CHAPTER 13 Simple Linear Regression

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546 CHAPTER THIRTEEN Simple Linear Regression a. Compute and interpret the coefficient of correlation, r. b. At the 0.05 level of significance, is there a significant linear relationship between the mileage as calculated by owners and by current government standards? 13.53 College basketball is big business, with coaches’ salaries, revenues, and expenses in millions of dollars. The data in the file colleges-basketball.xls represent the coaches’ salaries and revenues for college basketball at selected schools in a recent year (extracted from R. Adams, “Pay for Playoffs,” The Wall Street Journal, March 11–12, 2006, pp. P1, P8). a. Compute and interpret the coefficient of correlation, r. b. At the 0.05 level of significance, is there a significant linear relationship between a coach’s salary and revenue? 13.54 College football players trying out for the NFL are given the Wonderlic standardized intelligence test. The data in the file wonderlic.xls represent the average Wonderlic scores of football players trying out for the NFL and the graduation rates for football players at selected schools (extracted from S. Walker, “The NFL’s Smartest Team,” The Wall Street Journal, September 30, 2005, pp. W1, W10). a. Compute and interpret the coefficient of correlation, r. b. At the 0.05 level of significance, is there a significant linear relationship between the average Wonderlic score of football players trying out for the NFL and the graduation rates for football players at selected schools? c. What conclusions can you reach about the relationship between the average Wonderlic score of football players trying out for the NFL and the graduation rates for football players at selected schools? 13.8 ESTIMATION OF MEAN VALUES AND PREDICTION OF INDIVIDUAL VALUES This section presents methods of making inferences about the mean of Y and predicting individual values of Y. The Confidence Interval Estimate In Example 13.2 on page 519, you used the prediction line to predict the value of Y for a given X. The mean annual sales for stores with 4,000 square feet was predicted to be 7.644 millions of dollars ($7,644,000). This estimate, however, is a point estimate of the population mean. In Chapter 8, you studied the concept of the confidence interval as an estimate of the population mean. In a similar fashion, Equation (13.20) defines the confidence interval estimate for the mean response for a given X. CONFIDENCE INTERVAL ESTIMATE FOR THE MEAN OF Y Yˆ i ± tn−2SYX hi Yˆ − t S h ≤ µ ≤ Yˆ + t S h where h i n− 2 YX i Y | X = Xi i n−2 YX i 2 i 1 ( Xi − X) = + n SSX Yˆ i = predicted value of Y; Yˆ i = b0 + b1X S YX = standard error of the estimate n = sample size X i = given value of X i (13.20) µ YX | = Xi = mean value of Y when X = X i n ∑ SSX = ( X − X) i= 1 i 2
13.8: Estimation of Mean Values and Prediction of Individual Values 547 The width of the confidence interval in Equation (13.20) depends on several factors. For a given level of confidence, increased variation around the prediction line, as measured by the standard error of the estimate, results in a wider interval. However, as you would expect, increased sample size reduces the width of the interval. In addition, the width of the interval also varies at different values of X. When you predict Y for values of X close to X , the interval is narrower than for predictions for X values more distant from X . In the Sunflowers Apparel example, suppose you want to construct a 95% confidence interval estimate of the mean annual sales for the entire population of stores that contain 4,000 square feet (X = 4). Using the simple linear regression equation, Also, given the following: From Table E.3, t 12 = 2.1788. Thus, where Yˆ i = 0. 9645 + 1. 6699Xi = 0. 9645 + 1. 6699( 4) = 7. 6439 (millions of dollars) X = 2. 9214 S = 0. 9664 n ∑ YX SSX = ( X − X ) = 37. 9236 i= 1 i 2 Yˆ t S h i ± n −2 YX i h i 1 ( Xi − X) = + n SSX 2 so that ˆ 1 ( Xi − X) Yi ± tn−2SYX + n SSX 1 ( 4 − 2. 9214) = 7. 6439 ± ( 2. 1788)( 0. 9664) + 14 37. 9236 = 7. 6439 ± 0. 6728 2 2 so 6.9711 ≤ µ Y| X =4 ≤ 8.3167 Therefore, the 95% confidence interval estimate is that the mean annual sales are between $6,971,100 and $8,316,700 for the population of stores with 4,000 square feet. The Prediction Interval In addition to the need for a confidence interval estimate for the mean value, you often want to predict the response for an individual value. Although the form of the prediction interval is similar to that of the confidence interval estimate of Equation (13.20), the prediction interval is predicting an individual value, not estimating a parameter. Equation (13.21) defines the prediction interval for an individual response, Y, at a particular value, X i , denoted by . Y X = Xi
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 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 35: 13.7: Inferences About the Slope an
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

CHAPTER 13 Simple Linear Regression

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