business statistics session 17 (1)

8/7/2019 Business statistics Session 17 (1)

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Business statisticsSession 17

Simple correlation and regression


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Learning Objectives

Understand the concept of Correlation and regression Compute correlation coefficient Test the significance of correlation coefficient

Compute the equation of a simple regression line froma sample of data, and interpret the slope and interceptof the equation.

Understand the usefulness of residual analysis intesting the assumptions underlying regression

analysis and in examining the fit of the regression lineto the data. Compute a standard error of the estimate and interpret

its meaning.


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Regression and Correlation

Regression analysis is the process ofconstructing a mathematical model or

function that can be used to predict ordetermine one variable by anothervariable.

Correlation is a measure of the degree of

relatedness of two variables.


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Pearson Product-MomentCorrelation Coefficient

rS S X Y

S S X S S Y

X X Y Y

X YX Y

n

n n

X X Y Y

X X Y Y

!

!

!

2 2

2

2

2

2

e e1 1r


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Degrees of Correlation

Correlation is a measure of the degree ofrelatedness of variables

Coefficient of Correlation (r) - applicableonly if both variables being analyzed haveat least an interval level of data


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Degrees of Correlation

The term (r) is a measure of the linearcorrelation of two variables

The number ranges from -1 to 0 to +1 Closer to +1, the higher the correlation

between the dependent and the independentvariables

See the formula forPearson Product Momentcorrelation coefficient


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Correlation Coefficient

Covariance

COV(X,Y)=(Xi-X)*(Yi-Y)/N-1

Pearson product moment correlationcoefficient

rxy=Cov xy/Sx*Sy


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Correlation (contd.)

Population correlation (p) - If the databaseincludes an entire population

Sample correlation (r) - If measure isbased on a sample


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Three Degrees of Correlation

r < 0 r > 0

r = 0


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Computation ofrforthe Economics Example

Day

Interest

X

Futures

Index

Y1 7.43 221 55.205 48,841 1,642.03

2 7.48 222 55.950 49,284 1,660.563 8.00 226 64.000 51,076 1,808.00

4 7.75 225 60.063 50,625 1,743.75

5 7.60 224 57.760 50,176 1,702.40

6 7.63 223 58.217 49,729 1,701.49

7 7.68 223 58.982 49,729 1,712.64

8 7.67 226 58.829 51,076 1,733.42

9 7.59 226 57.608 51,076 1,715.34

10 8.07 235 65.125 55,225 1,896.45

11 8.03 233 64.481 54,289 1,870.99

12 8.00 241 64.000 58,081 1,928.00

Summations 92.93 2,725 720.220 619,207 21,115.07

X2 Y2 XY


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Computation ofrEconomics Example

r

X X Y Y

X YX Y

n

n n

!

!

!

22

2

2

2 2

2 1 1 1 5 0 79 2 9 3 2 7 2 5

1 2

7 2 0 2 2 1 2 6 1 9 2 0 7 1 29 2 9 3 2 7 2 5

8 1 5

, ..

. ,.

.


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Testing the Significance of theCorrelation Coefficient

Null hypothesis: Ho : p = 0

Alternative hypothesis: Ha : p 0

Test statistic

Example: n = 6 and r = .70

t=.706-2/1-.702

= 1.96

At E = .05 , n-2 = 4 degrees of freedom,

Critical value of t = 2.78

Since 1.96


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Simple Regression Analysis

Bivariate (two variables) linear regression-- the most elementary regression model dependent variable, the variable to be

predicted, usually called Y independent variable, the predictor or

explanatory variable, usually called X

Nonlinear relationships and regression

models with more than one independentvariable can be explored by using multipleregression models


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Regression Models Deterministic Regression Model

Y = F0 + F1X

Probabilistic Regression ModelY = F0 + F1X + I

F0 and F1 are population parameters

F0 and F1 are estimated by samplestatistics b0 and b1


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Equation of the SimpleRegression Line

YY

where

Y

b

bbb

ofvaluepredictedthe=slopesamplethe=

interceptsamplethe=:

1

0

10!


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Least Squares Analysis

Least squares analysis is a processwhereby a regression model is developedby producing the minimum sum of thesquared error values

The vertical distance from each point tothe line is the error of the prediction.

The least squares regression line is theregression line that results in the smallestsum of errors squared.


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Least Squares Analysis

1 2 2 2

2

2bY Y Y n Y

n

YY

n

n

!

!

!

0 1 1b b bYY

n n! !


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Solving forb1 and b0 of the Regression

Line: Airline Cost Example

Number ofPassengers Cost ($1,000)

X Y X2

XY

61 4.28 3,721 261.08

63 4.08 3,969 257.0467 4.42 4,489 296.1469 4.17 4,761 287.7370 4.48 4,900 313.6074 4.30 5,476 318.2076 4.82 5,776 366.3281 4.70 6,561 380.7086 5.11

7,396 4

39.46

91 5.13 8,281 466.8395 5.64 9,025 535.8097 5.56 9,409 539.32

X= 930 Y=56.69 2

X = 73,764 XY=4,462.22


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Residual Analysis: Airline Cost Example

Number of PredictedPassengers Cost ($1,000) Value Residual

X Y Y YY

61 4.28 4.053 .227

63 4.08 4.134 -.05467 4.42 4.297 .12369 4.17 4.378 -.20870 4.48 4.419 .06174 4.30 4.582 -.28276 4.82 4.663 .15781 4.70 4.867 -.16786 5.11 5.070 .040

91 5.13 5.274 -.14495 5.64 5.436 .20497 5.56 5.518 .042

! 001.)( YY


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Standard Error of the Estimate

Residuals represent errors of estimation forindividual points.

A more useful measurement of error is thestandard error of the estimate

The standard error of the estimate, denotedse,is a standard deviation of the error of the

regression model


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Standard Error of the Estimate forthe Airline Cost Example

1773.0

1031434.0

2

31434.0

2

!

!

!

!

!

n

SSE

SSE

S

YY

e


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Thank you

business statistics session 17 (1)

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