Prisons 2013 A report from the U.S. Department of Justice

Problem 58E Boomtowns 2011 In 2011, the website NewGeography .com listed its ranking of the best cities for job growth in the United States. Here is the magazine’s top 20 larger cites, along with their weighted job rating index. a) Make a suitable display of the weighted growth indices. b) Summarize the typical growth index among these cities with a median and mean. Why do they differ? c) Given what you know about the distribution, which of the measures in part b does the better job of summarizing the growth indices? Why? d) Summarize the spread of the growth index distribution with a standard deviation and with an IQR. e) Given what you know about the distribution, which of the measures in part d does the better job of summarizing the growth rates? Why? f) Suppose we subtract from each of the preceding growth rates the average U.S. large-city growth index of 49.23%, so that we can look at how much these indices exceed the U.S. rate. How would this change the values of the summary statistics you calculated above? (Hint: You need not recompute any of the summary statistics from scratch.) g) If we were to omit Austin-Round Rock-San Marcos, TX from the data, how would you expect the mean, median, standard deviation, and IQR to change? Explain your expectations for each. h) Write a brief report about all of these growth indices.

Problem 57E Math scores 2013 The National Center for Education Statistics (nces.ed.gov/nationsreportcard/) reports average mathematics achievement scores for eighth graders in all 50 states: a) Find the median, the IQR, the mean, and the standard deviation of these state averages. b) Which summary statistics would you report for these data? Why? c) Write a brief summary of the performance of eighth graders nationwide.

Problem 55E Zip codes Holes-R-Us, an Internet company that sells piercing jewelry, keeps transaction records on its sales. At a recent sales meeting, one of the staff presented a histogram of the zip codes of the last 500 customers, so that the staff might understand where sales are coming from. Comment on the usefulness and appropriateness of the display.

Problem 56E Zip codes revisited Here are some summary statistics to go with the histogram of the zip codes of 500 customers from the Holes-R-Us Internet Jewelry Salon that we saw in Exercise: Exercise Zip codes Holes-R-Us, an Internet company that sells piercing jewelry, keeps transaction records on its sales. At a recent sales meeting, one of the staff presented a histogram of the zip codes of the last 500 customers, so that the staff might understand where sales are coming from. Comment on the usefulness and appropriateness of the display. Count 500 Mean 64,970.0 StdDev 23,523.0 Median 64,871 IQR 44,183 Q1 46,050 Q3 90,233 What can these statistics tell you about the company’s sales?

Population growth 2010 The following data show the percentage change in population for the 50 states and the District of Columbia from the 2000 census to the 2010 census. Using appropriate graphical displays and summary statistics, write a report on the percentage change in population by state.

Problem 54E Final grades revisited After receiving many complaints about his final-grade histogram from students currently taking a Statistics course, the professor from Exercise distributed the following revised histogram: a) Comment on this display. ________________ b) Describe the distribution of grades. Exercise Final grades A professor (of something other than Statistics!) distributed the following histogram to show the distribution of grades on his 200-point final exam. Comment on the display.

Problem 60E Prisons 2013 A report from the U.S. Department of Justice (www.ojp.usdoj.gov/bjs/) reported the percent changes in federal prison populations in 20 northeastern and midwestern states from 2000 to 2012. Using appropriate graphical displays and summary statistics, write a report on the changes in prison populations.

Problem 1E Details Here are histograms for four manufactured sets of numbers. The histograms look rather different, but all four sets have the same 5-number summary, so the boxplots for all four sets are identical to the one shown. Using these plots as examples, explain some limitations in using boxplots to describe the shape of a distribution.

Problem 3E Outliers The 5-number summary for the run times in minutes of the 150 highest grossing movies of 2010 looks like this: Are there any outliers in these data? How can you tell?

Opposites In a way, boxplots are the opposite of histograms. A histogram divides the number line into equal intervals and displays the number of data values in each interval. A boxplot divides the data into equal parts and displays the portion of the number line each part covers. These two plots display the number of incarcerated prisoners in each state as of 2012. a) Explain how you could tell, by looking at a boxplot, where the tallest bars on the histogram would be located. b) Explain how both the boxplot and the histogram can indicate a skewed distribution. c) Identify one feature of the distribution that the histogram shows but a boxplot does not. d) Identify one feature of the distribution that the boxplot shows but the histogram does not.

