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1 Correlation and Regression Analysis – An Application Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Systems Engineering Program Department of Engineering Management, Information and Systems

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2 Montgomery, Peck, and Vining (2001) present data concerning the performance of the 28 National Football league teams in It is suspected that the number of games won(y) is related to the number of yards gained rushing by an opponent(x). The data are shown in the following table:

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3 Team Games Won (y) Yards Rushing by Opponent (x)Team Games Won (y) Yards Rushing by Opponent (x) Washington102205Detroit61901 Minnesota112096Green Bay52288 New England111847Houston52072 Oakland131903Kansas City52861 Pittsburgh101457Miami62411 Baltimore111848New Orleans42289 Los Angeles101564New york Giants32203 Dallas111821New York Jets32592 Atlanta42577Philadelphia42053 Buffalo22476St. Louis Chicago71984San Diego62048 Cincinnati101917San Francisco81786 Cleveland91761Seattle22876 Denver91709Tampa Bay02560

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Statistical analysis used to obtain a quantitative measure of the strength of the relationship between a dependent variable and one or more independent variables Correlation Analysis

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Scatter Plot

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Sample correlation coefficient Notes: -1 r 1 R=r 2 100% = coefficient of determination

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Stracener_EMIS 7370/STAT 5340_Fall 08_ R=r 2 100% =0.5447

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Stracener_EMIS 7370/STAT 5340_Fall 08_ To test for no linear association between x & y, calculate Where r is the sample correlation coefficient and n is the sample size. Correlation

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Conclude no linear association if then treat y 1, y 2, …, y n as a random sample Correlation

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Since t= < , we conclude that there is linear association between x and y and proceed with regression analysis Correlation Take α=0.05 and check from the T-table, we get

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Simple linear regression model where Y is the response (or dependent) variable 0 and 1 are the unknown parameters ~ N(0,) and data: (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ) Linear Regression Model

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Least squares estimates of 0 and 1

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Stracener_EMIS 7370/STAT 5340_Fall 08_ estimates of 1

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Stracener_EMIS 7370/STAT 5340_Fall 08_ estimates of 0

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Point estimate of the linear model is Least squares regression equation

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Regression Fitted Line Plot

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Point estimate of 2

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Stracener_EMIS 7370/STAT 5340_Fall 08_ (1 - ) 100% confidence interval for 0 is where and where Interval Estimates for y intercept ( 0 )

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Take=0.05, then 95 % confidence interval for 0 is Interval Estimates for y intercept ( 0 )

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Apply to the equation and we get the lower and upper bound for : Interval Estimates for y intercept ( 0 )

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Stracener_EMIS 7370/STAT 5340_Fall 08_ (1 - ) 100% confidence interval for 1 is where and where Interval Estimates for slope ( 1 )

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Interval Estimates for slope ( 1 )

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Confidence interval for conditional mean of Y, given x=2205 Given x equal to 2205, we can calculate the confidence interval of conditional mean of Y

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Confidence interval for conditional mean of Y, given x=2205 and

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Stracener_EMIS 7370/STAT 5340_Fall 08_

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Prediction interval for a single future value of Y, given x and

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Stracener_EMIS 7370/STAT 5340_Fall 08_ Given x= 2000, Prediction interval for a single future value of Y, given x=2000

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Stracener_EMIS 7370/STAT 5340_Fall 08_ and Prediction interval for a single future value of Y, given x=2000

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Stracener_EMIS 7370/STAT 5340_Fall 08_

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Stracener_EMIS 7370/STAT 5340_Fall 08_ XYXYX^2Y^2Y ^(Y-Y^)^2(x-xbar)^ SUM x-bar <-rS b b <-S^2b0l b <--Sb0u S b S b1l Y(2205)-> S b1u mu-l mu-u Y(2000)-> y-l y-u Excel Calculation

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Stracener_EMIS 7370/STAT 5340_Fall 08_ SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations28 ANOVA dfSSMSFSignificance F Regression E-06 Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95%Lower 95.0%Upper 95.0% Intercept E X Variable E RESIDUAL OUTPUT ObservationPredicted YResiduals Excel Regression Analysis Output

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