CHAPTER 13 Simple Linear Regression

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544 CHAPTER THIRTEEN Simple Linear Regression 13.45 In Problem 13.7 on page 523, you used PH Grade 13.40 You are testing the null hypothesis that PH Grade Test ing manager used shelf space for pet food to predict weekly sales. The data are stored in the file 13.48 In Problem 13.10 on page 523, you used hardness to predict the tensile strength of die-cast aluminum. The petfood.xls. From the results of that problem, b 1 = 7.4 and data are stored in the file hardness.xls. Using the results of S b1 = 1.59. that problem, a. At the 0.05 level of significance, is there evidence of a a. at the 0.05 level of significance, is there evidence of a linear relationship between shelf space and sales? linear relationship between hardness and tensile b. Construct a 95% confidence interval estimate of the strength? population slope, β 1 . b. construct a 95% confidence interval estimate of the population mine that b 1 = +4.5 and S b1 = 1.5. Using the results of that problem, a. What is the value of the t test statistic? a. at the 0.05 level of significance, is there evidence of a ASSIST there is no relationship between two variables, X and Y. From your sample of n = 18, you deter- ASSIST the weight of mail to predict the number of orders received. The data are stored in the file mail.xls. b. At the α = 0.05 level of significance, what are the critical values? c. Based on your answers to (a) and (b), what statistical decision should you make? linear relationship between the weight of mail and the number of orders received? b. construct a 95% confidence interval estimate of the population slope, β 1 . d. Construct a 95% confidence interval estimate of the 13.46 In Problem 13.8 on page 523, you used annual revenues to predict the value of a baseball franchise. The data population slope, β 1 . PH Grade ASSIST 13.41 You are testing the null hypothesis that there is no relationship between two variables, X and Y. From your sample of n = 20, you determine are stored in the file bbrevenue.xls. Using the results of that problem, a. at the 0.05 level of significance, is there evidence of a that SSR =60andSSE = 40. a. What is the value of the F test statistic? b. At the α = 0.05 level of significance, what is the critical value? linear relationship between annual revenue and franchise value? b. construct a 95% confidence interval estimate of the population slope, β 1 . c. Based on your answers to (a) and (b), what statistical 13.47 In Problem 13.9 on page 523, an agent for a real decision should you make? estate company wanted to predict the monthly rent for apartments, based on the size of the apartment. The data are stored d. Compute the correlation coefficient by first computing r 2 and assuming that b 1 is negative. in the file rent.xls. Using the results of that problem, e. At the 0.05 level of significance, is there a significant a. at the 0.05 level of significance, is there evidence of a correlation between X and Y? linear relationship between the size of the apartment and Applying the Concepts the monthly rent? b. construct a 95% confidence interval estimate of the population SELF 13.42 In Problem 13.4 on page 522, the market- slope, β 1 . slope, β 1 . 13.43 In Problem 13.5 on page 522, you used reported magazine newsstand sales to predict audited sales. The data are stored in the file circulation.xls. Using the results of that problem, b 1 = 0.5719 and S b1 = 0.0668. a. At the 0.05 level of significance, is there evidence of a linear relationship between reported sales and audited sales? b. Construct a 95% confidence interval estimate of the population slope, β 1 . 13.44 In Problem 13.6 on pages 522–523, the owner of a moving company wanted to predict labor hours, based on the number of cubic feet moved. The data are stored in the file moving.xls. Using the results of that problem, a. at the 0.05 level of significance, is there evidence of a linear relationship between the number of cubic feet moved and labor hours? b. construct a 95% confidence interval estimate of the population slope, β 1 . 13.49 The volatility of a stock is often measured by its beta value. You can estimate the beta value of a stock by developing a simple linear regression model, using the percentage weekly change in the stock as the dependent variable and the percentage weekly change in a market index as the independent variable. The S&P 500 Index is a common index to use. For example, if you wanted to estimate the beta for IBM, you could use the following model, which is sometimes referred to as a market model: (% weekly change in IBM) = β 0 + β 1 (% weekly change in S & P 500 index) + ε The least-squares regression estimate of the slope b 1 is the estimate of the beta value for IBM. A stock with a beta value of 1.0 tends to move the same as the overall market. A stock with a beta value of 1.5 tends to move 50% more than the overall market, and a stock with a beta value of 0.6