Problem 4E Adoptions The following table shows the mean and median ages in years of children adopted in different states in 2010. In all 50 states, the mean age is higher than the median. What might that imply about the shape of the distribution of ages of adopted children in each state?

Problem 5E Adoptions II Here is a histogram showing the total number of adoptions in each of the 50 states and the District of Columbia. Would you expect the mean number of adoptions or the median number of adoptions to be higher? Why?

Problem 7E Test score spreads The professor in question 6 had also calculated the IQR and standard deviation of the test scores before realizing her mistake. (She had entered the top score as 46 but it was actually 56.) What effect will correcting the error have on the IQR and the standard deviation?

Problem 6E Test score centers After entering the test scores from her Statistics class of 25 students, the instructor calculated the mean and the median of the scores. Upon checking, she discovered that she had entered the top score as 46, but it should have been 56. When she corrects this score, how will the mean and median be affected?

Problem 8E Fuel economy The boxplot shows the fuel economy ratings for 67 model year 2012 subcompact cars. Some summary statistics are also provided. The extreme outlier is the Mitsubishi i-MiEV, an electric car whose electricity usage is equivalent to 112 miles per gallon. If that electric car is removed from the data set, how will the standard deviation be affected? The IQR?

Problem 13E Thinking about shape Would you expect distributions of these variables to be uniform, unimodal, or bimodal? Symmetric or skewed? Explain why. a) The number of speeding tickets each student in the senior class of a college has ever had. ________________ b) Players’ scores (number of strokes) at the U.S. Open golf tournament in a given year. ________________ c) Weights of female babies born in a particular hospital over the course of a year. ________________ d) The length of the average hair on the heads of students in a large class.

Problem 15E Sugar in cereals The histogram displays the sugar content (as a percent of weight) of 49 brands of breakfast cereals. a) Describe this distribution. ________________ b) What do you think might account for this shape?

Singers The display shows the heights of some of the singers in a chorus, collected so that the singers could be positioned on stage with shorter ones in front and taller ones in back. a) Describe the distribution. b) Can you account for the features you see here?

More shapes Would you expect distributions of these variables to be uniform, unimodal, or bimodal? Symmetric or skewed? Explain why. a) Ages of people at a Little League game. b) Number of siblings of people in your class. c) Pulse rates of college-age males. d) Number of times each face of a die shows in 100 tosses.

Run times One of the authors collected the times (in minutes) it took him to run 4 miles on various courses during a 10-year period. Here is a histogram of the times. Describe the distribution and summarize the important features. What is it about running that might account for the shape you see?

Problem 17E Vineyards The histogram shows the sizes (in acres) of 36 vineyards in the Finger Lakes region of New York. a) Approximately what percentage of these vineyards are under 60 acres? ________________ b) Write a brief description of this distribution (shape, center, spread, unusual features).

Problem 19E Heart attack stays The histogram shows the lengths of hospital stays (in days) for all the female patients admitted to hospitals in New York during one year with a primary diagnosis of acute myocardial infarction (heart attack). a) From the histogram, would you expect the mean or median to be larger? Explain. ________________ b) Write a few sentences describing this distribution (shape, center, spread, unusual features). ________________ c) Which summary statistics would you choose to summarize the center and spread in these data? Why?

Super Bowl points How many points do football teams score in the Super Bowl? Here are the total numbers of points scored by both teams in each of the first 46 Super Bowl games: 44, 47, 23, 29, 30, 28, 21, 30, 22, 39, 46, 37, 67, 49, 37, 46, 43, 47, 55, 57, 60, 52, 60, 52, 35, 64, 38, 60, 70, 42, 73, 56, 56, 52, 39, 42, 37, 70, 60, 44, 32, 46, 32, 51, 47, 57, 37 a) Find the median. b) Find the quartiles. c) Write a description based on the 5-number summary

Problem 22E Super Bowl wins 2013 In the Super Bowl, by how many points does the winning team outscore the losers? Here are the winning margins for the first 48 Super Bowl games: 25, 19, 9, 16, 3, 21, 7, 17, 10, 4, 18, 17, 4, 12, 17, 5, 10, 29, 22, 36, 19, 32, 4, 45, 1, 13, 35, 17, 23, 10, 14, 7, 15, 7, 27, 3, 27, 3, 3, 11, 12, 3, 4, 14, 6, 6, 4, 3 a) Find the median. b) Find the quartiles. c) Write a description based on the 5-number summary.

