1. Correlation and Regression Analysis – An Application
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

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:

3. Yards Rushing by Opponent (x)
Team
Games Won (y)
Yards Rushing by Opponent (x)
Washington 10
2205
Detroit 6
1901
Minnesota 11
2096
Green Bay 5
2288
New England
1847
Houston
2072
Oakland 13
1903
Kansas City
2861
Pittsburgh
1457
Miami
2411
Baltimore
1848
New Orleans 4
2289
Los Angeles
1564
New york Giants 3
2203
Dallas
1821
New York Jets
2592
Atlanta
2577
Philadelphia
2053
Buffalo 2
2476
St. Louis
1979
Chicago 7
1984
San Diego
2048
Cincinnati
1917
San Francisco 8
1786
Cleveland 9
1761
Seattle
2876
Denver
1709
Tampa Bay
2560

4. 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

5. Scatter Plot

6. Sample correlation coefficient
Notes: -1  r  1
R=r2  100% = coefficient of determination

7. R=r2  100% =0.5447

8. Correlation To test for no linear association between x & y, calculate
Where r is the sample correlation coefficient and n
is the sample size.

9. Correlation Conclude no linear association if
then treat y1, y2, …, yn as a random sample

10. Correlation Take α=0.05 and check from the T-table, we get
Since t= < , we conclude that there
is linear association between x and y and proceed
with regression analysis

11. Linear Regression Model
Simple linear regression model
where
Y is the response (or dependent) variable
0 and 1 are the unknown parameters
 ~ N(0,) and data: (x1, y1), (x2, y2), ..., (xn, yn)

12. Least squares estimates of 0 and 1

13. estimates of 1

14. estimates of 0

15. Least squares regression equation
Point estimate of the linear model is
Least squares regression equation

16. Regression Fitted Line Plot

17. Point estimate of 2

18. Interval Estimates for y intercept (0)
(1 - )100% confidence interval for 0 is
where
and

19. Interval Estimates for y intercept (0)
Take =0.05, then 95% confidence interval for 0 is

20. Interval Estimates for y intercept (0)
Apply to the equation and we get the lower and upper
bound for :

21. Interval Estimates for slope (1)
(1 - )100% confidence interval for 1 is
where
and

22. Interval Estimates for slope (1)

23. 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

24. Confidence interval for conditional mean of Y, given x=2205
and

26. Prediction interval for a single future value of Y, given x
and

27. Prediction interval for a single future value of Y, given x=2000

28. Prediction interval for a single future value of Y, given x=2000
and

30. Excel Calculation X Y XY X^2 Y^2 Y ^ (Y-Y^)^2 (x-xbar)^2 2205 10 22050
100
2096 11
23056
121
1847
20317
1903 13
24739
169
1457
14570
1848
20328
1564
15640
298272
1821
20031
2577 4
10308 16
2476 2
4952
1984 7
13888 49
1917
19170
1761 9
15849 81
1709
15381
1901 6
11406 36
2288 5
11440 25
2072
10360
2861
14305
2411
14466
2289
9156
2203 3
6609
2592
7776
2053
8212
1979
19790
2048
12288
1786 8
14288 64
2876
5752
2560
SUM
59084
195
386127
1685
x-bar
9155
<-r
Sb0 b1
<-S^2
b0l b0
<--S
b0u
Sb1
Sb1l
Y(2205)->
Sb1u
mu-l
mu-u
Y(2000)->
y-l
y-u

31. Regression Statistics
Excel Regression Analysis Output
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 28
ANOVA df SS MS F
Significance F
Regression 1
7.381E-06
Residual 26
Total 27
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Lower 95.0%
Upper 95.0%
Intercept
1.46E-08
X Variable 1
7.38E-06
RESIDUAL OUTPUT
Observation
Predicted Y
Residuals 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25