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The Practice Of Statistics For Business And Economics 4th Edition Layth C. Alwan, Bruce A. Craig - Solutions

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 four 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 5 28, y⁄5 8.1 1 10.3x1 1 4.2x2, SE b1 5 5.0.(b) n 5 53, y⁄5 8.1 1 10.3x1 1 4.2x2, SE b1 5 5.0.(c) n 5 28, y⁄5 8.1 1 10.3x1 1 4.2x2 1 2.1x3, SE b1
Is Opening helpful when Budget and Theaters are available? We saw that Budget and Theaters are not useful in a model that contains the opening-weekend 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 Theater 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 0.4355. 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, Theaters; (c) Opening, Theaters; (d) Budget; (e) Opening;(f) Theaters. 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. The movie Kick-Ass was released during this same time period. It had a budget of $30.0 million and was shown in 3065 theaters, grossing $19.83 million during the first weekend. Use software to construct the following.(a) A 95% prediction interval based on the model
Prediction versus confidence intervals. For the movie revenue model, would confidence intervals for the mean response or prediction intervals be used more frequently? Explain your answer.
A simpler 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.
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 the movie’s budget (Budget), its standard error, the t statistic with degrees of freedom, and the P-value as reported by
Look at the data. Examine the data for total U.S. revenue, budget, opening-weekend revenue, and the number of opening-weekend theaters.That is, use graphs to display the distribution of each variable and the relationships between pairs of variables. Based on your examination, how would you describe
Architectural firm billings. A summary of firms engaged in commercial architecture in the Indianapolis, Indiana, area provides firm characteristics, including total annual billing in the current year, total annual billing in the previous year, the number of architects, the number of engineers, and
Predicting retail sales. Daily sales at a secondhand shop are recorded over a 25-day period.8 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 McDonald’s and Starbucks. Compare your results with what you obtained in that exercise. FFOOD
Rerun Exercise 11.25 without the data for McDonald’s and Starbucks. Compare your results with what you obtained in that exercise. FFOOD
Rerun Exercise 11.24 without the data for McDonald’s and Starbucks. Compare your results with what you obtained in that exercise. FFOOD
Rerun Exercise 11.23 without the data for McDonald’s and Starbucks. Compare your results with what you obtained in that exercise. FFOOD
Residuals. Refer to the fast-food data in Exercise 11.23. Find the residuals for the multiple regression used to predict market share based on the four explanatory variables. FFOOD(a) Give a graphical summary of the distribution of the residuals. Are there any outliers in this distribution?(b) Plot
Multiple regression equation. Refer to the fast-food data in Exercise 11.23. Run a multiple regression to predict market share using all four explanatory variables. FFOOD(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. FFOOD(a) Plot market share versus each of the explanatory variables.(b) Summarize these relationships. Are there any influential observations?(c) Find the correlation between each pair of variables.
Data analysis: Individual variables.Table 11.3 gives data on the current fast-food market share, along with the number of franchises, number of company-owned stores, annual sales ($ million) from three years ago, and whether it is a burger restaurant.7 Market share is expressed in percents, based
Predicting the price of a tablet. Refer to the previous exercise. Let’s use the model with Observation 11 removed. TABLTS(a) What is the predicted price for the second tablet?The characteristics are SIZE = 7.9, BATTERY = 12.8, WEIGHT = 0.7, EASE = 5, DISPLAY = 5, and VERSATILITY = 3.(b) The
Predicting the price of tablets: Multiple regression equation. Refer to the tablet data described in Exercise 11.19. TABLTS(a) Run a multiple regression to predict price using the six product characteristics. Give the equation for predicted price.(b) What is the value of the regression standard
Predicting the price of tablets: Pairs of variables. Refer to the tablet data described in Exercise 11.19. TABLTS(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 tablets: Individual variables. Suppose your company needs to buy some tablets. 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 20 tablets from Consumer
Understanding the fitted regression line. The fitted regression equation for a multiple regression is y⁄5 1.5 1 2.7x1 2 1.4x2(a) If x1 5 4 and x2 5 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 5 4 and x2 5 2 correspond to a
Describing a multiple regression. As part of a study, data from 282 students majoring in accounting at the College of Business Studies in Kuwait were obtained through a survey.5 The researchers were interested in finding determinants of academic performance measured by the student’s major grade
Compare the variability. Figure 11.2 (page 536) gives the standard deviation sy of the log profits of the BBC Global 30 companies. What is this value? The regression standard error s from Figure 11.6 also measures the variability of log profits, this time after taking into account the effect of
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 (page 534), the approximate values are s2 5 0.352 and s 5 0.593. Locate s2 and s in each of the four outputs in Figure 11.6 (pages
Examine the effect of Massachusetts. For the logarithm-transformed data, Massachusetts has far more assets than predicted by the regression equation. Delete Massachusetts 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, 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 the plots
Examine the effect of Ohio. The state of Ohio has far more assets than 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, you ran a multiple regression using the data in Table 11.2 (page 536). 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 (page 538), we considered the logarithm transformation for all variables in Table 11.2. Run the regression using the logarithm-transformed variables and report the least-squares equation. Note that the units differ from those in Exercise 11.9, so the
Try logs. The data file for Table 11.2 also contains the logarithms of each variable. Find the correlations and generate scatterplots for each pair of transformed variables. Interpret the results and then compare with your analysis of the original variables.==11.9. Predicting bank assets. Using the
Examining the pairs of relationships. Examine the relationship between each pair of variables in Table 11.2 (page 536). 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
Look at the data. Examine the data for assets, deposits, and equity 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 unusual in
Is there a problem? Refer to Exercise 11.4 (page 535). The 55 firms in the sample represented a range of industries, including retail and computer manufacturing.Suppose this resulted in the response variable, annual profits, having a bimodal distribution (see page 56 for a trimodal distribution).
