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The Practice Of Statistics For Business And Economics 3rd Edition David S. Moore, George P. McCabe, Layth C. Alwan, Bruce A. Craig, William M. Duckworth - Solutions
Predicting movie revenue. A plot of theater count versus box office revenue suggests that the relationship may be slightly curved. DATADATA DATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATA
Assessing collinearity in themovie revenue model. Many software packages will calculate VIF values for each explanatory variable. In this exercise, you will calculate the VIF values using several multiple regressions and then use them to see if there is collinearity among the movie explanatory
Online stock brokerages. Refer to the online stock brokerage data in Exercise 11.23 (page 589). Plot assets versus the number of accounts. Investigate the possibility that the relationship is curved by running a multiple regression to predict assets using the number of accounts and the square of
Write the model. For each of the following situations write a model for μy of the formμy = β0 + β1x1 + β2x2 +· · ·+βpx p where p is the number of explanatory variables. Be sure to give the value of p and, if necessary, explain how each of the x’s is coded.(a) A model where the
Differences in slopes and intercepts. Refer to the previous exercise. Verify that the coefficient of x1x2 is equal to the slope for Group B minus the slope for Group A in each of these cases. Also, verify that the coefficient of x1 is equal to the intercept for Group B minus the intercept for Group
Models with interactions. Suppose that x1 is an indicator variable with the value 0 for Group A and 1 for Group B, and x2 is a quantitative variable. Each of the models below describes a relationship between μy and the explanatory variables x1 and x2.For each model, substitute the value 0 for x1
Differences in means. Verify that the coefficient of x in each part of the previous exercise is equal to the mean for GroupB minus the mean for Group A. Do you think that this will be true in general? Explain your answer.
Models with indicator variables. Suppose that x is an indicator variable with the value 0 for GroupAand 1 for Group B.The following equations describe relationships between the value of μy and membership in Group A or B. For each equation, give the value of the mean response μy for Group A and
Quadratic models. Sketch each of the following quadratic equations for values of x between 0 and 5. Then describe the relationship between μy and x in your own words.(a) μy = 8 + 2x + 2x2(b) μy = 8 − 2x + 2x2(c) μy = 8 + 2x − 2x2(d) μy = 8 − 2x − 2x2
Residuals. Once we have chosen a model, we must examine the residuals for violations of the conditions of the multiple regression model. Examine the residuals from the model in Example 11.26.(a) Plot the residuals against SqFt. Do the residuals show a random scatter, or is there some systematic
How about the smaller homes? Would it make sense to do the same calculations as in the previous two exercises for homes that have 700 ft2? Explain why or why not.
Suppose the homes are larger? Consider two additional homes, both with 1600 ft2, one with an extra half bath and one without. Find the predicted prices and the difference. How does this difference compare with the difference you obtained in the previous exercise? Explain what you have found.
Comparing some predicted values. Consider two homes, both with 1400 ft2. Suppose the first has an extra half bath and the second does not. Find the predicted price for each home and then find the difference.
The home with three garages. There is only one home with 3 garage spaces. We might either place this house in the Garage = 2 group or remove it as unusual. Either decision leaves Garage with values 0, 1, and 2. Based on your plot in the previous exercise, which choice do you recommend?
What about garages? We have not yet examined the number of garage spaces as a possible explanatory variable for price. Make a scatterplot of price versus garage spaces. Describe the pattern. Use a “smooth” fit if your software has this capability. Otherwise, find the mean price for each
Modeling the means. Following the pattern in Example 11.21, use the output in Figure 11.18 to write the equations for the predicted mean price for(a) homes with 1 bathroom.(b) homes with 1.5 bathrooms.(c) homes with 2 bathrooms.(d) homes with 2.5 bathrooms.(e) How can we interpret the coefficient
Compare the means. Regression on a single indicator variable compares the mean responses in two groups. It is in fact equivalent to the pooled t test for comparing two means (Chapter 7, page 429). Use the pooled t test to compare the mean price of the homes that have three or more bedrooms with the
Find the means. Using the data set for Example 11.21, find the mean price for the homes that have three or more bedrooms and the mean price for those that do not.(a) Compare these sample means with the predicted values given in Example 11.21.(b) What is the difference between the mean price of the
Predicted values. Use the quadratic regression equation in Example 11.19 to predict the price of a home that has 1000 ft2. Do the same for a home that has 1500 ft2. Compare these predictions with the ones from an analysis that uses only SqFt as an explanatory variable.
