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PROBLEMS FOR SECTION 13.7
a. What is the value of the t test statistic?
b. At the α = 0.05 level of significance, what are the criti-cal values?
c. Based on your answers to (a) and (b), what statistical decision should you make?
Learning the Basics
13.39 You are testing the null hypothesis that there is no linear relationship between two variables, X and Y. From your sample of n = 10, you determine that r = 0.80.
13.40 You are testing the null hypothesis that there is no relationship between two variables, X and Y. From your sample of n = 18, you deter-mine that b1= +4.5 and = 1.5.
a. What is the value of the t test statistic?
b. At the α = 0.05 level of significance, what are the criti-cal values?
c. Based on your answers to (a) and (b), what statistical decision should you make?
d. Construct a 95% confidence interval estimate of the population slope, β1.
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 that SSR = 60 and SSE = 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?
c. Based on your answers to (a) and (b), what statistical decision should you make?
d. Compute the correlation coefficient by first computing r2and assuming that b1is negative.
e. At the 0.05 level of significance, is there a significant correlation between X and Y?
Applying the Concepts
13.42 In Problem 13.4 on page 522, the market-ing manager used shelf space for pet food to pre-dict weekly sales. The data are stored in the file
petfood.xls. From the results of that problem, b1= 7.4 and
= 1.59.
a. At the 0.05 level of significance, is there evidence of a linear relationship between shelf space and sales?
b. Construct a 95% confidence interval estimate of the population slope, β1.
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, b1= 0.5719 and = 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, β1.
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 pop-ulation slope, β1.
13.45 In Problem 13.7 on page 523, you used the weight of mail to predict the number of orders received. The data are stored in the file mail.xls. Using the results of that problem,
a. at the 0.05 level of significance, is there evidence of a linear relationship between the weight of mail and the number of orders received?
b. construct a 95% confidence interval estimate of the pop-ulation slope, β1.
13.46 In Problem 13.8 on page 523, you used annual rev-enues to predict the value of a baseball franchise. The data 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 linear relationship between annual revenue and fran-chise value?
b. construct a 95% confidence interval estimate of the pop-ulation slope, β1.
13.47 In Problem 13.9 on page 523, an agent for a real estate company wanted to predict the monthly rent for apart-ments, based on the size of the apartment. The data are stored in the file rent.xls. Using the results of that problem,
a. at the 0.05 level of significance, is there evidence of a linear relationship between the size of the apartment and the monthly rent?
b. construct a 95% confidence interval estimate of the pop-ulation slope, β1.
13.48 In Problem 13.10 on page 523, you used hardness to predict the tensile strength of die-cast aluminum. The data are stored in the file hardness.xls. Using the results of that problem,
a. at the 0.05 level of significance, is there evidence of a linear relationship between hardness and tensile strength?
b. construct a 95% confidence interval estimate of the pop-ulation slope, β1.
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 per-centage weekly change in the stock as the dependent vari-able 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) = β0+ β1(% weekly change in S & P 500 index) + ε
The least-squares regression estimate of the slope b1is 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
PH Grade
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 direc-tion 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 150% of the S&P 500 Index (RYNVX)
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
approxi-mately 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 per-formance of the S&P 500 Index. What is its approxi-mate 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
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.xlsrepresent 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 signif icance, is there a signif i-cant 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.xlsrepresent 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 gradu-ation 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 foot-ball 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 indi-vidual values of Y.