a. Develop a scatter diagram for these data.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Try to approximate the relationship between x and y by drawing a straight line through the data.
d. Develop the estimated regression equation by computing the values of b0 and b1 using equations (14.6) and (14.7).
e. Use the estimated regression equation to predict the value of y when x = 4.
Get solution
14.2 Given are five observations for two variables, x and y....
a. Develop a scatter diagram for these data.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Try to approximate the relationship between x and y by drawing a straight line through the data.
d. Develop the estimated regression equation by computing the values of b0 and b1 using equations (14.6) and (14.7).
e. Use the estimated regression equation to predict the value of y when x = 10.
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14.3
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14.4 The following data were collected on the height (inches) and weight (pounds) of women swimmers....
a. Develop a scatter diagram for these data with height as the independent variable.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Try to approximate the relationship between height and weight by drawing a straight line through the data.
d. Develop the estimated regression equation by computing the values of b0 and b1.
e. If a swimmer’s height is 63 inches, what would you estimate her weight to be?
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14.5
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14.6
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14.7
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14.8
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14.9
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14.10
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14.11
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14.12 A personal watercraft (PWC) is a vessel propelled by water jets, designed to be operated by a person sitting, standing, or kneeling on the vessel. In the early 1970s, Kawasaki Motors Corp. U.S.A. introduced the JET SKI® watercraft, the first commercially successful PWC. Today, jet ski is commonly used as a generic term for personal watercraft. The following data show the weight (rounded to the nearest 10 lbs.) and the price (rounded to the nearest $50) for 10 three-seater personal watercraft (http://www.jetskinews.com, 2006)....
a. Develop a scatter diagram for these data with weight as the independent variable.
b. What does the scatter diagram developed in part (a) indicate about the relationship between weight and price?
c. Use the least squares method to develop the estimated regression equation.
d. Predict the price for a three-seater PWC with a weight of 750 pounds.
e. The Honda AquaTrax F-12 weighs 750 pounds and has a price of $9500. Shouldn’t the predicted price you developed in part (d) for a PWC with a weight of 750 pounds also be $9500?
f. The Kawasaki SX-R 800 Jetski has a seating capacity of one and weighs 350 pounds. Do you think the estimated regression equation developed in part (c) should be used to predict the price for this model?
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14.13 To the Internal Revenue Service, the reasonableness of total itemized deductions depends on the taxpayer’s adjusted gross income. Large deductions, which include charity and medical deductions, are more reasonable for taxpayers with large adjusted gross incomes. If a taxpayer claims larger than average itemized deductions for a given level of income, the chances of an IRS audit are increased. Data (in thousands of dollars) on adjusted gross income and the average or reasonable amount of itemized deductions follow....
a. Develop a scatter diagram for these data with adjusted gross income as the independent variable.
b. Use the least squares method to develop the estimated regression equation.
c. Estimate a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of $52,500. If this taxpayer claimed itemized deductions of $20,400, would the IRS agent’s request for an audit appear justified? Explain.
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14.14
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14.15 The data from exercise 1 follow....The estimated regression equation for these data is ŷ = .20 + 2.60x.
a. Compute SSE, SST, and SSR using equations (14.8), (14.9), and (14.10).
b. Compute the coefficient of determination r2. Comment on the goodness of fit.
c. Compute the sample correlation coefficient.
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14.16 The data from exercise 2 follow....The estimated regression equation for these data is ŷ = 68 − 3x.
a. Compute SSE, SST, and SSR.
b. Compute the coefficient of determination r2. Comment on the goodness of fit.
c. Compute the sample correlation coefficient.
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14.17 The data from exercise 3 follow....The estimated regression equation for these data is ŷ = 7.6 + .9x. What percentage of the total sum of squares can be accounted for by the estimated regression equation? What is the value of the sample correlation coefficient?
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14.18 The following data are the monthly salaries y and the grade point averages x for students who obtained a bachelor’s degree in business administration with a major in information systems. The estimated regression equation for these data is ŷ = 1790.5 + 581.1x....
a. Compute SST, SSR, and SSE.
b. Compute the coefficient of determination r2. Comment on the goodness of fit.
c. What is the value of the sample correlation coefficient?
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14.19
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14.20 Consumer Reports provided extensive testing and ratings for more than 100 HDTVs. An overall score, based primarily on picture quality, was developed for each model. In general, a higher overall score indicates better performance. The following data show the price and overall score for the ten 42-inch plasma televisions (Consumer Reports, March 2006)....
a. Use these data to develop an estimated regression equation that could be used to estimate the overall score for a 42-inch plasma television given the price.
b. Compute r2. Did the estimated regression equation provide a good fit?
c. Estimate the overall score for a 42-inch plasma television with a price of $3200.