The relationship between SqFt and SqFt2. Using the data set for Example 11.19, plot SqFt2 versus SqFt. Describe the relationship. We know that it is not linear, but is it approximately linear? What is the correlation between SqFt and SqFt2? The plot and correlation demonstrate that these variables
Predicted values. Use the simple linear regression equation to obtain the predicted price for a home that has 1000 ft2. Do the same for a home that has 1500 ft2.
Plot the residuals. Obtain the residuals from the simple linear regression in the above example and plot them versus SqFt. Describe the plot. Does it suggest that the relationship might be curved?
Distributions. Make stemplots or histograms of the prices and of the square feet for the 44 homes in Table 11.5. Do the 7 homes excluded in Example 11.17 appear unusual for this location?
Compensation and human capital. A study of bank branch manager compensation collected data on the salaries of 82 managers at branches of a large eastern U.S. bank.17 Multiple regression models were used to predict how much these branch managers were paid. The researchers examined two sets of
Canada’s Small Business Financing Program. The Canada Small Business Financing Program (CSBFP) seeks to increase the availability of loans for establishing and improving small businesses. A survey was performed to better understand the experiences of small businesses when seeking loans and the
Direct versus indirect loans. The previous four exercises describe a study of loans for buying new cars. The authors conclude that banks take higher risks with indirect loans because they do not take into account borrower characteristics when setting the loan rate. Explain how the results of the
Auto dealer loans, continued. Table 11.4 gives the estimated regression coefficient and individual t statistic for each explanatory variable in the setting of the previous exercise. The t-values are given without the sign, assuming that all tests are two-sided.(a) What are the degrees of freedom of
Auto dealer loans. The previous two exercises describe auto loans made directly by a bank. The researchers also looked at 5664 loans made indirectly, that is, through an auto dealer.They again used multiple regression to predict the interest rate using the same set of 13 explanatory variables.(a)
Bank auto loans, continued. Table 11.3 gives the coefficients for the fitted model and the individual t statistic for each explanatory variable in the study described in the previous exercise.The t-values are given without the sign, assuming that all tests are two-sided.(a) State the null and
Bank auto loans. Banks charge different interest rates for different loans. A random sample of 2229 loans made by banks for the purchase of new automobiles was studied to identify variables that explain the interest rate charged. A multiple regression was run with interest rate as the response
Interpret the “demand-side” explanatory variables.Refer to the previous three exercises. Another set of variables included in the analysis were classified as “demand-side” factors that relate to the changing needs of individuals in the labor force.One of these was the change in the
Do the results for change in DC and the results for change in DB agree? Refer to the previous two exercises. Summarize the results and interpret the signs for the coefficient for P5 in the two models. Do the results appear to be consistent?Explain your answer.
Decreases in DB plans. Refer to the previous exercise. A model for the change in DB plans was also analyzed. The coefficient for P5 in this model was −1.186 with a standard error of 0.133. Answer the questions in parts (a) and (b) of the previous exercise for this analysis.
Pension benefits. The defined-benefit (DB) pension plan guarantees a certain income to employees when they retire. The defined-contribution (DC) plan specifies that a regular contribution will be made to an investment account, and upon retirement, the employee will receive the income from the
Effect of potential outliers. Statistical inference requires us to make some assumptions about our data.DATADATA DATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATA DATADATADATADATADATA DATA FILE
Effects of incentives on employee motivation in Japan.To improve worker motivation, organizations typically offer employees opportunities for promotion,wage increases, and engagement in challengingwork. The Japanese career system is typically lagging in terms of promotion and rapidwage
Inference basics. You run a multiple regression with 27 cases and 3 explanatory variables. The ANOVA table includes the sums of squares SSR = 16 and SSE = 120.(a) Find the F statistic for testing the null hypothesis that the regression coefficients for the 3 explanatory variables are all zero.Carry
Inference basics. You run a multiple regression with 65 cases and 4 explanatory variables.(a) What are the degrees of freedom for the F statistic for testing the null hypothesis that all four of the regression coefficients for the explanatory variables are zero?(b) Software output gives MSE = 42.
What’s wrong? In each of the following situations, explain what is wrong and why.(a) The multiple correlation gives the proportion of the variation in the response variable that is explained by the explanatory variables.(b) In a multiple regression with a sample size of 35 and 4 explanatory
What’s wrong? In each of the following situations, explain what is wrong and why.(a) One of the assumptions for multiple regression is that the distribution of each explanatory variable is Normal.(b) The smaller the P-value for the ANOVA F test, the greater the explanatory power of the model.(c)
Significance test for a regression coefficient. For each of the settings in the previous exercise, test the null hypothesis that the coefficient of x1 is zero versus the two-sided alternative.
