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Slides Prepared by JOHN S. LOUCKS St. Edward’s University

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Chapter 14 Simple Linear Regression

- Simple Linear Regression Model
- Least Squares Method
- Coefficient of Determination
- Model Assumptions
- Testing for Significance
- Excel’s Regression Tool
- Using the Estimated Regression Equation

for Estimation and Prediction

- Residual Analysis: Validating Model Assumptions
- Outliers and Influential Observations

The Simple Linear Regression Model

- Simple Linear Regression Model

y = 0 + 1x+

- Simple Linear Regression Equation

E(y) = 0 + 1x

- Estimated Simple Linear Regression Equation

y = b0 + b1x

^

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

for the ith observation

^

The Least Squares Method

- Slope for the Estimated Regression Equation
- y-Intercept for the Estimated Regression Equation

b0 = y - b1x

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

_

_

_

_

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 below.

Number of TV AdsNumber of Cars Sold

1 14

3 24

2 18

1 17

3 27

Example: Reed Auto Sales

- Slope for the Estimated Regression Equation

b1 = 220 - (10)(100)/5 = 5

24 - (10)2/5

- y-Intercept for the Estimated Regression Equation

b0 = 20 - 5(2) = 10

- Estimated Regression Equation

y = 10 + 5x

^

Using Excel to Develop a Scatter Diagramand Compute the Estimated Regression Equation

- Formula Worksheet (showing data)

Using Excel to Develop a Scatter Diagramand Compute the Estimated Regression Equation

- Producing a Scatter Diagram

Step 1 Select cells B1:C6

Step 2 Select the Chart Wizard

Step 3 When the Chart Type dialog box appears:

Choose XY (Scatter) in the Chart type list

Choose Scatter from the Chart sub-type display

Select Next >

Step 4 When the Chart Source Data dialog box appears

Select Next >

… continued

Using Excel to Develop a Scatter Diagramand Compute the Estimated Regression Equation

- Producing a Scatter Diagram

Step 5 When the Chart Options dialog box appears:

Select the Titles tab and then

Delete Cars Sold in the Chart title box

Enter TV Ads in the Value (X) axis box

Enter Cars Sold in the Value (Y) axis box

Select the Legend tab and then

Remove the check in the Show Legend box

Select Next >

… continued

Using Excel to Develop a Scatter Diagramand Compute the Estimated Regression Equation

- Producing a Scatter Diagram

Step 6 When the Chart Location dialog box appears:

Specify the location for the new chart

Select Finish to display the scatter diagram

Using Excel to Develop a Scatter Diagramand Compute the Estimated Regression Equation

- Adding the Trendline

Step 1 Position the mouse pointer over any data

point and right click to display the Chart menu

Step 2 Select the Add Trendline option

Step 3 When the Add Trendline dialog box appears:

On the Type tab select Linear

On the Options tab select the Displayequation on chart box

Click OK

Using Excel to Develop a Scatter Diagramand Compute the Estimated Regression Equation

- Scatter Diagram

^

The Coefficient of Determination- Relationship Among SST, SSR, SSE

SST = SSR + SSE

- Coefficient of Determination

r2 = SSR/SST

where:

SST = total sum of squares

SSR = sum of squares due to regression

SSE = sum of squares due to error

Example: Reed Auto Sales

- Coefficient of Determination

r2 = SSR/SST = 100/114 = .8772

The regression relationship is very strong since 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.

Using Excel to Computethe Coefficient of Determination

- Producing R2

Step 1 Position the mouse pointer over any data

point in the scatter diagram and right click

Step 2 When the Chart menu appears:

Select the Add Trendline option

Step 3 When the Add Trendline dialog box appears:

On the Options tab, select the Display R- squared value on chart box

Click OK

Using Excel to Computethe Coefficient of Determination

- Value Worksheet (showing R2)

The Correlation Coefficient

- Sample Correlation Coefficient

where:

b1 = the slope of the estimated regression

equation

Example: Reed Auto Sales

- Sample Correlation Coefficient

The sign of b1 in the equation is “+”.

rxy = +.9366

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.

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 s2, the variance of e in the regression model.

Testing for Significance

- An Estimate of s2

The mean square error (MSE) provides the estimate

of s2, and the notation s2 is also used.

s2 = MSE = SSE/(n-2)

where:

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.

Testing for Significance: t Test

- Hypotheses

H0: 1 = 0

Ha: 1 = 0

- Test Statistic
- Rejection Rule

Reject H0 if t < -tor t > t

where tis based on a t distribution with

n - 2 degrees of freedom.

Example: Reed Auto Sales

- t Test
- Hypotheses H0: 1 = 0

Ha: 1 = 0

- Rejection Rule

For = .05 and d.f. = 3, t.025 = 3.182

Reject H0 if t > 3.182

- Test Statistics

t = 5/1.08 = 4.63

- Conclusions

Reject H0

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.

