1. Chapter 12 Simple Linear Regression
Simple Linear Regression Model
Least Squares Method
Coefficient of Determination
Model Assumptions
Testing for Significance
Using the Estimated Regression Equation
for Estimation and Prediction
Computer Solution
Residual Analysis: Validating Model Assumptions

2. 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
b0 and b1 are called parameters of the model.
e is a random variable called the error term.

3. Simple Linear Regression Equation
The simple linear regression equation is:
E(y) = 0 + 1x
Graph of the regression equation is a straight line.
b0 is the y intercept of the regression line.
b1 is the slope of the regression line.
E(y) is the expected value of y for a given x value.

4. Simple Linear Regression Equation
Positive Linear Relationship
E(y)
Regression line
Intercept b0
Slope b1
is positive x

5. Simple Linear Regression Equation
Negative Linear Relationship
E(y)
Intercept b0
Regression line
Slope b1
is negative x

6. Simple Linear Regression Equation
No Relationship
E(y)
Regression line
Intercept b0
Slope b1
is 0 x

7. Estimated Simple Linear Regression Equation
The estimated simple linear regression equation is:
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 x value.

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

9. Least Squares Method Least Squares Criterion where:
yi = observed value of the dependent variable
for the ith observation
yi = estimated value of the dependent variable ^

10. The Least Squares Method
Slope for the Estimated Regression Equation

11. The Least Squares Method
y-Intercept for the Estimated Regression Equation
where:
xi = value of independent variable for ith observation
yi = value of dependent variable for ith observation
x = mean value for independent variable
y = mean value for dependent variable
n = total number of observations _ _

12. Example: Reed Auto Sales
Simple Linear Regression
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.

13. Example: Reed Auto Sales
Simple Linear Regression
Number of TV Ads Number of Cars Sold
1 14
3 24
2 18
1 17
3 27

14. Example: Reed Auto Sales
Slope for the Estimated Regression Equation
b1 = (10)(100)/5 = _____
24 - (10)2/5
y-Intercept for the Estimated Regression Equation
b0 = (2) = _____
Estimated Regression Equation
y = x ^

15. Example: Reed Auto Sales
Scatter Diagram ^

16. The Coefficient of Determination
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 ^

17. The Coefficient of Determination
The coefficient of determination is:
r2 = SSR/SST
where:
SST = total sum of squares
SSR = sum of squares due to regression

18. Example: Reed Auto Sales
Coefficient of Determination
r2 = SSR/SST = 100/114 =
The regression relationship is very strong because 88% of the variation in number of cars sold can be explained by the linear relationship between the number of TV ads and the number of cars sold.

19. The Correlation Coefficient
Sample Correlation Coefficient
where:
b1 = the slope of the estimated regression
equation

20. Example: Reed Auto Sales
Sample Correlation Coefficient
The sign of b1 in the equation is “+”.
rxy =

21. Model Assumptions Assumptions About the Error Term 
The error  is a random variable with mean of zero.
The variance of  , denoted by  2, is the same for all values of the independent variable.
The values of  are independent.
The error  is a normally distributed random variable.

22. Testing for Significance
To test for a significant regression relationship, we must conduct a hypothesis test to determine whether the value of b1 is zero.
Two tests are commonly used
t Test
F Test
Both tests require an estimate of s 2, the variance of e in the regression model.

23. Testing for Significance
An Estimate of s 2
The mean square error (MSE) provides the estimate
of s 2, and the notation s2 is also used.
s2 = MSE = SSE/(n-2)
where:

24. Testing for Significance
An Estimate of s
To estimate s we take the square root of s 2.
The resulting s is called the standard error of the estimate.

25. Testing for Significance: t Test
Hypotheses
H0: 1 = 0
Ha: 1 = 0
Test Statistic
where

26. Testing for Significance: t Test
Rejection Rule
Reject H0 if t < -tor t > t
where: t is based on a t distribution
with n - 2 degrees of freedom

27. Example: Reed Auto Sales
t Test
Hypotheses
H0: 1 = 0
Ha: 1 = 0
Rejection Rule
For  = .05 and d.f. = 3, t.025 = _____
Reject H0 if t > t.025 = _____

28. Example: Reed Auto Sales
t Test
Test Statistics
t = _____/_____ = 4.63
Conclusions
t = 4.63 > 3.182, so reject H0

29. Confidence Interval for 1
We can use a 95% confidence interval for 1 to test the hypotheses just used in the t test.
H0 is rejected if the hypothesized value of 1 is not included in the confidence interval for 1.

30. Confidence Interval for 1
The form of a confidence interval for 1 is:
where b1 is the point estimate
is the margin of error
is the t value providing an area
of a/2 in the upper tail of a
t distribution with n - 2 degrees
of freedom

31. Example: Reed Auto Sales
Rejection Rule
Reject H0 if 0 is not included in
the confidence interval for 1.
95% Confidence Interval for 1
= 5 +/ (1.08) = 5 +/- 3.44
or ____ to ____
Conclusion
0 is not included in the confidence interval.
Reject H0

32. Testing for Significance: F Test
Hypotheses
H0: 1 = 0
Ha: 1 = 0
Test Statistic
F = MSR/MSE

33. Testing for Significance: F Test
Rejection Rule
Reject H0 if F > F
where: F is based on an F distribution
with 1 d.f. in the numerator and
n - 2 d.f. in the denominator

34. Example: Reed Auto Sales
F Test
Hypotheses
H0: 1 = 0
Ha: 1 = 0
Rejection Rule
For  = .05 and d.f. = 1, 3: F.05 = ______
Reject H0 if F > F.05 = ______.

35. Example: Reed Auto Sales
F Test
Test Statistic
F = MSR/MSE = ____ / ______ = 21.43
Conclusion
F = > 10.13, so we reject H0.

36. Some Cautions about the Interpretation of Significance Tests
Rejecting H0: b1 = 0 and concluding that the relationship between x and y is significant does not enable us to conclude that a cause-and-effect relationship is present between x and y.
Just because we are able to reject H0: b1 = 0 and demonstrate statistical significance does not enable us to conclude that there is a linear relationship between x and y.

37. Using the Estimated Regression Equation for Estimation and Prediction
Confidence Interval Estimate of E(yp)
Prediction Interval Estimate of yp
yp + t/2 sind
where: confidence coefficient is 1 -  and
t/2 is based on a t distribution
with n - 2 degrees of freedom

38. Example: Reed Auto Sales
Point Estimation
If 3 TV ads are run prior to a sale, we expect the mean number of cars sold to be:
y = (3) = ______ cars ^

39. Example: Reed Auto Sales
Confidence Interval for E(yp)
95% confidence interval estimate of the mean number of cars sold when 3 TV ads are run is:
= ______ to _______ cars

40. Example: Reed Auto Sales
Prediction Interval for yp
95% prediction interval estimate of the number of cars sold in one particular week when 3 TV ads are run is:
= _____ to ______ cars

41. Residual Analysis Residual for Observation i yi – yi
Standardized Residual for Observation i
where:
and ^ ^ ^ ^

42. Example: Reed Auto Sales
Residuals

43. Example: Reed Auto Sales
Residual Plot

44. Residual Analysis
Residual Plot
Good Pattern
Residual x

45. Residual Analysis
Residual Plot
Nonconstant Variance
Residual x

46. Model Form Not Adequate
Residual Analysis
Residual Plot
Model Form Not Adequate
Residual x

47. End of Chapter 12