Confidence interval for a regression coefficient. In each of the following settings, give a 95% confidence interval for the coefficient of x1.(a) n = 28, ˆy = 10.3 + 14.7x1 + 17.3x2, SEb1= 7.4.(b) n = 53, ˆy = 10.3 + 14.7x1 + 17.3x2, SEb1= 7.4.(c) n = 28, ˆy = 10.3 + 14.7x1 + 17.3x2 + 2.1x3,
Is Opening helpful when Budget and Theaters are available? We saw that Budget and Theaters are not useful in a model that contains the openingweekend revenue. Now, let’s examine the other version of this question. Does Opening help explain USRevenue in a model that contains Budget and Theaters?
Are Budget and Theaters useful predictors of USRevenue? Run the multiple regression to predict movie revenue using all three predictors. Then run the model using only Budget and Theaters.(a) The R2 for the second model is 0.4461. Does your work confirm this?(b) Make a table giving the Budget and
R2 for different models. Use each of the following sets of explanatory variables to predict U.S. box office revenue: (a) Budget, Opening; (b)Budget, Theater; (c) Opening, Theater; (d) Budget; (e) Opening; (f) Theater. Make a table giving the model and the value of R2 for each. Summarize what you
F test for the model without Theaters. Rerun the multiple regression using the movie’s budget and opening weekend revenue to predict U.S. box office revenue. Report the F statistic, the associated degrees of freedom, and the P-value.How do these differ from the corresponding values for the model
Predicting U.S. movie revenue. Talladega Nights: The Ballad of Ricky Bobby was released August 4, 2006. It had a budget of $73.0 million and was shown in 3803 theaters grossing $47.042 million during the first weekend. Use software to construct the following.(a) A 95% prediction interval based on
Prediction versus confidence intervals. For the movie revenue model,would confidence or prediction intervals be used more frequently? Explain your answer.
Asimpler model. In the multiple regression analysis using all three variables, opening-weekend theater count, Theaters, appears to be the least helpful(given that the other two explanatory variables are in the model). Do a new analysis using only the movie’s budget and opening-weekend revenue.
The model matters. Carry out the simple linear regression of U.S.box office revenue on opening-weekend theater count, Theaters. What is the P-value of the t test for the hypothesis that Theaters does not help predict U.S. box office revenue? Explain carefully to someone who knows no statistics why
Reading software outputs. Carefully examine the outputs from the three software packages given in Figure 11.11. Make a table giving the estimated regression coefficient for opening-weekend revenue (Opening), its standard error, the t statistic with degrees of freedom, and the P-value as reported by
Architectural firm billings. A summary of firms engaged in commercial architecture in the Indianapolis, Indiana, area provides firm characteristics including total annual billing and the number of architects, engineers, and staff employed in the firm.10 Consider developing a model to predict total
Predicting retail sales. Daily sales at a secondhand shop are recorded over a 25-day period.9 The daily gross sales and total number of items sold are broken down into items paid by check, cash, and credit card. The owners expect that the daily numbers of cash items, check items, and credit card
Rerun Exercise 11.26 without the data for Schwab and Fidelity. Compare your results with what you obtained in that exercise.
Rerun Exercise 11.25 without the data for Schwab and Fidelity. Compare your results with what you obtained in that exercise.
Rerun Exercise 11.24 without the data for Schwab and Fidelity. Compare your results with what you obtained in that exercise.
Rerun Exercise 11.23 without the data for Schwab and Fidelity. Compare your results with what you obtained in that exercise.
Residuals. Refer to the brokerage data in Exercise 11.23.Find the residuals for the multiple regression used to predict market share with number of accounts and assets as explanatory variables.(a) Give a graphical summary of the distribution of the residuals.Are there any outliers in this
Multiple regression equation. Refer to the brokerage data in Exercise 11.23. Run a multiple regression to predict market share using number of accounts and assets as explanatory variables.(a) Give the equation for predicted market share.(b) What is the value of the regression standard error s?
