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business
business statistics in practice
Business Statistics Plus Pearson Mylab Statistics With Pearson Etext 3rd Edition Norean R Sharpe ,Richard D De Veaux ,Paul Velleman - Solutions
=+57. GDP 2013. The scatterplot shows the gross domestic product (GDP) of the United States in trillions of dollars plotted against years since 1950.12 96 3GDP 1950 1965 1980 1995 2010 Year A linear model fit to the relationship looks like this:Dependent variable is: GDP($T)R squared = 96.7% R
=+b) What does the coefficient of Traps mean in this model? Does it predict that licensing more traps would cause an increase in the price of lobster? Suggest some alternative explanations.M16_SHAR8696_03_SE_C16.indd 579 14/07/14 7:38 AM 580 CHAPTER 16 Understanding Residuals
=+a) Are the assumptions and conditions for regression inference satisfied?
=+56. Lobsters 2012, part 4. Of course, what matters most to the individual entrepreneur—the licensed commercial lobster fisher—is the price of lobster. Here’s an analysis relating that price ($/lb) to the number of traps (millions) since 1950:Dependent variable is: Price/lb R squared =
=+b) Interpret the slope coefficient. Do more fishers cause a higher valued harvest? Suggest alternative explanations.
=+59. Logs (not logarithms). Many professions use tables to determine key quantities. The value of a log is based on the number of board feet of lumber the log may contain. (A board foot is the equivalent of a piece of wood 1 inch thick, 12 inches wide, and 1 foot long. For example, a 2″ *
=+44. New homes. A real estate agent collects data to develop a model that will use the Size of a new home (in square feet)to predict its Sale Price (in thousands of dollars). Which of these is most likely to be the slope of the regression line:0.008, 0.08, 0.8, or 8? Explain.
=+43. Colorblind. Although some women are colorblind, this condition is found primarily in men. An advertisement for socks marked so they were easy for someone who was colorblind to match started out “There’s a strong correlation between sex and colorblindness.” Explain in statistics terms
=+d) Explain why it’s not safe to interpret the y-intercept.
=+) What does the slope say about marriage ages since 1975?
=+) Is this linear model appropriate for the post-1975 data?Explain.
=+) Why is R2 higher for the first model (in Exercise 40)?
=+42. Age at first marriage 2011, part 3. Has the trend of decreasing difference in age at first marriage seen in Exercise 40 gotten stronger recently? Here are the scatterplot and residual plot for the data from 1975 through 2011, along with a regression for just those years.2.50 2.75 2.25 2.00
=+d) Given these two models, what would you predict the interest rate on 3-month Treasury bills will be in 2020?
=+c) Do you trust this newer predicted value? Explain.
=+b) What does this model estimate the interest rate to have been in 2000? How does this compare to the rate you predicted in Exercise 39?
=+a) How does this model compare to the one in Exercise 39?
=+41. Modern interest rates. In Exercise 39 you investigated the federal rate on 3-month Treasury bills between 1950 and M16_SHAR8696_03_SE_C16.indd 576 14/07/14 7:38 AM Exercises 577 1980. The scatterplot below shows that the trend changed dramatically after 1980, so we’ve built a new regression
=+e) Describe reasons why you might not place much faith in that prediction.
=+a) What transformation of Board Feet makes this relationship linear?
=+) Predict the average age difference in 2020.
=+a) What is the correlation between Age Difference and Year?b) Interpret the slope of this line.
=+b) Based on a linear regression using this transformation, How much lumber would you estimate that a log 10 inches in diameter contains?
=+40. Age at first marriage 2011, part 2. In Exercise 21 we looked at the age at which women married as one of the variables considered by those selling wedding services. Another variable of concern is the difference in age of the two partners.The graph shows the ages of both men and women at
=+e) Would you expect this prediction to have been accurate?Explain.
=+c) What does this model predict for the interest rate in the year 2000?
=+65. Lobsters 2012, part 5. How has the number of licensed lobster fishers changed? Here’s a plot of the number of Fishers vs. Year:1950.0 1970.0 1990.0 2010.0 6000 7500 9000 10,500 Year Fishers This plot isn’t straight. Would a transformation help? If so, which one? If not, why not?
=+) What is the correlation between Rate and Year?b) Interpret the slope and intercept.
=+39. Historical interest rates. Here’s a plot showing the federal rate on 3-month Treasury bills from 1950 to 1980, and a regression model fit to the relationship between the Rate(in %) and Years since 1950. (www.gpoaccess.gov/eop/)3 69 12 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Rate (%)Years since
=+g) Do you think this is the appropriate model for that association? Explain.
=+f) Do you think there appears to be a strong association between Speed and Fuel Efficiency? Explain.
=+e) What was the actual Fuel Efficiency when the bus was driven at 55 mph?
=+d) What Fuel Efficiency does the model predict when the bus is driven at 45 mph?
=+c) When this model predicts high Fuel Efficiency, what can you say about those predictions?