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14.21 An important application of regression analysis in accounting is in the estimation of cost. By collecting data on volume and cost and using the least squares method to develop an estimated regression equation relating volume and cost, an accountant can estimate the cost associated with a particular manufacturing volume. Consider the following sample of production volumes and total cost data for a manufacturing operation....
a. Use these data to develop an estimated regression equation that could be used to predict the total cost for a given production volume.
b. What is the variable cost per unit produced?
c. Compute the coefficient of determination. What percentage of the variation in total cost can be explained by production volume?
d. The company’s production schedule shows 500 units must be produced next month. What is the estimated total cost for this operation?
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14.22
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14.23 The data from exercise 1 follow....
a. Compute the mean square error using equation (14.15).
b. Compute the standard error of the estimate using equation (14.16).
c. Compute the estimated standard deviation of b1 using equation (14.18).
d. Use the t test to test the following hypotheses (α = .05):...
e. Use the F test to test the hypotheses in part (d) at a .05 level of significance. Present the results in the analysis of variance table format.
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14.24 The data from exercise 2 follow....
a. Compute the mean square error using equation (14.15).
b. Compute the standard error of the estimate using equation (14.16).
c. Compute the estimated standard deviation of b1 using equation (14.18).
d. Use the t test to test the following hypotheses (α = .05):...
e. Use the F test to test the hypotheses in part (d) at a .05 level of significance. Present the results in the analysis of variance table format.
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14.25 The data from exercise 3 follow....
a. What is the value of the standard error of the estimate?
b. Test for a significant relationship by using the t test. Use α = .05.
c. Use the F test to test for a significant relationship. Use α = .05. What is your conclusion?
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14.26 In exercise 18 the data on grade point average and monthly salary were as follows....
a. Does the t test indicate a significant relationship between grade point average and monthly salary? What is your conclusion? Use α = .05.
b. Test for a significant relationship using the F test. What is your conclusion? Use α = .05.
c. Show the ANOVA table.
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14.27 Outside Magazine tested 10 different models of day hikers and backpacking boots. The following data show the upper support and price for each model tested. Upper support was measured using a rating from 1 to 5, with a rating of 1 denoting average upper support and a rating of 5 denoting excellent upper support (Outside Magazine Buyer’s Guide, 2001)....
a. Use these data to develop an estimated regression equation to estimate the price of a day hiker and backpacking boot given the upper support rating.
b. At the .05 level of significance, determine whether upper support and price are related.
c. Would you feel comfortable using the estimated regression equation developed in part (a) to estimate the price for a day hiker or backpacking boot given the upper support rating?
d. Estimate the price for a day hiker with an upper support rating of 4.
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14.28
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14.29 Refer to exercise 21, where data on production volume and cost were used to develop an estimated regression equation relating production volume and cost for a particular manufacturing operation. Use α = .05 to test whether the production volume is significantly related to the total cost. Show the ANOVA table. What is your conclusion?
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14.30
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14.31 In exercise 20, data on x = price ($) and y = overall score for ten 42-inch plasma televisions tested by Consumer Reports provided the estimated regression equation ŷ = 12.0169 + .0127x. For these data SSE = 540.04 and SST = 982.40. Use the F test to determine whether the price for a 42-inch plasma television and the overall score are related at the .05 level of significance.
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14.32 The data from exercise 1 follow....
a. Use equation (14.23) to estimate the standard deviation of ŷp when x = 4.
b. Use expression (14.24) to develop a 95% confidence interval for the expected value of y when x = 4.
c. Use equation (14.26) to estimate the standard deviation of an individual value of y when x = 4.
d. Use expression (14.27) to develop a 95% prediction interval for y when x = 4.
Get solution
14.33 The data from exercise 2 follow....
a. Estimate the standard deviation of ŷp when x = 8.
b. Develop a 95% confidence interval for the expected value of y when x = 8.
c. Estimate the standard deviation of an individual value of y when x = 8.
d. Develop a 95% prediction interval for y when x = 8.
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14.34 The data from exercise 3 follow....Develop the 95% confidence and prediction intervals when x = 12. Explain why these two intervals are different.