Describing a multiple regression. As part of a study, data from 55 Fortune 500 companies were obtained.4 Based on these data, the researchers described the relationship between a company’s annual profits and the age and facial width-toheight ratio of its CEO.(a) What is the response variable?(b)
Assets, interest-bearing deposits, and equity capital. Table 11.2 gives data for insured commercial banks, by state or other area.3 The cases are the 50 states, the District of Columbia, Guam, and Puerto Rico. Bank assets, interestbearing deposits, and equity capital are given in billions of
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.
Agricultural productivity. Few sectors of the economy have increased their productivity as rapidly as agriculture. Let’s describe this increase. Productivity is defined as output per unit input. “Total factor productivity” (TFP) takes all inputs (labor, capital, fuels, and so on) into
Growth in grocery store size. Here are data giving the median store size (in square feet) by year for grocery stores.23 GROCERY Year Store size Year Store size Year Store size 1993 33.0 2000 44.6 2007 47.5 1994 35.1 2001 44.0 2008 46.8 1995 37.2 2002 44.0 2009 46.2 1996 38.6 2003 44.0 2010 46.0
Check the outliers. The plot you generated in Exercise 10.85 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.22 Here are the data.Employees Rooms Employees Rooms 1200 1388 275 424 180 348 105 240 350 294 435 601 250 413 585
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.21 The correlation
Predicting college debt: One last measure. Refer to Exercises 10.75, 10.77, and 10.79.Given the in-state cost prior to and after aid, another measure is the average amount of need-based aid. Create this new variable by subtracting these two costs, and investigate its relationship with average
Significance tests and confidence intervals. The significance test for the slope in a simple linear regression gave a value t 5 2.12 with 28 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 database of tornadoes, floods, and other weather phenomena.Table 10.6 summarizes the annual number of tornadoes in the United States between 1953 and 2013.20 TWISTER(a) Make a
Predicting college debt: Other measures. Refer to Exercise 10.75. Let’s now look at AvgDebt and its relationship with all six measures available in the data set. In addition to the in-state cost after aid (InCostAid), we have the admittance rate(Admit), the four-year graduation rate (Grad),
More on predicting college debt. Refer to the previous exercise. The University of Minnesota has an in-state cost of $14,933 and an average debt of $29,702. Texas A&M University has an in-state cost of $9007 and an average debt of $22,955.BESTVAL(a) Using your answer to part (a) of the previous
Predicting college debt. Refer to Exercise 10.75. Figure 10.21 contains Minitab output for the simple linear regression of AvgDebt on InCostAid. BESTVAL(a) State the least-squares regression line.(b) The University of Oklahoma is one school in this sample. It has an in-state cost of $12,960 and
Can we consider this an SRS? Refer to the previous exercise. The report states that Kiplinger’s rankings focus on traditional four-year public colleges with broad-based curricula. Each year, they start with more than 500 schools and then narrow the list down to roughly 120 based on academic
College debt versus adjusted in-state costs.Kiplinger’s “Best Values in Public Colleges” provides a ranking of U.S. public colleges based on a combination of various measures of academics and affordability.19 We’ll consider a random collection of 40 colleges from Kiplinger’s 2014 report
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 5 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 5 b0 1 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
Squared correlation. SAS gives the squared correlation r2 as “R-Square.” How can you obtain r2 from the ANOVA table? Do this, and verify that your result agrees with R-Square.