13.7: Inferences About the Slope and Correlation Coefficient 545 tends to move only 60% as much as the overall market. Stocks with negative beta values tend to move in a direction opposite that of the overall market. The following table gives some beta values for some widely held stocks: Company Ticker Symbol Beta AT&T T 0.80 IBM IBM 1.20 Disney Company DIS 1.40 Alcoa AA 2.26 LSI Logic LSI 3.61 Source: Extracted from finance.yahoo.com, May 31, 2006. a. For each of the five companies, interpret the beta value. b. How can investors use the beta value as a guide for investing? 13.50 Index funds are mutual funds that try to mimic the movement of leading indexes, such as the S&P 500 Index, the NASDAQ 100 Index, or the Russell 2000 Index. The beta values for these funds (as described in Problem 13.49) are therefore approximately 1.0. The estimated market models for these funds are approximately (% weekly change in index fund) = 0.0 + 1.0 (% weekly change in the index) Leveraged index funds are designed to magnify the movement of major indexes. An article in Mutual Funds (L. O’Shaughnessy, “Reach for Higher Returns,” Mutual Funds, July 1999, pp. 44–49) described some of the risks and rewards associated with these funds and gave details on some of the most popular leveraged funds, including those in the following table: Name (Ticker Symbol) Fund Description Potomac Small Cap 125% of Russell 2000 Index Plus (POSCX) Rydex “Inv” Nova (RYNVX) 150% of the S&P 500 Index ProFund UltraOTC Double (200%) the NASDAQ 100 “Inv” (UOPIX) Index Thus, estimated market models for these funds are approximately (% weekly change in POSCX) = 0.0 + 1.25 (% weekly change in the Russell 2000 Index) (% weekly change in RYNVX) = 0.0 + 1.50 (% weekly change in the S&P 500 Index) (% weekly change in UOPIX fund) = 0.0 + 2.0 (% weekly change in the NASDAQ 100 Index) Thus, if the Russell 2000 Index gains 10% over a period of time, the leveraged mutual fund POSCX gains approximately 12.5%. On the downside, if the same index loses 20%, POSCX loses approximately 25%. a. Consider the leveraged mutual fund ProFund UltraOTC “Inv” (UOPIX), whose description is 200% of the performance of the S&P 500 Index. What is its approximate market model? b. If the NASDAQ gains 30% in a year, what return do you expect UOPIX to have? c. If the NASDAQ loses 35% in a year, what return do you expect UOPIX to have? d. What type of investors should be attracted to leveraged funds? What type of investors should stay away from these funds? 13.51 The data in the file coffeedrink.xls represent the calories and fat (in grams) of 16-ounce iced coffee drinks at Dunkin’ Donuts and Starbucks: Product Calories Fat Dunkin’ Donuts Iced Mocha Swirl latte (whole milk) 240 8.0 Starbucks Coffee Frappuccino blended coffee 260 3.5 Dunkin’ Donuts Coffee Coolatta (cream) 350 22.0 Starbucks Iced Coffee Mocha Espresso (whole milk and whipped cream) 350 20.0 Starbucks Mocha Frappuccino blended coffee (whipped cream) 420 16.0 Starbucks Chocolate Brownie Frappuccino blended coffee (whipped cream) 510 22.0 Starbucks Chocolate Frappuccino Blended Crème (whipped cream) 530 19.0 Source: Extracted from “Coffee as Candy at Dunkin’ Donuts and Starbucks,” Consumer Reports, June 2004, p. 9. 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 calories and fat? 13.52 There are several methods for calculating fuel economy. The following table (contained in the file mileage.xls) indicates the mileage as calculated by owners and by current government standards: Government Vehicle Owner Standards 2005 Ford F-150 14.3 16.8 2005 Chevrolet Silverado 15.0 17.8 2002 Honda Accord LX 27.8 26.2 2002 Honda Civic 27.9 34.2 2004 Honda Civic Hybrid 48.8 47.6 2002 Ford Explorer 16.8 18.3 2005 Toyota Camry 23.7 28.5 2003 Toyota Corolla 32.8 33.1 2005 Toyota Prius 37.3 56.0
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 31 and 32: 13.7: Inferences About the Slope an
Page 33: 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

CHAPTER 13 Simple Linear Regression

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