Data analysis: pairs of variables. Refer to the previous exercise.(a) Plot market share versus number of accounts, market share versus assets, and number of accounts versus assets.(b) Summarize these relationships. Are there any influential observations?(c) Find the correlation between each pair of
Data analysis: individual variables. The following table gives data on market share, number of accounts, and assets held by 10 online stock brokerages.8 Brokerage Market share Accounts Assets Charles Schwab 27.5 2500 219.0 E* Trade 12.9 909 21.1 TD Waterhouse 11.6 615 38.8 Datek 10.0 205 5.5
Predicting the price of circular saws. Refer to the previous exercise.(a) What is the predicted price for the second saw? The characteristics are WEIGHT=12, AMPS=15, DEPTH=2.25, SPEED=5, POWER=5, EASE=4, and CONSTRUCTION=5.(b) The stated price for this model is $110. Is the predicted price above or
Predicting the price of circular saws: multiple regression equation. Refer to the circular saw data described in Exercise 11.19.(a) Run a multiple regression to predict price using the seven product characteristics. Give the equation for predicted price.(b) What is the value of the regression
Predicting the price of circular saws: pairs of variables. Refer to the circular saw data described in Exercise 11.19.(a) Examine the relationship between each pair of variables using correlation and a scatterplot.(b) Which characteristic is most strongly correlated with price?Is any pair of
Predicting the price of circular saws: individual variables.Suppose your construction company needs to buy some circular saws. To help in the purchasing decision, you decide to develop a model to predict the selling price. You decide to obtain price and product characteristic information on 19
Understanding the fitted regression line. The fitted regression equation for a multiple regression isˆy = 2.5 + 3.8x1 − 2.3x2(a) If x1 = 4 and x2 = 2, what is the predicted value of y?(b) For the answer to part (a) to be valid, is it necessary that the values x1 = 4 and x2 = 2 correspond to a
Describing a multiple regression. As part of a study, data from 106 Bryant College actuarial graduates were obtained.6 The researchers were interested in describing how students’ overall math grade point averages are explained by SAT Math andVerbal scores, class rank, and Bryant College’s
Compare the variability. Figure 11.2 (page 578) gives the standard deviation sy of the profits of the DJU companies. What is this value? The regression standard error s from Figure 11.6 also measures the variability of profits, this time after taking into account the effect of assets and sales on
Reading software output. Regression software usually reports both s2 and the regression standard error s. For the assets, sales, and profits data of Case 11.1, the approximate values are s2 = 0.125 and s = 0.354. Locate s2 and s in each of the four outputs in Figure 11.6 (pages 582–584). Give the
Examine the effect of Alaska. For the logarithm-transformed data, Alaska has far more assets than is predicted by the regression equation. Delete Alaska from the data set and rerun the multiple regression using the transformed data. Describe how the regression coefficients change.
Residuals for the log analysis. In Exercise 11.10 (page 582), you carried out multiple regression using the logarithms of all the variables in Table 11.2. Obtain the residuals from this regression and examine them as you did in Exercise 11.11.Summarize your conclusions and compare your plots with
Examine the effect of Delaware. The state of Delaware has far more assets than is predicted by the regression equation. Delete this observation and rerun the multiple regression. Describe how the regression coefficients change.
Examine the residuals. In Exercise 11.9 (page 582), you ran a multiple regression using the data in Table 11.2. Obtain the residuals from this regression and plot them versus each of the explanatory variables. Also, examine the Normality of the residuals using a histogram or stemplot. If possible,
Regression after transforming. In Exercise 11.8 we considered the logarithm transformation for all variables in Table 11.2. Run the regression using the logarithmtransformed variables and report the least-squares equation. Note that the units differ from those in Exercise 11.9, so the results
Predicting bank assets. Using the bank data in Table 11.2, do the regression to predict assets using deposits and number of banks. Give the least squares regression equation.
Try logs. The data file for Table 11.2 contains the logarithms of each variable.The logarithm transformation pulls in the long tail of a skewed distribution. It is common to use logarithms to make economic and financial data more symmetric before doing inference. Find the correlations and generate
Examining the pairs of relationships. Examine the relationship between each pair of variables in Table 11.2. That is, compute correlations and construct scatterplots.Based on these summaries, describe these relationships. Are there any states or other areas that you consider unusual in any way?
Look at the data. Examine the data for deposits, assets, and number of banks given in Table 11.2. That is, use graphs to display the distribution of each variable.Based on your examination, how would you describe the data? Are there any states or other areas that you consider to be outliers or
Is there a problem? Refer to Exercise 11.4. The researchers selected the 25 highest and 25 lowest Fortune 500 companies for their study. Suppose this resulted in the response variable, annual profits, having a bimodal distribution. Since this distribution is not Normal, will this necessarily be a
Describing a multiple regression. As part of a study, data from 50 Fortune 500 companies were obtained.5 Based on these data, the researchers described the relationship between a company’s annual profits and the age, attractiveness, and facial maturity level of its CEO.(a) What is the response
Deposits, assets, and number of banks. Table 11.2 gives data for insured commercial banks, by state or other area.4 The cases are the 50 states, the District of Columbia, and Puerto Rico. Bank assets and deposits are given in billions of dollars.We are interested in describing how assets are
Needs of the Department of Mathematics. The Department of Mathematics has 1 department head, 57.5 faculty, 2 managers, 49.75 administrators and lecturers, 198.74 graduate assistants, and 10.64 clerical and service workers.(a) Find the office needs for the Mathematics Department that are predicted
Check the formula. The table that appears before Example 11.3 shows that the predicted office space needed by the Chemistry Department is 45,959.0 ft2. Verify that the formula given in Example 11.3 gives the same predicted value.