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14.35 In exercise 18, the data on grade point average x and monthly salary y provided the estimated regression equation ŷ = 1790.5 + 581.1x.
a. Develop a 95% confidence interval for the mean starting salary for all students with a 3.0 GPA.
b. Develop a 95% prediction interval for the starting salary for Joe Heller, a student with a GPA of 3.0.
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14.36 In exercise 10, data on x = temperature rating (F°) and y = price ($) for 11 sleeping bags manufactured by Bergans of Norway provided the estimated regression equation ŷ = 359.2668 − 5.2772x. For these data s = 37.9372.
a. Develop a point estimate of the price for a sleeping bag with a temperature rating of 30.
b. Develop a 95% confidence interval for the mean overall temperature rating for all sleeping bags with a temperature rating of 30.
c. Suppose that Bergans developed a new model with a temperature rating of 30. Develop a 95% prediction interval for the price of this new model.
d. Discuss the differences in your answers to parts (b) and (c).
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14.37 In exercise 13, data were given on the adjusted gross income x and the amount of itemized deductions taken by taxpayers. Data were reported in thousands of dollars. With the estimated regression equation ŷ = 4.68 + .16x, the point estimate of a reasonable level of total itemized deductions for a taxpayer with an adjusted gross income of $52,500 is $13,080.
a. Develop a 95% confidence interval for the mean amount of total itemized deductions for all taxpayers with an adjusted gross income of $52,500.
b. Develop a 95% prediction interval estimate for the amount of total itemized deductions for a particular taxpayer with an adjusted gross income of $52,500.
c. If the particular taxpayer referred to in part (b) claimed total itemized deductions of $20,400, would the IRS agent’s request for an audit appear to be justified?
d. Use your answer to part (b) to give the IRS agent a guideline as to the amount of total itemized deductions a taxpayer with an adjusted gross income of $52,500 should claim before an audit is recommended.
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14.38 Refer to Exercise 21, where data on the production volume x and total cost y for a particular manufacturing operation were used to develop the estimated regression equation ŷ = 1246.67 + 7.6x.
a. The company’s production schedule shows that 500 units must be produced next month. What is the point estimate of the total cost for next month?
b. Develop a 99% prediction interval for the total cost for next month.
c. If an accounting cost report at the end of next month shows that the actual production cost during the month was $6000, should managers be concerned about incurring such a high total cost for the month? Discuss.
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14.39 Almost all U.S. light-rail systems use electric cars that run on tracks built at street level. The Federal Transit Administration claims light-rail is one of the safest modes of travel, with an accident rate of .99 accidents per million passenger miles as compared to 2.29 for buses. The following data show the miles of track and the weekday ridership in thousands of passengers for six light-rail systems (USA Today, January 7, 2003)....
a. Use these data to develop an estimated regression equation that could be used to predict the ridership given the miles of track.
b. Did the estimated regression equation provide a good fit? Explain.
c. Develop a 95% confidence interval for the mean weekday ridership for all light-rail systems with 30 miles of track.
d. Suppose that Charlotte is considering construction of a light-rail system with 30 miles of track. Develop a 95% prediction interval for the weekday ridership for the Charlotte system. Do you think that the prediction interval you developed would be of value to Charlotte planners in anticipating the number of weekday riders for their new light-rail system? Explain.
d. Compute the standardized residuals.
e. Develop a plot of the standardized residuals against y. What conclusions can you draw from this plot?
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14.40 The commercial division of a real estate firm is conducting a regression analysis of the relationship between x, annual gross rents (in thousands of dollars), and y, selling price (in thousands of dollars) for apartment buildings. Data were collected on several properties recently sold and the following computer output was obtained....
a. How many apartment buildings were in the sample?
b. Write the estimated regression equation.
c. What is the value of ...?
d. Use the F statistic to test the significance of the relationship at a .05 level of significance.
e. Estimate the selling price of an apartment building with gross annual rents of $50,000.
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14.41 Following is a portion of the computer output for a regression analysis relating y = maintenance expense (dollars per month) to x = usage (hours per week) of a particular brand of computer terminal....
a. Write the estimated regression equation.
b. Use a t test to determine whether monthly maintenance expense is related to usage at the .05 level of significance.
c. Use the estimated regression equation to predict monthly maintenance expense for any terminal that is used 25 hours per week.
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14.42 A regression model relating x, number of salespersons at a branch office, to y, annual sales at the office (in thousands of dollars) provided the following computer output from a regression analysis of the data....
a. Write the estimated regression equation.
b. How many branch offices were involved in the study?
c. Compute the F statistic and test the significance of the relationship at a .05 level of significance.
d. Predict the annual sales at the Memphis branch office. This branch employs 12 salespersons.