The regression standard error. SAS labels the regression standard error s as “Root MSE.” How can you obtain s from the ANOVA table? Do this, and verify that your result agrees with Root MSE.
F versus t. How do the ANOVA F statistic and its P-value relate to the t statistic for the slope and its P-value? Identify these results on the output and verify their relationship (up to roundoff error).
Predicting profitability. A company not covered by the Fortune survey has reputation score x 5 7. Will a 95% prediction interval for this company’s profitability be wider or narrower than the confidence interval found in the previous exercise? Explain why we should expect this. Then give the 95%
Predicting profitability. An additional calculation shows that the variance of the reputation scores for these 154 firms is s2x 5 0.8101. SAS labels the regression standard error s as “Root MSE” and the sample mean of the responses y as “Dep Mean.” Starting from these facts, give a 95%
Estimating the slope. Explain clearly what the slope 1 of the population regression line tells us in this setting. Give a 99% confidence interval for this slope.
Significance in two senses.(a) Is there good evidence that reputation helps explain profitability? (State hypotheses, give a test statistic and P-value, and state a conclusion.)(b) What percent of the variation in profitability among these companies is explained by regression on reputation?(c) Use
Predicting the return for a future year.Suppose the S&P annual return for a future year is 0%.Using the information from the previous four exercises, construct the appropriate 95% interval. Also, explain why this interval is or is not the same interval constructed in Exercise 10.63.
Inference for the intercept? The mean of the S&P 500 returns for these years is 11.97. From this and information from the previous exercises, find the standard error for the least-squares intercept b0. Use this to construct a 95% confidence interval. Finally, explain why the intercept 0 is
Estimating the standard error of the slope. The standard deviation of the S&P 500 returns for these years is 18.70%. From this and your work in the previous exercise, find the standard error for the least-squares slope b1. Give a 90% confidence interval for the slope 1 of the population regression
s and r2. What are the values of the regression standard error s and the squared correlation r2?
The ANOVA table. Complete the analysis of variance table by filling in the “Residual Error”row and the other missing items in the DF, MS, and F columns.
ANOVA by-products.(a) The output gives r2 5 0.4815. How can you obtain this from the ANOVA table?(b) The output gives the regression standard error as s 5 2.1407. How can you obtain this from the ANOVA table?
The ANOVA table. Use the numerical results in the Excel output to verify each of these relationships.(a) The ANOVA equation for sums of squares.(b) How to obtain the total degrees of freedom and the residual degrees of freedom from the number of observations.(c) How to obtain each mean square from
A significant relationship? The output reports two tests of the null hypothesis that regressing on inflation does help to explain the return on T-bills.State the hypotheses carefully, give the two test statistics, show how they are related, and give the common P-value.
Predicting T-bill return. Figure 10.16 (page 514) uses statistical software to predict the return on Treasury bills in a year when the inflation rate is 2.25%. Let’s do this without specialized software. Figure 10.10 (page 499) contains Excel regression output. Use a calculator or software to
T-bills and inflation. Figure 10.10 (page 499) gives the Excel output for regressing the annual return on Treasury bills on the annual rate of inflation. The data appear in Table 10.1 (page 498). Starting with the regression standard error s 5 2.1407 from the output and the variance of the
Selling a large house. Among the houses for which we have data in Table 10.5 (page 509), just four have floor areas of 1800 square feet or more.Give a 90% confidence interval for the mean selling price of houses with floor areas of 1800 square feet or more.
Beer and blood alcohol. Exercise 10.34(page 509) gives data from measuring the blood alcohol content (BAC) of students 30 minutes after they drank an assigned number of cans of beer. Steve thinks he can drive legally 30 minutes after he drinks five beers. The legal limit is BAC 5 0.08. Give a 90%
Size and selling price of houses. Table 10.5(page 509) gives data on the size in square feet of a random sample of houses sold in a Midwest city along with their selling prices. HSIZE(a) Find the mean size x of these houses and also their mean selling price y. Give the equation of the least-squares
Two confidence intervals. The data used for Exercise 10.47 include 195 men 30 years old. The mean income of these men is y 5 $49,880 and the standard deviation of these 195 incomes is sy 5 $38,250.(a) Use the one-sample t procedure to give a 95%confidence interval for the mean income y of
T-bills and inflation. Figure 10.16 (page 514)gives part of a regression analysis of the data in Table 10.1 relating the return on Treasury bills to the rate of inflation. The output includes prediction of the T-bill return when the inflation rate is 2.25%.(a) Use the output to give a 90%
Predicting income from age, continued. Use the computer outputs in Figure 10.12 (page 507) and Exercise 10.47 to give a 99% confidence interval for the mean income of all 40-year-old men.
