Chapter 12b

1
Chapter 12b

• Testing for significancethe t-test
• Developing confidence intervals for estimates of
  ß1.
• Testing for significancethe f-test
• Using Excels regression tool.

2
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
and
Both the t test and F test require an estimate
of s 2, the variance of e in the regression
model.
3
Testing for Significance

• An Estimate of s

The mean square error (MSE) provides the
estimate of s 2, and the notation s2 is also used.
s 2 MSE SSE/(n - 2)
where
4
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.

5
Testing for Significance t Test

• Hypotheses
• Test Statistic

6
Testing for Significance t Test

• Rejection Rule

Reject H0 if t lt -t????or t gt t????
where t??? is based on a t
distribution with n - 2 degrees of freedom
7
Testing for Significance t Test

• Using the Test Statistic

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 t gt 3.182
(3 degrees of freedom)
8
Testing for Significance t Test

• Using the Test Statistic

5. Compute the value of the test statistic.
6. Determine whether to reject H0.
Because t 4.63 gt 3.182, we reject H0.
At the .05 level of significance, the
sample evidence indicates that there is a
significant relationship between the number of
TV ads aired and the number of cars sold.
9
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.

10
Confidence Interval for ?1

• The form of a confidence interval for ?1 is

b1 is the point estimator
11
Confidence Interval for ?1

• Rejection Rule
• 95 Confidence Interval for ?1
• Conclusion

Reject H0 if 0 is not included in the confidence
interval for ?1.
or 1.56 to 8.44
0 is not included in the confidence interval.
Reject H0
12
Testing for Significance F Test

• Hypotheses
• Test Statistic

F MSR/MSE
13
Testing for Significance F Test

• Rejection Rule

Reject H0 if F gt F?
where F? is based on an F distribution with 1
degree of freedom in the numerator and n - 2
degrees of freedom in the denominator
14
Testing for Significance F Test

• Using the Test Statistic

1. Determine the hypotheses.
2. Specify the level of significance.
a .05
3. Select the test statistic.
F MSR/MSE
4. State the rejection rule.
Reject H0 if F gt 10.13
(1 d.f. in numerator, 3 d.f. in denominator)
15
Testing for Significance F Test

• Using the Test Statistic

5. Compute the value of the test statistic.
F MSR/MSE 100/4.667 21.43
6. Determine whether to reject H0.
Because F 21.43 gt 10.13, we reject H0.
At the .05 level of significance, the
statistical evidence is sufficient to conclude
that we have a significant relationship between
the number of TV ads aired and the number of cars
sold.
16
Some Cautions about theInterpretation 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.

17
Using Excels Regression Tool

• Up to this point, you have seen how Excel can
  be
• used for various parts of a regression
  analysis.

• Excel also has a comprehensive tool in its
  Data
• Analysis package called Regression.

• The Regression tool can be used to perform a
• complete regression analysis.

18
Using Excels Regression Tool

• Formula Worksheet (showing data)

19
Using Excels Regression Tool

• Performing the Regression Analysis

Step 1 Select the Tools pull-down menu
Step 2 Choose the Data Analysis option
Step 3 Choose Regression from the list of
Analysis Tools
20
Using Excels Regression Tool

• Performing the Regression Analysis

Step 4 When the Regression dialog box
appears Enter C1C6 in the Input Y
Range box Enter B1B6 in the Input X
Range box Select Labels
Select Confidence Level Enter 95 in
the Confidence Level box Select Output
Range Enter A9 (any cell) in the Ouput
Range box Click OK to begin the
regression analysis
21
Using Excels Regression Tool

• Regression Dialog Box

22
Using Excels Regression Tool

• Value Worksheet

Data
Regression Statistics Output
Estimated Regression Equation Output
ANOVA Output
23
Using Excels Regression Tool

• Estimated Regression Equation Output (left
  portion)

Note Columns F-I are not shown.
24
Using Excels Regression Tool

• Estimated Regression Equation Output (right
  portion)

Note Columns C-E are hidden.
25
Using Excels Regression Tool

• ANOVA Output

26
Using Excels Regression Tool

• Regression Statistics Output

27
Using the Estimated Regression Equationfor
Estimation and Prediction

• Confidence Interval Estimate of E(yp)
• Prediction Interval Estimate of yp

where confidence coefficient is 1 - ? and t?/2
is based on a t distribution with n - 2 degrees
of freedom
28
Point Estimation

• If 3 TV ads are run prior to a sale, we expect
  the mean number of cars sold to be

29
Using Excel to Develop Confidenceand Prediction
Interval Estimates

• Formula Worksheet (confidence interval portion)

30
Using Excel to Develop Confidenceand Prediction
Interval Estimates

• Value Worksheet (confidence interval portion)

31
Confidence Interval for E(yp)
The 95 confidence interval estimate of the
mean number of cars sold when 3 TV ads are run is
25 4.61 20.39 to 29.61 cars
32
Using Excel to Develop Confidenceand Prediction
Interval Estimates

• Formula Worksheet (prediction interval portion)

33
Using Excel to Develop Confidenceand Prediction
Interval Estimates

• Value Worksheet (prediction interval portion)

34
Prediction Interval for yp
The 95 prediction interval estimate of the
number of cars sold in one particular week when 3
TV ads are run is
25 8.28 16.72 to 33.28 cars