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14.43 Health experts recommend that runners drink 4 ounces of water every 15 minutes they run. Although handheld bottles work well for many types of runs, all-day cross-country runs require hip-mounted or over-the-shoulder hydration systems. In addition to carrying more water, hip-mounted or over-the-shoulder hydration systems offer more storage space for food and extra clothing. As the capacity increases, however, the weight and cost of these larger-capacity systems also increase. The following data show the weight (ounces) and the price for 26 hip-mounted or over-the-shoulder hydration systems (Trail Runner Gear Guide, 2003).......
a. Use these data to develop an estimated regression equation that could be used to predict the price of a hydration system given its weight.
b. Test the significance of the relationship at the .05 level of significance.
c. Did the estimated regression equation provide a good fit? Explain.
d. Assume that the estimated regression equation developed in part (a) will also apply to hydration systems produced by other companies. Develop a 95% confidence interval estimate of the price for all hydration systems that weigh 10 ounces.
e. Assume that the estimated regression equation developed in part (a) will also apply to hydration systems produced by other companies. Develop a 95% prediction interval estimate of the price for the Back Draft system produced by Eastern Mountain Sports. The Back Draft system weighs 10 ounces.
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14.44
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14.45
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14.46 The following data were used in a regression study....
a. Develop an estimated regression equation for these data.
b. Construct a plot of the residuals. Do the assumptions about the error term seem to be satisfied?
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14.47 Data on advertising expenditures and revenue (in thousands of dollars) for the Four Seasons Restaurant follow....
a. Let x equal advertising expenditures and y equal revenue. Use the method of least squares to develop a straight line approximation of the relationship between the two variables.
b. Test whether revenue and advertising expenditures are related at a .05 level of significance.
c. Prepare a residual plot of y − ŷ versus ŷ. Use the result from part (a) to obtain the values of ŷ.
d. What conclusions can you draw from residual analysis? Should this model be used, or should we look for a better one?
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14.48 Refer to exercise 9, where an estimated regression equation relating years of experience and annual sales was developed.
a. Compute the residuals and construct a residual plot for this problem.
b. Do the assumptions about the error terms seem reasonable in light of the residual plot?
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14.49
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14.50 Consider the following data for two variables, x and y....
a. Compute the standardized residuals for these data. Do the data include any outliers? Explain.
b. Plot the standardized residuals against ŷ. Does this plot reveal any outliers?
c. Develop a scatter diagram for these data. Does the scatter diagram indicate any outliers in the data? In general, what implications does this finding have for simple linear regression?
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14.51 Consider the following data for two variables, x and y....
a. Compute the standardized residuals for these data. Do the data include any outliers? Explain.
b. Compute the leverage values for these data. Do there appear to be any influential observations in these data? Explain.
c. Develop a scatter diagram for these data. Does the scatter diagram indicate any influential observations? Explain.
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14.52 The following data show the media expenditures ($ millions) and the shipments in bbls. (millions) for 10 major brands of beer....
a. Develop the estimated regression equation for these data.
b. Use residual analysis to determine whether any outliers and/or influential observations are present. Briefly summarize your findings and conclusions.
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14.53 Health experts recommend that runners drink 4 ounces of water every 15 minutes they run. Runners who run three to eight hours need a larger-capacity hip-mounted or over-the-shoulder hydration system. The following data show the liquid volume (fl oz) and the price for 26 Ultimate Direction hip-mounted or over-the-shoulder hydration systems (Trail Runner Gear Guide, 2003)....
a. Develop the estimated regression equation that can be used to predict the price of a hydration system given its liquid volume.
b. Use residual analysis to determine whether any outliers or influential observations are present. Briefly summarize your findings and conclusions.
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14.54
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14.55 Does a high value of r2 imply that two variables are causally related? Explain.
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14.56 In your own words, explain the difference between an interval estimate of the mean value of y for a given x and an interval estimate for an individual value of y for a given x.
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14.57 What is the purpose of testing whether β1 = 0? If we reject β1 = 0, does it imply a good fit?
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14.58 The data in the following table show the number of shares selling (millions) and the expected price (average of projected low price and projected high price) for 10 selected initial public stock offerings....
a. Develop an estimated regression equation with the number of shares selling as the independent variable and the expected price as the dependent variable.
b. At the .05 level of significance, is there a significant relationship between the two variables?
c. Did the estimated regression equation provide a good fit? Explain.
d. Use the estimated regression equation to estimate the expected price for a firm considering an initial public offering of 6 million shares.