Predict what? The two 95% intervals for the income of 30-year-olds given in Exercise 10.47 are very different. Explain briefly to someone who knows no statistics why the second interval is so much wider than the first. Start by looking at 30-year-olds in Figure 10.11(page 506).
Predicting income from age. Figures 10.11 and 10.12 (pages 506 and 507) analyze data on the age and income of 5712 men between the ages of 25 and 65.Here is Minitab output predicting the income for ages 30, 40, 50, and 60 years:Predicted Values Fit SE Fit 95% CI 95% PI 51638 948 (49780, 53496)
Is the price right? Refer to Exercise 10.31(page 508), where the relationship between the size of a home and its selling price is examined. HSIZE(a) Suppose that you have a client who is thinking about purchasing a home in this area that is 1500 square feet in size. The asking price is $140,000.
Compare the estimates. Case 18 in Table 10.3(Purdue) has a 2000 tuition of $3872 and a 2008 tuition of $7750. A predicted 2013 tuition amount based on 2008 tuition was computed in Exercise 10.43, while one based on the 2000 tuition was computed in Exercise==10.44. Compare these two estimates and
Predicting 2013 tuition from 2000 tuition. Refer to Exercise 10.21 (page 506). TUIT(a) Find a 95% confidence interval for the mean tuition amount corresponding to a 2000 tuition of $3872.(b) Find a 95% prediction interval for a future response corresponding to a 2000 tuition of $3872.(c) Write a
Predicting 2013 tuition from 2008 tuition.Refer to Exercise 10.15 (pages 504–505). TUIT(a) Find a 95% confidence interval for the mean tuition amount corresponding to a 2008 tuition of $7750.(b) Find a 95% prediction interval for a future response corresponding to a 2008 tuition of $7750.(c)
More on assessment value versus sales price.Refer to Exercises 10.13 and 10.14 (pages 503–504).Suppose we’re interested in determining whether the population regression line differs from y 5 x. We’ll look at this three ways. HSALES(a) Construct a 95% confidence interval for each property in
More on public university tuition. Refer to Exercise 10.15 (pages 504–505). TUIT(a) The tuition at BusStat U was $8800 in 2008. Find the 95% prediction interval for its tuition in 2013.(b) The tuition at Moneypit U was $15,700 in 2008. Find the 95% prediction interval for its tuition in 2013.(c)
Predicting the return on Treasury bills. Table 10.1 (page 498) gives data on the rate of inflation and the percent return on Treasury bills for 55 years.Figures 10.9 and 10.10 analyze these data. You think that next year’s inflation rate will be 2.25%. Figure 10.16 displays part of the Minitab
Predicting the average log income. In Example 10.7 (pages 511–512)software predicts the mean log income of entrepreneurs with 16 years of education to be y⁄5 10.0560. We also see that the standard error of this estimated mean is SE ⁄ 5 0.167802. These results come from data on 100
Highway MPG and CO2 Emissions. Let’s investigate the relationship between highway miles per gallon (MPGHWY) and CO2 emissions for premium gasoline cars as reported by Natural Resources Canada.17 PREM(a) Make a scatterplot of the data and describe the pattern.(b) Plot MPGHWY versus the logarithm
Computer memory. The capacity of memory commonly available at retail has increased rapidly over time.(a) Make a scatterplot of the data. Growth is much faster than linear.(b) Plot the logarithm of capacity against year. Are these points closer to a straight line?(c) Regress the logarithm of DRAM
Influence? Your scatterplot in Exercise 10.31 shows one house whose selling price is quite high for its size. Rerun the analysis without this outlier. Does this one house influence r2, the location of the least-squares line, or the t statistic for the slope in a way that would change your
Beer and blood alcohol. How well does the number of beers a student drinks predict his or her blood alcohol content (BAC)? Sixteen student volunteers at Ohio State University drank a randomly assigned number of 12-ounce cans of beer. Thirty minutes later, a police officer measured their BAC.Student
Do larger houses have higher prices? We expect that there is a positive correlation between the sizes of houses in the same market and their selling prices. HSIZE(a) Use the data in Table 10.5 to test this hypothesis.(State hypotheses, find the sample correlation r and the t statistic based on it,

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