=+b) Explain why it’s silly to attach any meaning to the y-intercept.
=+b) Use the re-expression ina) for the scatterplot against HDI. Comment.c) Would you feel confident using this relationship to predict the number of Internet Users based on the HDI?M16_SHAR8696_03_SE_C16.indd 581 14/07/14 7:38 AM 582 CHAPTER 16 Understanding Residuals
=+a) Interpret the slope of this line in context.
=+38. Fuel economy. A bus company suffering losses for several months would like to find ways to make profits. In order to reduce fuel costs, the company hired a research firm to study the optimal speed for their buses to drive. Researchers drove a bus for 200 miles at speeds ranging from 35 to 75
=+g) Would this model be more successful if the temperature were expressed in degrees Celsius? Explain.
=+f) Do you think the home owner should use this model?Explain.
=+e) During one of the months on which the model was based, the temperature did average 10°. What were the actual heating costs for that month?
=+d) What heating cost does the model predict for a month that averages 10°?
=+) During months when the temperature stays around freezing, would you expect cost predictions based on this model to be accurate, too low, or too high? Explain.
=+b) Interpret the y-intercept of the line in this context.
=+a) Interpret the slope of the line in this context.
=+a) Internet Users is a count of subscribers (per 1000). What re-expression is often useful for counts? Examine the histogram of Internet Users and the histogram of Internet Users using the re-expression you suggested. Comment.
=+37. Heating cost. Small businesses must track every expense.A flower shop owner tracked her costs for heating and related it to the average daily Fahrenheit temperature, finding the model Cost = 133 - 2.13 Temp. The residuals plot for her data is shown.$20$10$0–$10–$20 Residual 0 10 20 30
=+36. What’s the effect? Published reports about violence in computer games have become a concern to developers and distributors of these games. One firm commissioned a study of violent behavior in elementary-school children. The researcher asked the children’s parents how much time each
=+35. What’s the cause? A researcher gathering data for a pharmaceutical firm measures blood pressure and the percentage of body fat for several adult males and finds a strong positive association. Describe three different possible cause-and-effect relationships that might be present.
=+34. The extra point, part 2. The original five points in Exercise 33 produce a regression line with slope 0. Match each of the green points (a–e) with the slope of the line after that one point is added:1) -0.45 2) -0.30 3) 0.00 4) 0.05 5) 0.85
=+33. The extra point. The scatterplot shows five blue data points at the left. Not surprisingly, the correlation for these points is r = 0. Suppose one additional data point is added at one of the five positions suggested below in green. Match each point (a–e) with the correct new correlation
=+64. Human Development Index 2012, part 4. In Exercise 63 we examined the relationship between log(GDPPC) and HDI.The number of Internet users (per 1000 people) is also positively associated with economic progress in a country.Here’s a scatterplot of Internet users (subscribers per 1000 people)
=+32. More unusual points. Each of the following scatterplots a–d shows a cluster of points and one “stray” point. For each, answer questions 1–4:
=+31. Unusual points. Each of the four scatterplots a–d that follow shows a cluster of points and one “stray” point. For each, answer questions 1–4:1) In what way is the point unusual? Does it have high leverage, a large residual, or both?2) Do you think that point is an influential
=+e) What cautions would you state about your conclusion?
=+d) Use technology to compute the Durbin-Watson statistic and comment.
=+c) If the Center had a goal of achieving an average tracking error of 50 nautical miles by 2015, will they make it?Explain.
=+a) Interpret the slope and intercept of the regression model.b) Interpret se in this context.
=+30. Tracking hurricanes 2012. Like many businesses, The National Hurricane Center also participates in a program to improve the quality of data and predictions by government agencies. They report their errors in predicting the path of hurricanes. The following scatterplot shows the trend in
=+f) There’s a point near the middle of this time span with a large negative residual. Can you explain this outlier?
=+b) What does the value of R2 say about how successful the model is?c) Interpret se in this context.d) Compute the Durbin-Watson statistic and comment.e) Would you use this model to predict the numbers of passengers in 2010 (YearsSince1990 = 20)? Explain.
=+29. Oakland passengers. Much attention has been paid to the challenges faced by the airline industry. Patterns in customer demand are an important variable to watch. The scatterplot below, created in Excel, shows the number of passengers departing from Oakland (CA) airport month by month since
=+c) The film Harry Potter and the Goblet of Fire is the purple point in the upper right. If it were omitted from this analysis, how might that change your conclusions about dramas?M16_SHAR8696_03_SE_C16.indd 573 14/07/14 7:38 AM 574 CHAPTER 16 Understanding Residuals
=+63. Human Development Index 2012, part 3. In Exercise 23 we saw that the United Nations Development Programme(UNDP) uses the Human Development Index (HDI) in an attempt to summarize the progress in health, education, and economics of a country with one number. The gross domestic product per
=+) How do the gross receipts of adventure films (the red line on top) differ from those of comedies (the green line below the red)? Discuss both the slopes and the intercepts.