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14.59
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14.60
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14.61 Jensen Tire & Auto is in the process of deciding whether to purchase a maintenance contract for its new computer wheel alignment and balancing machine. Managers feel that maintenance expense should be related to usage, and they collected the following information on weekly usage (hours) and annual maintenance expense (in hundreds of dollars)....
a. Develop the estimated regression equation that relates annual maintenance expense to weekly usage.
b. Test the significance of the relationship in part (a) at a .05 level of significance.
c. Jensen expects to use the new machine 30 hours per week. Develop a 95% prediction interval for the company’s annual maintenance expense.
d. If the maintenance contract costs $3000 per year, would you recommend purchasing it? Why or why not?
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14.62 In a manufacturing process the assembly line speed (feet per minute) was thought to affect the number of defective parts found during the inspection process. To test this theory, managers devised a situation in which the same batch of parts was inspected visually at a variety of line speeds. They collected the following data....
a. Develop the estimated regression equation that relates line speed to the number of defective parts found.
b. At a .05 level of significance, determine whether line speed and number of defective parts found are related.
c. Did the estimated regression equation provide a good fit to the data?
d. Develop a 95% confidence interval to predict the mean number of defective parts for a line speed of 50 feet per minute.
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14.63 A sociologist was hired by a large city hospital to investigate the relationship between the number of unauthorized days that employees are absent per year and the distance (miles) between home and work for the employees. A sample of 10 employees was chosen, and the following data were collected....
a. Develop a scatter diagram for these data. Does a linear relationship appear reasonable? Explain.
b. Develop the least squares estimated regression equation.
c. Is there a significant relationship between the two variables? Use α = .05.
d. Did the estimated regression equation provide a good fit? Explain.
e. Use the estimated regression equation developed in part (b) to develop a 95% confidence interval for the expected number of days absent for employees living 5 miles from the company.
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14.64 The regional transit authority for a major metropolitan area wants to determine whether there is any relationship between the age of a bus and the annual maintenance cost. A sample of 10 buses resulted in the following data....
a. Develop the least squares estimated regression equation.
b. Test to see whether the two variables are significantly related with α = .05.
c. Did the least squares line provide a good fit to the observed data? Explain.
d. Develop a 95% prediction interval for the maintenance cost for a specific bus that is 4 years old.
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14.65 A marketing professor at Givens College is interested in the relationship between hours spent studying and total points earned in a course. Data collected on 10 students who took the course last quarter follow....
a. Develop an estimated regression equation showing how total points earned is related to hours spent studying.
b. Test the significance of the model with α = .05.
c. Predict the total points earned by Mark Sweeney. He spent 95 hours studying.
d. Develop a 95% prediction interval for the total points earned by Mark Sweeney.
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14.66
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14.67 The Transactional Records Access Clearinghouse at Syracuse University reported data showing the odds of an Internal Revenue Service audit. The following table shows the average adjusted gross income reported and the percent of the returns that were audited for 20 selected IRS districts....
a. Develop the estimated regression equation that could be used to predict the percent audited given the average adjusted gross income reported.
b. At the .05 level of significance, determine whether the adjusted gross income and the percent audited are related.
c. Did the estimated regression equation provide a good fit? Explain.
d. Use the estimated regression equation developed in part (a) to calculate a 95% confidence interval for the expected percent audited for districts with an average adjusted gross income of $35,000.
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14.68 The Australian Public Service Commission’s State of the Service Report 2002–2003 reported job satisfaction ratings for employees. One of the survey questions asked employees to choose the five most important workplace factors (from a list of factors) that most affected how satisfied they were with their job. Respondents were then asked to indicate their level of satisfaction with their top five factors. The following data show the percentage of employees who nominated the factor in their top five, and a corresponding satisfaction rating measured using the percentage of employees who nominated the factor in the top five and who were “very satisfied” or “satisfied” with the factor in their current workplace (www.apsc.gov.au/stateoftheservice)....
a. Develop a scatter diagram with Top Five (%) on the horizontal axis and Satisfaction Rating (%) on the vertical axis.
b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?
c. Develop the estimated regression equation that could be used to predict the Satisfaction Rating (%) given the Top Five (%).
d. Test for a significant relationship at the .05 level of significance.
e. Did the estimated regression equation provide a good fit? Explain.
f. What is the value of the sample correlation coefficient?
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