Interpreting percentiles (a) Mrs. Munson is concerned

Interpreting percentiles (a) Mrs. Munson is concerned about how her daughters height and weight compare with those of other girls her age. She uses an online calculator to determine that her daughter is at the 87th percentile for weight and the 67th percentile for height. Explain to Mrs. Munson what this means. (b) According to the Los Angeles Times, speed limits on California highways are set at the 85th percentile of vehicle speeds on those stretches of road. Explain to someone who knows little statistics what that means.

Comparing performance: percentiles Peter is a star runner on the track team, and Molly is one of the best sprinters on the swim team. Both athletes qualify for the league championship meet based on their performance during the regular season. (a) In the track playoff s, Peter records a time that would fall at the 80th percentile of all his race times that season. But his performance places him at the 50th percentile in the league championship meet. Explain how this is possible. (b) Molly swims a bit slowly for her in the league swim meet, recording a time that would fall at the 50th percentile of all her meet times that season. But her performance places Molly at the 80th percentile in this event at the league meet. Explain how this could happen.

Measuring bone density Individuals with low bone density have a high risk of broken bones (fractures). Physicians who are concerned about low bone density (osteoporosis) in patients can refer them for specialized testing. Currently, the most common method for testing bone density is dual-energy X-ray absorptiometry (DEXA). A patient who undergoes a DEXA test usually gets bone density results in grams per square centimeter (g/cm2 ) and also in standardized units. Francine, who is 25 years old, has her bone density measured using DEXA. Her results indicate a bone density in the hip of 948 g/cm2 , which converts to a standardized score of z 5 21.45. In the reference population of 25-year-old women like Francine,2 the mean bone density in the hip is 956 g/cm2 . (a) Francine has not taken a statistics class in a few years. Explain to her in simple language what the standardized score tells her about her bone density. (b) Use the information provided to calculate the standard deviation of bone density in the reference population.

Comparing bone density Refer to the previous exercise. One of Francines friends, Louise, has the bone density in her hip measured using DEXA. Louise is 35 years old. Her bone density is also reported as 948 g/cm2 , but her standardized score is z 5 0.50. Th e mean bone density in the hip for the reference population of 35-year-old women is 944 g/cm2 . (a) Whose bones are healthierFrancines or Louises? Justify your answer. (b) Calculate the standard deviation of bone density in Louises reference population. How does this compare with your answer to Exercise 3.3(b)? Does this make sense to you? Exercises 3.5 and 3.6 refer to information found in Table 3.1 and Figure 3.2 about the salaries of the 2008 World Champion Philadelphia Phillies baseball team. Table 3.1 shows the salaries for each member of the Phillies baseball team on the opening day of the 2008 season. Figure 3.2 gives a dotplot and summary statistics for the salary data. Table 3.1 Opening-Day Salaries for the Philadelphia Phillies, 2008 Player Salary ($) Player Salary ($) Player Salary ($) Burrell, Pat 14,250,000 Romero, J. C. 3,250,000 Kendrick, Kyle 445,000 Howard, Ryan 10,000,000 Feliz, Pedro 3,000,000 Dobbs, Greg 440,000 Myers, Brett 8,583,333 Helms, Wes 2,400,000 Ruiz, Carlos 425,000 Rollins, Jimmy 8,000,000 Werth, Jayson 1,700,000 Condrey, Clayton 420,000 Eaton, Adam 7,958,333 Madson, Ryan 1,400,000 Coste, Chris 415,000 Utley, Chase 7,785,714 Durbin, Chad 900,000 Rosario, Francisco 395,000 Lidge, Brad 6,350,000 Taguchi, So 900,000 Zagurski, Mike 392,500 Moyer, Jamie 6,000,000 Bruntlett, Eric 600,000 Lahey, Tim 390,000 Gordon, Tom 5,500,000 Hamels, Cole 500,000 Mathieson, Scott 390,000 Jenkins, Geoff 5,000,000 Victorino, Shane 480,000 Source: USA Today online salary data base, http://content.usatoday.com/sports/baseball/salaries/default.aspx.

Baseball salaries, I Brad Lidge played a crucial role as the Phillies closer; that is, he pitched the end of many games throughout the season. (a) Find the percentile corresponding to Lidges salary. Explain what this value means. (b) Find the z-score corresponding to Lidges salary. Explain what this value means.

Baseball salaries, II Did Ryan Madson have a high salary or a low salary compared with the rest of the team? Justify your answer using Madsons percentile and z-score.

A uniform distribution Figure 3.7 (on the facing page) shows the density curve for a uniform distribution. Th is curve has height 1 over the interval from 0 to 1 and is zero outside that range. Use this density curve to answer the following questions. (a) Why is the height of the curve equal to 1? (b) Recall that the mean of a density curve is the balance point. What is the value of the mean of this uniform distribution? (c) What is the median? What property of this density curve tells you that the mean and median are related? (d) What percent of the observations lie between 0 and 0.4? Explain.