Problem 25E Mistake A clerk entering salary data into a company spreadsheet accidentally put an extra “0” in the boss’s salary, listing it as $2,000,000 instead of $200,000. Explain how this error will affect these summary statistics for the company payroll: a) measures of center: median and mean. ________________ b) measures of spread: range, IQR, and standard deviation.

Problem 26E Sick days During contract negotiations, a company seeks to change the number of sick days employees may take, saying that the annual “average” is 7 days of absence per employee. The union negotiators counter that the “average” employee misses only 3 days of work each year. Explain how both sides might be correct, identifying the measure of center you think each side is using and why the difference might exist.

E-mails A university teacher saved every e-mail received from students in a large Introductory Statistics class during an entire term. He then counted, for each student who had sent him at least one e-mail, how many e-mails each student had sent. a) From the histogram, would you expect the mean or the median to be larger? Explain. b) Write a few sentences describing this distribution (shape, center, spread, unusual features). c) Which summary statistics would you choose to summarize the center and spread in these data? Why?

Tornadoes Here are the annual numbers of deaths from tornadoes in a certain region: 124 90 30 45 565 60 39 44 70 89 115 Find these statistics: a) mean b) median and quartiles c) range and IQR

Problem 23E Summaries Here are costs of nine compact refrigerators rated very good or excellent by Consumer Reports on their website www.consumerreports.org. $ 150 150 160 180 150 140 120 130 120 Find these statistics: a) mean b) median and quartiles c) range and IQR

Problem 27E Standard deviation I For each lettered part, a through c, examine the two given sets of numbers. Without doing any calculations, decide which set has the larger standard deviation and explain why. Then check by finding the standard deviations by hand.

Problem 28E Standard deviation II For each lettered part, a through c, examine the two given sets of numbers. Without doing any calculations, decide which set has the larger standard deviation and explain why. Then check by finding the standard deviations by hand. Set 1 Set 2 a) 4, 7, 7, 7, 10 4, 6, 7, 8, 10 b) 100, 140, 150, 160, 200 10, 50, 60, 70, 110 c) 10, 16, 18, 20, 22, 28 48, 56, 58, 60, 62, 70

Problem 29E Pizza prices The histogram shows the distribution of the prices of plain pizza slices (in $) for 156 weeks in Dallas, TX. Which summary statistics would you choose to summarize the center and spread in these data? Why?

Problem 32E Neck sizes again Look again at the histogram of men’s neck sizes in Exercise 30. a) Is the mean closer to 14, 15, or 16 inches? Why? b) Is the standard deviation closer to 1 inch, 3 inches, or 5 inches? Explain. Exercise 30: Neck size The histogram shows the neck sizes (in inches) of 250 men recruited for a health study in Utah. Which summary statistics would you choose to summarize the center and spread in these data? Why?

Problem 30E Neck size The histogram shows the neck sizes (in inches) of 250 men recruited for a health study in Utah. Which summary statistics would you choose to summarize the center and spread in these data? Why?

Movie lengths 2010 The histogram shows the running times in minutes of the 150 top-grossing feature films released in 2010. a) You plan to see a movie this weekend. Based on these movies, how long do you expect a typical movie to run? b) Would you be surprised to find that your movie ran for \(2 \frac {1}{2}\) hours (150 minutes)? c) Which would you expect to be higher: the mean or the median run time for all movies? Why?

Problem 31E Pizza prices again Look again at the histogram of the pizza prices in Exercise 29. a) Is the mean closer to $2.40, $2.60, or $2.80? Why? b) Is the standard deviation closer to $0.15, $0.50, or $1.00? Explain. Exercise 29: Pizza prices The histogram shows the distribution of the prices of plain pizza slices (in $) for 156 weeks in Dallas, TX. Which summary statistics would you choose to summarize the center and spread in these data? Why?

Movie lengths II 2010 Exercise 33 looked at the running times of movies released in 2010. The standard deviation of these running times is 16.6 minutes, and the quartiles are \(Q_1 = 98\) minutes and \(Q_3 = 116\) minutes. a) Write a sentence or two describing the spread in running times based on i) the quartiles. ii) the standard deviation. b) Do you have any concerns about using either of these descriptions of spread? Explain.