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

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 +/- 3.182(1.08) = 5 +/- 3.44

or 1.56 to 8.44

- Conclusion

Reject H0

Testing for Significance: F Test

- Hypotheses

H0: 1 = 0

Ha: 1 = 0

- Test Statistic

F = MSR/MSE

- 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.

- F Test
- Hypotheses H0: 1 = 0

Ha: 1 = 0

- Rejection Rule
- For = .05 and d.f. = 1, 3: F.05 = 10.13
- Reject H0 if F > 10.13.
- Test Statistic
- F = MSR/MSE = 100/4.667 = 21.43
- Conclusion
- We can reject H0.

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.

Using Excel’s 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.

Using Excel’s Regression Tool

- Formula Worksheet (showing data)

Using Excel’s 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

… continued

Using Excel’s Regression Tool

- Performing the Regression Analysis

Step 4 When the Regression dialog box appears:

Enter C1:C6 in the Input Y Range box

Enter B1:B6 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

Using Excel’s Regression Tool

- Value Worksheet

Data

Regression Statistics Output

ANOVA Output

Regression Equation Output

Using Excel’s Regression Tool

- Estimated Regression Equation Output (left portion)

Note: Columns F-I are not shown.

Using Excel’s Regression Tool

- Estimated Regression Equation Output (right portion)

Note: Columns C-E are hidden.

Using Excel’s Regression Tool

- ANOVA Output

Using Excel’s Regression Tool

- Regression Statistics Output

Using the Estimated Regression Equationfor Estimation and Prediction

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

yp+t/2 sind

where the confidence coefficient is 1 - and

t/2 is based on a t distribution with n - 2 d.f.

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 = 10 + 5(3) = 25 cars

- Confidence Interval for E(yp)

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

- 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: 25 + 8.28 = 16.72 to 33.28 cars

^

Using Excel to Develop Confidence and Prediction Interval Estimates

- Formula Worksheet (confidence interval portion)

Using Excel to Develop Confidence and Prediction Interval Estimates

- Value Worksheet (confidence interval portion)

Using Excel to Develop Confidence and Prediction Interval Estimates

- Formula Worksheet (prediction interval portion)

Using Excel to Develop Confidence and Prediction Interval Estimates

- Value Worksheet (prediction interval portion)

Residual Analysis

- If the assumptions about the error term e appear questionable, the hypothesis tests about the significance of the regression relationship and the interval estimation results may not be valid.
- The residuals provide the best information about e.
- Much of the residual analysis is based on an examination of graphical plots.

Residual Plot Against x

- If the assumption that the variance of e is the same for all values of x is valid, and the assumed regression model is an adequate representation of the relationship between the variables:

The residual plot should give an overall

impression of a horizontal band of points

- Residuals

Using Excel’s Regression Tool to Construct a Residual Plot

- Producing a Residual Plot
- The steps outlined earlier to obtain the regression output are performed with one change.
- When the Regression dialog box appears, we must also select the Residual Plot option.
- The output will include two new items:
- A plot of the residuals against the independent variable, and
- A list of predicted values of y and the corresponding residual values.

Using Excel’s Regression Tool to Construct a Residual Plot

- Value Worksheet (showing Residual Plot)

Using Excel’s Regression Tool to Construct a Residual Plot

- Value Worksheet (Residual Output portion)

Standardized Residual Plot

- The standardized residual plot can provide insight about the assumption that the error term e has a normal distribution.
- If this assumption is satisfied, the distribution of the standardized residuals should appear to come from a standard normal probability distribution.

Using Excel to Construct a Standardized Residual Plot

- Excel’s Regression tool be used to obtain the standardized residuals.
- The steps described earlier in order to conduct a regression analysis are performed with one change:
- When the Regression dialog box appears, we must select the Standardized Residuals option
- The Standardized Residuals option does not automatically produce a standardized residual plot.

Using Excel to Construct a Standardized Residual Plot

- Value Worksheet

Using Excel to Construct a Standardized Residual Plot

- Excel’s Chart Wizard can be used to construct the standardized residual plot.
- A scatter diagram is developed in which:
- The values of the independent variable are placed on the horizontal axis
- The values of the standardized residuals are placed on the vertical axis

Using Excel to Construct a Standardized Residual Plot

- Standardized Residual Plot

Standardized Residual Plot

- All of the standardized residuals are between –1.5 and +1.5 indicating that there is no reason to question the assumption that e has a normal distribution.

Outliers and Influential Observations

- Detecting Outliers
- An outlier is an observation that is unusual in comparison with the other data.
- Minitab classifies an observation as an outlier if its standardized residual value is < -2 or > +2.
- This standardized residual rule sometimes fails to identify an unusually large observation as being an outlier.
- This rule’s shortcoming can be circumvented by using studentized deleted residuals.
- The |i th studentized deleted residual| will be larger than the |i th standardized residual|.

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