CHAPTER 13 Simple Linear Regression

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518 CHAPTER THIRTEEN Simple Linear Regression VISUAL EXPLORATIONS Exploring Simple Linear Regression Coefficients Use the Visual Explorations Simple Linear Regression procedure to produce a prediction line that is as close as possible to the prediction line defined by the least-squares solution. Open the Visual Explorations.xla add-in workbook and select VisualExplorations ➔ Simple Linear Regression (Excel 97-2003) or Add-ins ➔ Visual Explorations ➔ Simple Linear Regression (Excel 2007). (See Section E1.6 to learn about using add-ins.) When a scatter plot of the Sunflowers Apparel data of Table 13.1 on page 515 with an initial prediction line appears (shown below), click the spinner buttons to change the values for b 1 , the slope of the prediction line, and b 0 , the Y intercept of the prediction line. Try to produce a prediction line that is as close as possible to the prediction line defined by the least-squares estimates, using the chart display and the Difference from Target SSE value as feedback (see page 525 for an explanation of SSE). Click Finish when you are done with this exploration. At any time, click Reset to reset the b 1 and b 0 values, Help for more information, or Solution to reveal the prediction line defined by the least-squares method. workbook and select VisualExplorations ➔ Simple Linear Regression with your worksheet data (97-2003) or Add-ins ➔ Visual Explorations ➔ Simple Linear Regression with your worksheet data (2007). In the procedure’s dialog box (shown below), enter your Y variable cell range as the Y Variable Cell Range and your X variable cell range as the X Variable Cell Range. Click First cells in both ranges contain a label, enter a title as the Title, and click OK. When the scatter plot with an initial prediction line appears, use the instructions in the first part of this section to try to produce the prediction line defined by the least-squares method. Using Your Own Regression Data To use Visual Explorations to find a prediction line for your own data, open the Visual Explorations.xla add-in
13.2: Determining the Simple Linear Regression Equation 519 Return to the Using Statistics scenario concerning the Sunflowers Apparel stores. Example 13.2 illustrates how you use the prediction equation to predict the mean annual sales. EXAMPLE 13.2 PREDICTING MEAN ANNUAL SALES, BASED ON SQUARE FOOTAGE Use the prediction line to predict the mean annual sales for a store with 4,000 square feet. SOLUTION You can determine the predicted value by substituting X = 4 (thousands of square feet) into the simple linear regression equation: Yˆ i = 0. 9645 + 1. 6699Xi Yˆ = 0. 9645 + 1. 6699( 4) = 7. 644 or $ 7, 644, 000 i Thus, the predicted mean annual sales of a store with 4,000 square feet is $7,644,000. Predictions in Regression Analysis: Interpolation Versus Extrapolation When using a regression model for prediction purposes, you need to consider only the relevant range of the independent variable in making predictions. This relevant range includes all values from the smallest to the largest X used in developing the regression model. Hence, when predicting Y for a given value of X, you can interpolate within this relevant range of the X values, but you should not extrapolate beyond the range of X values. When you use the square footage to predict annual sales, the square footage (in thousands of square feet) varies from 1.1 to 5.8 (see Table 13.1 on page 515). Therefore, you should predict annual sales only for stores whose size is between 1.1 and 5.8 thousands of square feet. Any prediction of annual sales for stores outside this range assumes that the observed relationship between sales and store size for store sizes from 1.1 to 5.8 thousand square feet is the same as for stores outside this range. For example, you cannot extrapolate the linear relationship beyond 5,800 square feet in Example 13.2. It would be improper to use the prediction line to forecast the sales for a new store containing 8,000 square feet. It is quite possible that store size has a point of diminishing returns. If that is true, as square footage increases beyond 5,800 square feet, the effect on sales might become smaller and smaller. Computing the Y Intercept, b 0 , and the Slope, b 1 For small data sets, you can use a hand calculator to compute the least-squares regression coefficients. Equations (13.3) and (13.4) give the values of b 0 and b 1 , which minimize n ∑ i= 1 n 2 i i ∑ i 0 1 i i= 1 ( Y − Yˆ ) = [ Y − ( b + b X )] 2 COMPUTATIONAL FORMULA FOR THE SLOPE, b 1 SSXY b1 = (13.3) SSX where ⎛ n ⎞⎛ n ⎞ ⎜ X Y n n ∑ i⎟⎜∑ i⎟ ⎝ i= 1 ⎠⎝ i= 1 ⎠ SSXY = ∑ ( Xi − X)( Yi − Y) = ∑ XiYi − n i= 1 ⎛ ⎜ n n ⎝ SSX = ( X − X) = X − ∑ i= 1 i ∑ i= 1 2 2 i= 1 i i= 1 n ∑ ⎞ X i⎟ ⎠ n 2
Page 1 and 2: CHAPTER 13 Simple Linear Regression
Page 3 and 4: 13.1: Types of Regression Models 51
Page 5 and 6: 13.2: Determining the Simple Linear
Page 7: 13.2: Determining the Simple Linear
Page 15 and 16: 13.3: Measures of Variation 525 REG
Page 17 and 18: 13.3: Measures of Variation 527 COM
Page 19 and 20: 13.4: Assumptions 529 13.15 If SSR
Page 21 and 22: 13.5: Residual Analysis 531 FIGURE
Page 23 and 24: 13.5: Residual Analysis 533 Equal V
Page 25 and 26: 13.6: Measuring Autocorrelation: Th
Page 27 and 28: 13.6: Measuring Autocorrelation: Th
Page 29 and 30: 13.7: Inferences About the Slope an
Page 37 and 38: 13.8: Estimation of Mean Values and
Page 39 and 40: 13.8: Estimation of Mean Values and
Page 41 and 42: 13.9: Pitfalls in Regression and Et
Page 43 and 44: 13.9: Pitfalls in Regression and Et
Page 45 and 46: Key Equations 555 You have learned
Page 47 and 48: Chapter Review Problems 557 (GPA),
Page 49 and 50: Chapter Review Problems 559 f. At t
Page 51 and 52: Chapter Review Problems 561 sured f
Page 53 and 54: References 563 in other months. In
Page 55 and 56: E13.2: Creating Scatter Plots and A
Page 57 and 58: E13.6: Example: Sunflowers Apparel
Page 59:
E13.6: Example: Sunflowers Apparel
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CHAPTER 13 Simple Linear Regression

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