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

2. Sampling Distribution of b1
Expected value of b1:
E(b1) =b1
Variance of b1:
Var(b1) = σ2/SSx

3. Estimate of σ2 s 2 = MSE = SSE/(n - 2) where:
The mean square error (MSE) provides the estimate of σ2.
s 2 = MSE = SSE/(n - 2)
where:

4. Sample variance of b1 Estimate of variance of b1:
Standard error of b1:
s is called the standard error of the estimate.

5. Interval Estimate of b1:
(1-a)100% confidence interval for b1 is:
Where ta/2 is the value from t distribution with (n-2) degrees of freedom such that probability in the upper tail is a/2.

6. Example: Reed Auto Sales
s2 = MSE = SSE/(n - 2) = 8.2/3 =2.73
95% confidence interval for b1:
We can say we 95% confidence that b1 will lie between 1.87 and 7.13.

7. Testing for Significance: t Test
Hypotheses
Test Statistic
Where b1 is the slope estimate and SE(b1) is the standard error of b1.

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

9. Testing for Significance: t Test
1. Determine the hypotheses.
2. Specify the level of significance.
a = .05
3. Select the test statistic.
4. State the rejection rule.
Reject H0 if p-value < .05
or t ≤ or t ≥ 3.182

10. Testing for Significance: t Test
5. Compute the value of the test statistic.
6. Determine whether to reject H0.
t = 5.42 > ta/2 = We can reject H0.

11. 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.

12. Multiple Regression Model
The equation that describes how the dependent variable y is related to the independent variables x1, x2, xp and an error term is called the multiple regression model.
y = b0 + b1x1 + b2x bpxp + e
where:
b0, b1, b2, , bp are the parameters, and
e is a random variable called the error term

13. Estimated Multiple Regression Equation
A simple random sample is used to compute sample statistics b0, b1, b2, , bp that are used as the point estimators of the parameters b0, b1, b2, , bp.
The estimated multiple regression equation is: ^
y = b0 + b1x1 + b2x bpxp

14. Interpreting the Coefficients
In multiple regression analysis, we interpret each
regression coefficient as follows:
bi represents an estimate of the change in y
corresponding to a 1-unit increase in xi when all
other independent variables are held constant.

15. Multiple Regression Model
Example: Car Sales
Suppose we believe that number of cars sold (y) is
not only related to the number of ads (x1), but also to the minimum down payment required at the (x2). The regression model can be given by:
y = 0 + 1x1 + 2x2 + 
where
y = number of cars sold
x1 = number of ads
x2 = minimum down payment required (‘000)

16. Estimated Regression Equation
y = x x2
Interpretation?
Estimated values of y?
Error?
Prediction?

17. Multiple 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

18. Multiple Coefficient of Determination
R2 = SSR/SST
R2 = 84.63/89.2 =
Adjusted Multiple Coefficient of Determination
Standard Error of Estimate

19. Testing for Significance: t Test
Hypotheses
Test Statistics
Rejection Rule
Reject H0 if p-value < a or
if t < -tor t > t where t
is based on a t distribution
with n - p - 1 degrees of freedom.

20. Example: Testing for significance of coefficients
Hypotheses
For  = .05 and d.f. = ?, t.025 =
Rejection Rule
Test Statistics

21. Testing for Significance of Regression: F Test
H0: 1 = 2 = = p = 0
Ha: One or more of the parameters
is not equal to zero.
Hypotheses
Test Statistics
F = MSR/MSE
Rejection Rule
Reject H0 if p-value < a or if F > F,
where F is based on an F distribution
with p d.f. in the numerator and
n - p - 1 d.f. in the denominator.

22. Multiple Regression Model
Example 2: Programmer Salary Survey
A software firm collected data for a sample
of 20 computer programmers. A suggestion
was made that regression analysis could
be used to determine if salary was related
to the years of experience and the score
on the firm’s programmer aptitude test.
The years of experience, score on the aptitude
test, and corresponding annual salary ($1000s) for a
sample of 20 programmers is shown on the next
slide.

23. Multiple Regression Model
Exper.
Score
Salary
Exper.
Score
Salary 4 7 1 5 8 10 6 78
100 86 82 84 75 80 83 91 24 43
23.7
34.3
35.8 38
22.2
23.1 30 33 9 2 10 5 6 8 4 3 88 73 75 81 74 87 79 94 70 89 38
26.6
36.2
31.6 29 34
30.1
33.9
28.2 30

24. Multiple Regression Model
Suppose we believe that salary (y) is
related to the years of experience (x1) and the score on
the programmer aptitude test (x2) by the following
regression model:
y = 0 + 1x1 + 2x2 + 
where
y = annual salary ($1000)
x1 = years of experience
x2 = score on programmer aptitude test

25. Solving for b0, b1 and b2:

26. Anova Table Source of Variation Sum of Squares Degrees of Freedom
Mean Square
F-statistic
Regression
500.34 ……
……..
……….
Error
…….
Total
599.8

27. Estimated Regression Equation
SALARY = (EXPER) (SCORE)
b1 = implies that salary is expected to increase by $1,404 for each additional year of experience (when the variable score on programmer attitude test is held constant).
b2 = implies that salary is expected to increase by $251 for each additional point scored on the programmer aptitude test (when the variable years of experience is held constant).

28. Prediction Bob’s estimated salary is $28,358.
Suppose Bob had an experience of 4 years and had a score of 78 on the aptitude test. What would you estimate (or expect) his score to be?
= *(4) (78) =
Bob’s estimated salary is $28,358.

29. Error
Bob’s actual salary is $ How much error we made in estimating his salary based on his experience and score?
So, we shall overestimate Bob’s salary.

30. Multiple 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

31. Multiple Coefficient of Determination
R2 = SSR/SST
R2 = / =
Adjusted Multiple Coefficient of Determination

32. Testing for Significance: t Test
Hypotheses
Test Statistics
Rejection Rule
Reject H0 if p-value < a or
if t < -tor t > t where t
is based on a t distribution
with n - p - 1 degrees of freedom.

33. Example Hypotheses For  = .05 and d.f. = 17, t.025 = 2.11
Reject H0 if p-value < .05 or if t > 2.11
Rejection Rule
Test Statistics
Since t=7.07 > t0.025 =2.11, we reject H0.

34. Testing for Significance of Regression: F Test
H0: 1 = 2 = = p = 0
Ha: One or more of the parameters
is not equal to zero.
Hypotheses
Test Statistics
F = MSR/MSE
Rejection Rule
Reject H0 if p-value < a or if F > F,
where F is based on an F distribution
with p d.f. in the numerator and
n - p - 1 d.f. in the denominator.

35. Example F = 42.8 > F0.05 = 3.59, so we can reject H0. Hypotheses
Ha: One or both of the parameters
is not equal to zero.
For  = .05 and d.f. = 2, 17; F.05 = 3.59
Reject H0 if p-value < .05 or F > 3.59
Rejection Rule
Test Statistics
F = MSR/MSE
= /5.86 = 42.8
F = 42.8 > F0.05 = 3.59, so we can reject H0.