Predicting success in a statistics course. Can a pretest on mathematics skills predict success in a statistics course? The 55 students in an introductory statistics class took a pretest at the beginning of the semester. The least-squares regression line for predicting the score y on the final exam
Some unusual observations. Refer to the previous exercise.Two corporations that were not included in the simple random sample are the School System of Hammond and the Northwest Indiana Special Education Cooperative. Hammond is a very large corporation with 137 thousand students and 83 million
Expenditures of school districts. There are 315 school corporations in the state of Indiana. Various information about these corporations is available on the Internet.25 Here are data on the numbers of students (NStu), expressed in thousands, and the total yearly expenditures (Exp), expressed in
Check the outliers. The plot that you generated in Exercise 10.83 has two observations that appear to be outliers.(a) Identify these points on a plot of the data.(b) Rerun the analysis with the other 12 hotels and summarize the effect of the two possible outliers on the results that you gave in
How can we use the results? Refer to the previous exercise.(a) If one hotel had 100 more rooms than another, how many additional employees would you expect that hotel to have?(b) Give a 95% confidence interval for your answer in part (a).(c) The study collected these data from 14 hotels in Toronto.
Hotel sizes and numbers of employees. A human resources study of hotels collected data on the size, measured by number of rooms, and the number of employees for 14 hotels in Canada.23 Here are the data: DATADATA DATADATADATA DATADATADATADATADATA DATADATADATADATADATA DATADATADATADATADATA
Brand equity and sales. Brand equity is one of the most important assets of a business. It includes brand loyalty, brand awareness, perceived quality, and brand image. One study examined the relationship between brand equity and sales using simple linear regression analysis.22 The correlation
What can we infer? The study described in the previous exercise analyzed data that were collected from a Chevys Freshmex Restaurant in suburban Phoenix during busy times at the restaurant on Fridays, Saturdays, and Sundays.(a) To what extent do you think that the results that you obtained in the
Correlations for restaurant spending. Astudy of spending by customers in restaurants examined 1413 transactions.21 Five of the variables measured were check (total check amount divided by the number of customers at the table), minutes (the amount of time spent dining), table (the size of the
Bias in Medicare payments. Payments to hospitals for pharmaceuticals under Medicare’s Outpatient Prospective Payment System (OPPS) are based on estimates of costs. These are determined by multiplying the cost of the pharmaceutical to the hospital by a department-specific average markup. Some
Significance tests and confidence intervals. The significance test for the slope in a simple linear regression gave a value t = 4.12 with 50 degrees of freedom.Would the 95% confidence interval for the slope include the value zero? Give a reason for your answer.
Plot indicates model assumptions. Construct a plot with data and a regression line that fits the simple linear regression model framework. Then construct another plot that has the same slope and intercept but a much smaller value of the regression standard error s.
Yearly number of tornadoes. The Storm Prediction Center of the National Oceanic and Atmospheric Administration maintains a list of tornadoes, floods, and other weather phenomena.Table 10.8 summarizes the annual number of tornadoes in the United States between 1953 and 2008.19 DATADATA DATADATADATA
Are the two fuel efficiency measurements similar? Refer to Exercise 7.45 (page 416). The driver wants to determine if these two mpg calculations are different.Fill-up: 1 2 3 4 5 6 7 8 9 10 Computer: 41.5 50.7 36.6 37.3 34.2 45.0 48.0 43.2 47.7 42.2 Driver: 36.5 44.2 37.2 35.6 30.5 40.5 40.0 41.0
What’s wrong? For each of the following, explain what is wrong and why.(a) The parameters of the simple linear regression model are b0, b1, and s.(b) To test H0: b1 = 0, use a t test.(c) For any value of the explanatory variable x, the confidence interval for the mean response will be wider than
What’s wrong? For each of the following, explain what is wrong and why.(a) The slope describes the change in x for a unit change in y.(b) The population regression line is y = b0 + b1x.(c) A 95% confidence interval for the mean response is the same width regardless of x.
Correlation. The regression in Figure 10.19 takes reputation as explaining profitability. We could as well take reputation as in part explained by profitability. We would then reverse the roles of the variables, regressing REPUTAT on PROFIT. Both regressions lead to the same conclusions about the
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