Problem 34E Golf drives 2013 The display shows the average drive distance (in yards) for 155 professional golfers during a week on the men’s PGA tour in 2013. a) Describe this distribution. b) Approximately what proportion of professional male golfers drive, on average, 280 yards or less? c) Estimate the mean by examining the histogram. d) Do you expect the mean to be smaller than, approximately equal to, or larger than the median? Why?

Problem 36E Golf drives II 2013 Exercise 34 looked at distances PGA golfers can hit the ball. The standard deviation of these average drive distances is 11.2 yards, and the quartiles are Q1 = 282.5 yards and Q3 = 295.6 yards. a) Write a sentence or two describing the spread in distances based on i) the quartiles. ii) the standard deviation. b) Do you have any concerns about using either of these descriptions of spread? Explain. Exercise 34: Golf drives 2013 The display shows the average drive distance (in yards) for 155 professional golfers during a week on the men’s PGA tour in 2013. a) Describe this distribution. b) Approximately what proportion of professional male golfers drive, on average, 280 yards or less? c) Estimate the mean by examining the histogram. d) Do you expect the mean to be smaller than, approximately equal to, or larger than the median? Why?

Problem 37E Movie earnings 2013 The histogram shows total gross earnings (in millions of dollars) of the top 200 major release movies in 2013. An industry publication reports that the typical movie makes 29.6 million, but a watchdog group concerned with rising ticket prices says that the average earnings are closer to 49.3 million. What statistic do you think each group is using? Explain.

Problem 38E Cold weather A meteorologist preparing a talk about global warming compiled a list of weekly low temperatures (in degrees Fahrenheit) he observed at his southern Florida home last year. The coldest temperature for any week was 36°F, but he inadvertently recorded the Celsius value of 2°. Assuming that he correctly listed all the other temperatures, explain how this error will affect these summary statistics: a) measures of center: mean and median. b) measures of spread: range, IQR, and standard deviation.

Problem 41E Gasoline 2014 In January 2014, 16 gas stations in eastern Wisconsin posted these prices for a gallon of regular gasoline: a) Make a stem-and-leaf display of these gas prices. Use split stems; for example, use two 3.2 stems—one for prices between $3.20 and $3.24 and the other for prices from $3.25 to $3.29. b) Describe the shape, center, and spread of this distribution. c) What unusual feature do you see?

Problem 39E Payroll A small warehouse employs a supervisor at $1200 a week, an inventory manager at $700 a week, six stock boys at $400 a week, and four drivers at $500 a week. a) Find the mean and median wage. ________________ b) How many employees earn more than the mean wage? ________________ c) Which measure of center best describes a typical wage at this company: the mean or the median? ________________ d) Which measure of spread would best describe the payroll: the range, the IQR, or the standard deviation? Why?

Problem 40E Singers The frequency table shows the heights (in inches) of 130 members of a choir. Height Count Height Count 60 2 69 5 61 6 70 11 62 9 71 8 63 7 72 9 64 5 73 4 65 20 74 2 66 18 75 4 67 7 76 1 68 12 a) Find the median and IQR. ________________ b) Find the mean and standard deviation. ________________ c) Display these data with a histogram. ________________ d) Write a few sentences describing the distribution.

Problem 42E The Great One During his 20 seasons in the NHL, Wayne Gretzky scored 50% more points than anyone who ever played professional hockey. He accomplished this amazing feat while playing in 280 fewer games than Gordie Howe, the previous record holder. Here are the number of games Gretzky played during each season: 79, 80, 80, 80, 74, 80, 80, 79, 64, 78, 73, 78, 74, 45, 81, 48, 80, 82, 82, 70 a) Create a stem-and-leaf display for these data, using split stems. ________________ b) Describe the shape of the distribution. ________________ c) Describe the center and spread of this distribution. ________________ d) What unusual feature do you see? What might explain this?

Problem 43E States The stem-and-leaf display shows populations of the 50 states, in millions of people, according to the 2010 census. a) What measures of center and spread are most appropriate? b) Without doing any calculations, which must be larger: the median or the mean? Explain how you know. c) From the stem-and-leaf display, find the median and the interquartile range. d) Write a few sentences describing this distribution.