=+a) In what ways is the relationship between run times and U.S. Gross similar for the four kinds of films?
=+28. Movie revenues. How does what a movie earns relate to its run time? Will audiences pay more for a longer film?Does the relationship depend on the type of film?The scatterplot shows the relationship for the films in Exercise 27 between U.S. Gross earnings and Run Time.Dramas are plotted with
=+c) In what way are dramas different from other genres of movies with respect to this relationship?
=+b) In what way are dramas and other movies similar with respect to this relationship?
=+a) What are the units for the slopes of these lines?
=+27. Movie budgets. Here’s a scatterplot of the production budgets (in millions of dollars) vs. the running time (in minutes) for a collection of major movies. Dramas are plotted in red and all other genres are plotted in blue. A separate least squares regression line has been fitted to each
=+b) Does this model allow the intern to make accurate predictions? Explain.
=+a) Does this mean that a linear model is not appropriate?Explain.
=+26. Bad model? An intern who has created a linear model is disappointed to find that her R2 value is a very low 13%.
=+b) Does this model allow the consultant to make accurate predictions? Explain.
=+a) Is this reasoning correct? Explain.
=+25. Good model? In justifying his choice of a model, a consultant says “I know this is the correct model because R2 = 99.4%.”
=+a) Explain why fitting a linear model to these data might be misleading.
=+24. Human Development Index 2012, part 2. The United Nations Development Programme (UNDP) uses the Human Development Index (HDI) in an attempt to summarize in one number the progress in health, education, and economics of a country. The number of Internet accounts per 1000 people is positively
=+c) There are two outliers, Qatar and Luxembourg with very high GDPPC’s and Equitorial Guinea with a very low HDI for its GDPPC. Will setting these points aside improve the model substantially? Explain.
=+b) If you fit a linear model to the data, what do you think a scatterplot of residuals versus predicted HDI will look like?
=+a) Explain why fitting a linear model to these data would be misleading.
=+62. Productivity. In repetitive tasks, significant productivity gains can occur within a few days. The average amount of time (in minutes) required for a new employee to install electronic components in a television was measured for her first 15 days. Create a model for this relationship using
=+23. Human Development Index (HDI) 2012. The United Nations Development Programme (UNDP) collects data in the developing world to help countries solve global and national development challenges. In the UNDP annual Human Development Report, you can find data on over 100 variables for each of 197
=+22. Burnout. The HR director of a company has noticed an increasing number of employees suffering from “burnout”, in which physical and emotional fatigue hurt job performance. The HR director has read that the more time a person spends socializing with coworkers, the higher the chance of
=+d) Do you think a linear model is appropriate for these data? Explain.
=+b) Is the association strong?c) Is the correlation high? Explain.
=+21. Age at first marriage 2011. Weddings are one of the fastest growing businesses; about $40 billion is spent on weddings in the United States each year. But demographics may be changing, and this could affect wedding retailers’marketing plans. Is there evidence that the age at which women
=+b) What would you suggest?Chapter Exercises
=+a) Would you recommend this transformation? Why or why not?
=+20. For the regression in Exercise 19:A student tries taking the reciprocal of customers and produces the plot shown below:0.00012 0.00010 0.00008 0.00002 0 15 0.00006 0.00004 5 10 20 25 30 35 1/Customers Month
=+b) What power in the ladder of powers does that correspond to?
=+a) What re-expression might you suggest for the number of customers?
=+19. A quickly growing company shows the following scatterplot of customers vs. time (in months).M16_SHAR8696_03_SE_C16.indd 571 14/07/14 7:37 AM 572 CHAPTER 16 Understanding Residuals Customers 0 5 10 15 20 25 30 35 Month 20,000 100,000 80,000 60,000 40,000 140,000 120,000
=+18. One possible model for the manufacturing process of Exercise 17 is the following:Dependent variable is: Log(Cost per unit)R squared = 90.1% R squared (adjusted) = 89.3%s = 0.0841 with 15 - 2 = 13 degrees of freedom Variable Coefficient SE(Coeff) t-ratio P-value Intercept 0.7618 0.0405 18.8
=+What should be done to make the relationship more nearly linear?
=+17. A small company has developed an improved process for making solar panels. The company needs to set its prices and wants those prices to reflect the efficiencies of producing larger batches. The data show the following:#Units Cost per Unit 10 7.389 20 6.049 30 4.953 40 4.055 50 3.320 60 2.718
=+16. A study of homes looking at the relationship between the Price of a luxury mobile home unit and its Age produced the following scatterplot. A regression was fit to the data as shown belowPrice 0 20 40 60 80 100 Age 200,000 250,000 300,000 350,000 On the basis of this plot, would you advice
=+15. A scatterplot of Salary against Years Experience for some employees, and the scatterplot of residuals against predicted Salary from the regression line are shown in the figures. On the basis of these plots, would you recommend a re-expression of either Salary or Years Experience? Explain.
=+b) What do you conclude from this test?Section 16.6
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