Another uniform distribution Refer to the previous exercise. Now lets consider a uniform distribution over the interval from 0 to 2. (a) Sketch a graph of the density curve. What is the height of the curve? Why? (b) What percent of the observations lie between 1 and 1.4? Explain. (c) Find the median and the quartiles for this distribution. Show your work.

From histogram to density curve, I Copy the distribution in Figure 3.8 onto your paper. Th en sketch a smooth curve that describes the distribution well. Mark your best guess for the mean and median of the distribution.

From histogram to density curve, II Copy the distribution in Figure 3.9 onto your paper. Th en sketch a smooth curve that describes the distribution well. Mark your best guess for the mean and median of the distribution.

Mean and median Figure 3.10 (on the next page) shows two density curves. Briefl y describe the overall shape of each distribution. Two points are marked on each curve to help you locate the mean and the median. For each curve, give the letter that corresponds to (i) the median and (ii) the mean.

Percentiles and density curves Joey told his dad that he scored at the 98th percentile on a national standardized test. Th e scores on this test are approximately Normally distributed. Sketch a Normal density curve and show the approximate location of Joeys score in the distribution.

Finding means and medians Figure 3.11 (on the next page) displays three density curves, each with three points indicated. At which of these points on each curve do the mean and the median fall?

Cholesterol: good or bad? Martin came home very excited aft er a visit to his doctor. He announced proudly to his wife, My doctor says my cholesterol level is at the 90th percentile among men like me. Th at means Im better off than about 90% of similar men. How should his wife, who is a statistician, respond to Martins statement?

Unemployment in the states, I Each month the Bureau of Labor Statistics announces the unemployment rate for the previous month. Unemployment rates are economically important and politically sensitive. Unemployment may vary greatly among the states because types of work are unevenly distributed across the country. Table 3.2 presents the unemployment rates for each of the 50 states in September 2008. Figure 3.11 Th ree density curves: can you locate the mean and median? AB (a) (b) (c) C A B C AB C Table 3.2 Unemployment Rates by State, September 2008 State Percent State Percent State Percent Alabama 5.3 Louisiana 5.2 Ohio 7.2 Alaska 6.8 Maine 5.6 Oklahoma 3.8 Arizona 5.9 Maryland 4.6 Oregon 6.4 Arkansas 4.9 Massachusetts 5.2 Pennsylvania 5.7 California 7.7 Michigan 8.7 Rhode Island 8.8 Colorado 5.2 Minnesota 5.9 South Carolina 7.3 Connecticut 6.1 Mississippi 7.8 South Dakota 3.2 Delaware 4.8 Missouri 6.4 Tennessee 7.2 Florida 6.6 Montana 4.6 Texas 5.1 Georgia 6.5 Nebraska 3.5 Utah 3.5 Hawaii 4.5 Nevada 7.3 Vermont 5.2 Idaho 5.0 New Hampshire 4.1 Virginia 4.3 Illinois 6.9 New Jersey 5.8 Washington 5.8 Indiana 6.2 New Mexico 4.0 West Virginia 4.5 Iowa 4.2 New York 5.8 Wisconsin 5.0 Kansas 4.8 North Carolina 7.0 Wyoming 3.3 Kentucky 7.1 North Dakota 3.6 Source: Bureau of Labor Statistics Web site, www.bls.gov. (a) Make a histogram of these data. Be sure to label and scale your axes. (b) Calculate numerical summaries for this data set. Describe the shape, center, and spread of the distribution of unemployment rates. (c) Determine the percentile for Illinois. Explain in simple terms what this says about the unemployment rate in Illinois relative to the other states. (d) Which state is at the 40th percentile? Calculate the z-score for this state.

Unemployment in the states, II Refer to the previous exercise. Th e December 2000 unemployment rates for the 50 states had a symmetric, singlepeaked distribution with a mean of 3.47% and a standard deviation of about 1%. Th e unemployment rate for Illinois that month was 4.5%. Th ere were 42 states with lower unemployment rates than Illinois. (a) Write a sentence comparing the actual rates of unemployment in Illinois in December 2000 and September 2008. (b) Compare the percentiles for the Illinois unemployment rate in these same two months in a sentence or two. (c) Compare the z-scores for the Illinois unemployment rate in these same two months in a sentence or two.

Male heights Consider the height distribution for 15-year-old males. (a) Find its mean and standard deviation. Show your method clearly. (b) What height would correspond to a z-score of 2.5? Show your work.

Female heights Consider the height distribution for 16-year-old females. (a) Find its mean and standard deviation. Show your method clearly. (b) What height would correspond to a z-score of 1.5? Show your work.

Is Paul tall? