PART II REVIEW EXERCISES

Review exercises are short and straightforward exercises that help you solidify the basic ideas and skills in each part of this book. We have provided “hints’’ that indicate where you can find the relevant material for the odd-numbered problems.

Question

II.1. Poverty in the states. Table II.1 gives the percentages of people living below the poverty line in the 26 states east of the Mississippi River. Make a stemplot of these data. Is the distribution roughly symmetric, skewed to the right, or skewed to the left? Which states (if any) are outliers? (Hint: See page 253.)

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TABLE II.1 Percentages of state residents living in poverty, 2012–2013 two-year average
State Percent State Percent State Percent
Alabama 16.4 Connecticut 10.8 Delaware 13.7
Florida 15.1 Georgia 17.2 Illinois 12.9
Indiana 13.4 Kentucky 18.9 Maine 12.5
Maryland 10.1 Massachusetts 11.6 Michigan 14.1
Mississippi 22.2 New Hampshire 8.6 New Jersey 10.2
New York 15.9 North Carolina 17.9 Ohio 14.5
Pennsylvania 13.1 Rhode Island 13.6 South Carolina 16.3
Tennessee 18.4 Vermont 10.0 Virginia 10.5
West Virginia 17.0 Wisconsin 11.2
Source: www.census.gov/hhes/www/poverty/data/index.html

Question

II.2. Quarterbacks. Table II.2 gives the total passing yards for National Football League starting quarterbacks during the 2014 season. (These are the quarterbacks with the most passing yards on each team.) Make a histogram of these data. Does the distribution have a clear shape: roughly symmetric, clearly skewed to the left, clearly skewed to the right, or none of these? Which quarterbacks (if any) are outliers?

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TABLE II.2 Passing yards for NFL quarterbacks in 2014
Quarterback Yards Quarterback Yards
Blake Bortels 2908 Josh McCown 2206
Tom Brady 4109 Zach Mettenberger 1412
Drew Brees 4952 Cam Newton 3127
Teddy Bridgewater 2919 Kyle Orton 3018
Derek Carr 3270 Philip Rivers 4286
Kirk Cousins 1710 Aaron Rodgers 4381
Jay Cutler 3812 Ben Roethlisberger 4952
Andy Dalton 3398 Tony Romo 3705
Austin Davis 2001 Matt Ryan 4694
Ryan Fitzpatrick 2483 Mark Sanchez 2418
Joe Flacco 3986 Alex Smith 3265
Brian Hoyer 3326 Geno Smith 2525
Colin Kaepernick 3369 Matthew Stafford 4257
Andrew Luck 4761 Drew Stanton 1711
Eli Manning 4410 Ryan Tannehill 4045
Peyton Manning 4727 Russell Wilson 3475
Source: www.pro-football-reference.com/years/2014/passing.htm.

Question

II.3. Poverty in the states. Give the five-number summary for the data on poverty from Table II.1. (Hint: See page 272.)

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II.4. Quarterbacks. Give the five-number summary for the data on passing yards for NFL quarterbacks from Table II.2.

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II.5. Poverty in the states. Find the mean percentage of state residents living in poverty from the data in Table II.1. If we removed Mississippi from the data, would the mean increase or decrease? Why? Find the mean for the 25 remaining states to verify your answer. (Hint: See page 277.)

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II.6. Big heads? The army reports that the distribution of head circumference among male soldiers is approximately Normal with mean 22.8 inches and standard deviation 1.1 inches. Use the 68–95–99.7 rule to answer these questions.

  1. (a) Between what values do the middle 95% of head circumferences fall?

  2. (b) What percentage of soldiers have head circumferences greater than 23.9 inches?

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II.7. SAT scores. The scale for SAT exam scores is set so that the distribution of scores is approximately Normal with mean 500 and standard deviation 100. Answer these questions without using a table.

  1. (a) What is the median SAT score? (Hint: See page 299.)

  2. (b) You run a tutoring service for students who score between 400 and 600 and hope to attract many students. What percentage of SAT scores are between 400 and 600? (Hint: See page 300.)

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II.8. Explaining correlation. You have data on the yearly wine consumption (liters of alcohol from drinking wine per person) and yearly deaths from cirrhosis of the liver for several developed countries. Say as specifically as you can what the correlation r between yearly wine consumption and yearly deaths from cirrhosis of the liver measures.

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Question

II.9. Data on snakes. For a biology project, you measure the length (inches) and weight (ounces) of 12 snakes of the same variety. What units of measurement do each of the following have?

  1. (a) The mean length of the snakes. (Hint: See page 277.)

  2. (b) The first quartile of the snake lengths. (Hint: See page 270.)

  3. (c) The standard deviation of the snake lengths. (Hint: See page 277.)

  4. (d) The correlation between length and snake weight. (Hint: See page 325.)

Question

II.10. More data on snakes. For a biology project, you measure the length (inches) and weight (ounces) of 12 snakes of the same variety.

  1. (a) Explain why you expect the correlation between length and weight to be positive.

  2. (b) The mean length turns out to be 20.8 inches. What is the mean length in centimeters? (There are 2.54 centimeters in an inch.)

  3. (c) The correlation between length and weight turns out to be r = 0.6. If you were to measure length in centimeters instead of inches, what would be the new value of r?

Figure II.3 plots the average brain weight in grams versus average body weight in kilograms for many species of mammals. There are many small mammals whose points at the lower left overlap. Exercises II.11 through II.16 are based on this scatterplot.

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Figure II.3: Figure II.3 Scatterplot of the average brain weight (grams) against the average body weight (kilograms) for 96 species of mammals, Exercises II.11 through II.16.

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Question

II.11. Dolphins and hippos. The points for the dolphin and hippopotamus are labeled in Figure II.3. Read from the graph the approximate body weight and brain weight for these two species. (Hint: See page 318.)

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II.12. Dolphins and hippos. One reaction to this scatterplot is “Dolphins are smart; hippos are dumb.’’ What feature of the plot lies behind this reaction?

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II.13. Outliers. African elephants are much larger than any other mammal in the data set but lie roughly in the overall straight-line pattern. Dolphins, humans, and hippos lie outside the overall pattern. The correlation between body weight and brain weight for the entire data set is r = 0.86.

  1. (a) If we removed elephants, would this correlation increase, decrease, or not change much? Explain your answer. (Hint: See page 326.)

  2. (b) If we removed dolphins, hippos, and humans, would this correlation increase, decrease, or not change much? Explain your answer. (Hint: See page 326.)

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II.14. Brain and body. The correlation between body weight and brain weight is r = 0.86. How well does body weight explain brain weight for mammals? Compute r2 to answer this question, and briefly explain what r2 tells us.

Question

II.15. Prediction. The line on the scatterplot in Figure II.3 is the least-squares regression line for predicting brain weight from body weight. Suppose that a new mammal species with body weight 600 kilograms is discovered hidden in the rain forest. Predict the brain weight for this species. (Hint: See page 340.)

Question

II.16. Slope. The line on the scatter-plot in Figure II.3 is the least-squares regression line for predicting brain weight from body weight. The slope of this line is one of the numbers below. Which number is the slope? Why?

  1. (a) b = 0.5.

  2. (b) b = 1.3.

  3. (c) b = 3.2.

From Rex Boggs in Australia comes an unusual data set: before showering in the morning, he weighed the bar of soap in his shower stall. The weight goes down as the soap is used. The data appear in Table II.3 (weights in grams). Notice that Mr. Boggs forgot to weigh the soap on some days. Exercises II.17, II.18, and II.19 are based on the soap data set.

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TABLE II.3 Weight (grams) of a bar of soap used to shower
Day Weight Day Weight Day Weight
1 124 8 84 16 27
2 121 9 78 18 16
5 103 10 71 19 12
6 96 12 58 20 8
7 90 13 50 21 6
Source: Rex Boggs.

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Question

II.17. Scatterplot. Plot the weight of the bar of soap against day. Is the overall pattern roughly straight-line? Based on your scatterplot, is the correlation between day and weight close to 1, positive but not close to 1, close to 0, negative but not close to −1, or close to −1? Explain your answer. (Hint: See page 325.)

Question

II.18. Regression. The equation for the least-squares regression line for the data in Table II.3 is

weight = 133.2 − 6.31 × day

  1. (a) Explain carefully what the slope b = −6.31 tells us about how fast the soap lost weight.

  2. (b) Mr. Boggs did not measure the weight of the soap on Day 4. Use the regression equation to predict that weight.

  3. (c) Draw the regression line on your scatterplot from the previous exercise.

Question

II.19. Prediction? Use the regression equation in the previous exercise to predict the weight of the soap after 30 days. Why is it clear that your answer makes no sense? What’s wrong with using the regression line to predict weight after 30 days? (Hint: See page 345.)

Question

II.20. Keeping up with the Joneses. The Jones family had a household income of $30,000 in 1980, when the average CPI (1982–84 = 100) was 82.4. The average CPI for 2014 was 236.7. How much must the Joneses earn in 2014 to have the same buying power they had in 1980?

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II.21. Affording a Mercedes. A Mercedes-Benz 190 cost $24,000 in 1981, when the average CPI (1982–84 = 100) was 90.9. The average CPI for 2014 was 236.7. How many 2014 dollars must you earn to have the same buying power as $24,000 had in 1981? (Hint: See page 372.)

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II.22. Affording a Steinway. A Steinway concert grand piano cost $13,500 in 1976. A similar Steinway cost $163,600 in August 2015. Has the cost of the piano gone up or down in real terms? Using Table 16.1 and the fact that the August 2015 CPI was 238.7, give a calculation to justify your answer.

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Question

II.23. The price of gold. Some people recommend that investors buy gold “to protect against inflation.’’ Here are the prices of an ounce of gold at the end of the year for the years between 1985 and 2013. Using Table 16.1, make a graph that shows how the price of gold changed in real terms over this period. Would an investment in gold have protected against inflation by holding its value in real terms?

Year: 1985 1987 1989 1991 1993
Gold price: $327 $484 $399 $353 $392
Year: 1995 1997 1999 2001 2003
Gold price: $387 $290 $290 $277 $416
Year: 2005 2007 2009 2011 2013
Gold price: $513 $834 $1088 $1531 $1204

(Hint: See page 372.)

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II.23 While the value of gold dropped from 1987 to 2001 (in terms of 2013 dollars), an investment in gold most certainly holds its value today.

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Figure II.4: Figure II.4 Bar graphs of the number of heart attack victims admitted and discharged from hospitals in Ontario, Canada, on each day of the week, Exercise II.24.

Question

II.24. Never on Sunday? The Canadian province of Ontario carries out statistical studies to monitor how Canada’s national health care system is working in the province. The bar graphs in Figure II.4 come from a study of admissions and discharges from community hospitals in Ontario. They show the number of heart attack patients admitted and discharged on each day of the week during a two-year period.

  1. (a) Explain why you expect the number of patients admitted with heart attacks to be roughly the same for all days of the week. Do the data show that this is true?

  2. (b) Describe how the distribution of the day on which patients are discharged from the hospital differs from that of the day on which they are admitted. What do you think explains the difference?

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Question

II.25. Drive time. Professor Moore, who lives a few miles outside a college town, records the time he takes to drive to the college each morning. Here are the times (in minutes) for 42 consecutive weekdays, with the dates in order along the rows:

8.25 7.83 8.30 8.42 8.50 8.67 8.17
9.00 9.00 8.17 7.92 9.00 8.50 9.00
7.75 7.92 8.00 8.08 8.42 8.75 8.08
9.75 8.33 7.83 7.92 8.58 7.83 8.42
7.75 7.42 6.75 7.42 8.50 8.67 10.17
8.75 8.58 8.67 9.17 9.08 8.83 8.67
  1. (a) Make a histogram of these drive times. Is the distribution roughly symmetric, clearly skewed, or neither? Are there any clear outliers? (Hint: See pages 247 and 249.)

  2. (b) Make a line graph of the drive times. (Label the horizontal axis in days, 1 to 42.) The plot shows no clear trend, but it does show one unusually low drive time and two unusually high drive times. Circle these observations on your plot. (Hint: See page 223.)

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Question

II.26. Drive time outliers. In the previous exercise, there are three outliers in Professor Moore’s drive times to work. All three can be explained. The low time is the day after Thanksgiving (no traffic on campus). The two high times reflect delays due to an accident and icy roads. Remove these three observations. To summarize normal drive times, use a calculator to find the mean and standard deviation s of the remaining 39 times.

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image II.27. House prices. An April 15, 2014, article in the Los Angeles Times reported that the median housing price in Southern California was about $400,000. Would the mean housing price be higher, about the same, or lower? Why? (Hint: See page 281.)

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TABLE II.4 Percentage of votes for President Obama in 2012
State Percent State Percent State Percent
Alabama 38.4 Louisiana 40.6 Ohio 50.7
Alaska 40.8 Maine 56.3 Oklahoma 33.2
Arizona 44.6 Maryland 62.0 Oregon 54.2
Arkansas 36.9 Massachusetts 60.7 Pennsylvania 52.0
California 60.2 Michigan 54.2 Rhode Island 62.7
Colorado 51.5 Minnesota 52.3 South Carolina 44.1
Connecticut 58.1 Mississippi 43.8 South Dakota 39.9
Delaware 58.6 Missouri 44.4 Tennessee 39.1
Florida 50.0 Montana 41.7 Texas 41.4
Georgia 45.5 Nebraska 38.0 Utah 24.8
Hawaii 70.6 Nevada 52.4 Vermont 66.6
Idaho 32.6 New Hampshire 52.0 Virginia 51.2
Illinois 57.6 New Jersey 58.4 Washington 56.2
Indiana 43.9 New Mexico 53.0 West Virginia 35.5
Iowa 52.0 New York 63.4 Wisconsin 52.8
Kansas 38.0 North Carolina 48.4 Wyoming 27.8
Kentucky 37.8 North Dakota 38.7
Source: uselectionatlas.org/.

Question

II.28. The 2012 election. Barack Obama was elected president in 2012 with 51.1% of the popular vote. His Republican opponent, Mitt Romney, received 47.2% of the vote, with minor candidates taking the remaining votes. Table II.4 gives the percentage of the popular vote won by President Obama in each state. Describe these data with a graph, a numerical summary, and a brief verbal description.

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II.29. Statistics for investing. Joe’s retirement plan invests in stocks through an “index fund’’ that follows the behavior of the stock market as a whole, as measured by the Standard & Poor’s 500 index. Joe wants to buy a mutual fund that does not track the index closely. He reads that monthly returns from Fidelity Technology Fund have correlation r = 0.77 with the S&P 500 index and that Fidelity Real Estate Fund has correlation r = 0.37 with the index.

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  1. (a) Which of these funds has the closer relationship to returns from the stock market as a whole? How do you know? (Hint: See page 325.)

  2. (b) Does the information given tell Joe anything about which fund has had higher returns? (Hint: See page 328.)