Problem 44E Wayne Gretzky In Exercise, you examined the Number of games played by hockey great Wayne Gretzky during his 20-year career in the NHL. a) Would you use the median or the mean to describe the center of this distribution? Why? ________________ b) Find the median. ________________ c) Without actually finding the mean, would you expect it to be higher or lower than the median? Explain. Exercise The Great One During his 20 seasons in the NHL, Wayne Gretzky scored 50% more points than anyone who ever played professional hockey. He accomplished this amazing feat while playing in 280 fewer games than Gordie Howe, the previous record holder. Here are the number of games Gretzky played during each season: 79, 80, 80, 80, 74, 80, 80, 79, 64, 78, 73, 78, 74, 45, 81, 48, 80, 82, 82, 70 a) Create a stem-and-leaf display for these data, using split stems. ________________ b) Describe the shape of the distribution. ________________ c) Describe the center and spread of this distribution. ________________ d) What unusual feature do you see? What might explain this?

Problem 45E A-Rod 2013 Alex Rodriguez (known to fans as A-Rod) was the youngest player ever to hit 500 home runs. Here is a stem-and-leaf display of the number of home runs hit by A-Rod during the 1994–2013 seasons. (www.baseball -reference.com/players/r/rodrial01.shtml.) Describe the distribution, mentioning its shape and any unusual features.

Acid rain Two researchers measured the pH (a scale on which a value of 7 is neutral and values below 7 are acidic) of water collected from rain and snow over a 6-month period in Allegheny County, Pennsylvania. Describe their data with a graph and a few sentences: 4.57 5.62 4.12 5.29 4.64 4.31 4.30 4.39 4.45 5.67 4.39 4.52 4.26 4.26 4.40 5.78 4.73 4.56 5.08 4.41 4.12 5.51 4.82 4.63 4.29 4.60

Problem 49E A-Rod again 2013 Students were asked to make a histogram of the number of home runs hit by Alex Rodriguez from 1994 to 2013 (see Exercise 45). One student submitted the following display (he noted that A Rod had hit no home runs in 1994): a) Comment on this graph. b) Create your own histogram of the data.

Return of the birds 2013 Students were given the assignment to make a histogram of the data on bird counts reported in Exercise 46. One student submitted the following display: a) Comment on this graph. b) Create your own histogram of the data.

Bird species 2013 The Cornell Lab of Ornithology holds an annual Christmas Bird Count (www.birdsource.org), in which bird watchers at various locations around the country see how many different species of birds they can spot. Here are the number of species counted from the 20 sites with the most species in 2013: 184 98 101 126 150 166 82 136 124 118 133 83 86 101 105 97 88 131 128 106 a) Create a stem-and-leaf display of these data. b) Write a brief description of the distribution. Be sure to discuss the overall shape as well as any unusual features.

Problem 48E Horsepower Create a stem-and-leaf display for these horse powers of autos reviewed by Consumer Reports one year, and describe the distribution: 155 103 130 80 65 142 125 129 71 69 125 115 138 68 78 150 133 135 90 97 68 105 88 115 110 95 85 109 115 71 97 110 65 90 75 120 80 70

Problem 47E Major Hurricanes 2013 The following data give the number of hurricanes classified as major hurricanes in the Atlantic Ocean each year from 1944 through 2013, as reported by NOAA (www.nhc.noaa.gov): 3, 3, 1, 2, 4, 3, 8, 5, 3, 4, 2, 6, 2, 2, 5, 2, 2, 7, 1, 2, 6, 1, 3, 1, 0, 5, 2, 1, 0, 1, 2, 3, 2, 1, 2, 2, 2, 3, 1, 1, 1, 3, 0, 1, 3, 2, 1, 2, 1, 1, 0, 5, 6, 1, 3, 5, 3, 4, 2, 3, 6, 7, 2, 2, 5, 2, 5, 4, 2, 0 a) Create a dotplot of these data. b) Describe the distribution.

Marijuana 2007 In 2007 the Council of Europe published a report entitled The European School Survey Project on Alcohol and Other Drugs (www.espad.org). Among other issues, the survey investigated the percentages of 16-year-olds who had used marijuana. Shown here are the results for 34 European countries. Create an appropriate graph of these data, and describe the distribution.

Problem 53E Final grades A professor (of something other than Statistics!) distributed the following histogram to show the distribution of grades on his 200-point final exam. Comment on the display.