1. Econ 3790: Business and Economics Statistics
Instructor: Yogesh Uppal

2. Today’s Lecture Covariance and Simple Correlation Coefficient
Simple Linear Regression

3. Covariance
Covariance between x and y is a measure of relationship between x and y.

4. Reed Auto periodically has
Covariance
Example: Reed Auto Sales
Reed Auto periodically has
a special week-long sale.
As part of the advertising
campaign Reed runs one or
more television commercials
during the weekend preceding the sale. Data from a
sample of 5 previous sales are shown on the next slide.

5. Covariance Example: Reed Auto Sales Number of TV Ads Number of
Cars Sold 1 3 2 14 24 18 17 27

6. Covariance x y 1 14 -1
-5.6
5.6 3 24
4.4 2 18
-1.6 17
-2.6
2.6 25
5.4
Total=10
Total = 98
SSxy=18

7. Simple Correlation Coefficient
Simple Population Correlation Coefficient
If r < 0, a negative relationship between x and y.
If r > 0, a positive relationship between x and y.

8. Simple Correlation Coefficient
Since population standard deviations of x and y are not known, we use their sample estimates to compute an estimate of r.

9. Simple Correlation Coefficient
Example: Reed Auto Sales x y
SSxx
SSyy 1 14 -1
-5.6
31.36 3 24
4.4
19.36 2 18
-1.6
2.56 17
-2.6
6.76 25
5.4
29.16
Total=10
Total=98
Total=4
Total= 89.2

10. Simple Correlation Coefficient

11. Chapter 14 Simple Linear Regression
Simple Linear Regression Model
Residual Analysis
Coefficient of Determination
Testing for Significance
Using the Estimated Regression Equation
for Estimation and Prediction

12. Simple Linear Regression Model
The equation that describes how y is related to x and
an error term is called the regression model.
The simple linear regression model is:
y = b0 + b1x +e
where:
b0 and b1 are called parameters of the model,
e is a random variable called the error term.

13. Simple Linear Regression Equation
Positive Linear Relationship y x
Regression line
Intercept b0
Slope b1
is positive

14. Simple Linear Regression Equation
Negative Linear Relationship y x
Intercept b0
Regression line
Slope b1
is negative

15. Simple Linear Regression Equation
No Relationship y x
Regression line
Intercept b0
Slope b1
is 0

16. Interpretation of b0 and b1
b0 (intercept parameter): is the value of y when x = 0.
b1 (slope parameter): is the change in y given x changes by 1 unit.

17. Estimated Simple Linear Regression Equation
The estimated simple linear regression equation
The graph is called the estimated regression line.
b0 is the y intercept of the line.
b1 is the slope of the line.
is the estimated value of y for a given value of x.

18. provide point estimates of
Estimation Process
Regression Model
y = b0 + b1x +e
Unknown Parameters
b0, b1
Sample Data:
x y
x y1
xn yn
Estimated
Regression Equation
b0 and b1
provide point estimates of

19. Least Squares Method
Slope for the Estimated Regression Equation

20. Least Squares Method y-Intercept for the Estimated Regression Equation_
x = mean value for independent variable _
y = mean value for dependent variable

21. Estimated Regression Equation
Example: Reed Auto Sales
Slope for the Estimated Regression Equation
y-Intercept for the Estimated Regression Equation
Estimated Regression Equation

22. Scatter Diagram and Regression Line

23. Estimate of Residuals x y 1 14 15.1 -1.1 3 24 24.1 -0.1 2 18 19.6 -1.617
1.9 25
0.9

24. Residual Plot Against x

25. Decomposition of total sum of squares
Relationship Among SST, SSR, SSE
SST = SSR SSE
where:
SST = total sum of squares
SSR = sum of squares due to regression
SSE = sum of squares due to error

26. Decomposition of total sum of squares
-1.1
1.21
-4.5
20.25
-0.1
0.01
4.5
-1.6
2.56
1.9
3.61
0.9
0.81
SSE=8.2
SSR=81
Check if SST= SSR + SSE

27. Coefficient of Determination
The coefficient of determination is:
r2 = SSR/SST
r2 = SSR/SST = 81/89.2 =
The regression relationship is very strong; about 91%
of the variability in the number of cars sold can be
explained by the number of TV ads.
The coefficient of determination (r2) is also the square of
the correlation coefficient (r).

28. Sample Correlation Coefficient