

Using a regression model without knowledge of

the subject matter

Extrapolating outside the relevant range

Strategies for Avoiding


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Strategies for Avoidingthe Pitfalls of Regression

Start with a scatter diagram of X vs. Y to

observe possible relationship

Perform residual analysis to check the

assumptions

Plot the residuals vs. X to check for violations of

assumptions such as homoscedasticity

Use a histogram, stem-and-leaf display, box-and-whisker plot, or normal probability plot of the

residuals to uncover possible non-normality


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Chapter Summary

Described inference about the slope

Discussed correlation -- measuring the

strength of the associationAddressed estimation of mean values and

prediction of individual values

Discussed possible pitfalls in regression andrecommended strategies to avoid them



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Thank You

Documents

BB ABS 10th Univariate and Bivariate Corr SR MR 81Pages Final