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Systems Engineering Program. Department of Engineering Management, Information and Systems. EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS. Correlation and Regression Analysis – An Application. Dr. Jerrell T. Stracener, SAE Fellow.
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Systems Engineering Program Department of Engineering Management, Information and Systems EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS Correlation and Regression Analysis – An Application Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering
Montgomery, Peck, and Vining (2001) present data concerning the performance of the 28 National Football league teams in 1976. 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:
Correlation Analysis • Statistical analysis used to obtain a quantitative • measure of the strength of the relationship between • a dependent variable and one or more independent • variables
Sample correlation coefficient Notes: -1 r 1 R=r2 100% = coefficient of determination
Correlation To test for no linear association between x & y, calculate Where r is the sample correlation coefficient and n is the sample size.
Correlation Conclude no linear association if then treat y1, y2, …, yn as a random sample
Correlation Take α=0.05 and check from the T-table, we get Since t=-5.5766 < -2.0555, we conclude that there is linear association between x and y and proceed with regression analysis
Linear Regression Model Simple linear regression model where Y is the response (or dependent) variable 0 and 1are the unknown parameters ~ N(0,) and data: (x1, y1), (x2, y2), ..., (xn, yn)
Point estimate of the linear model is Least squares regression equation
(1 -)100% confidence interval for 0is where and where Interval Estimates for y intercept (0)
Interval Estimates for y intercept (0) Take =0.05, then 95% confidence interval for 0is
Interval Estimates for y intercept (0) Apply to the equation and we get the lower and upper bound for :
Interval Estimates for slope(1) (1 -)100% confidence interval for 1is where and where
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
Confidence interval for conditional mean of Y, given x=2205 and
Prediction interval for a single future value of Y, given x and
Prediction interval for a single future value of Y, given x=2000 Given x= 2000,
Prediction interval for a single future value of Y, given x=2000 and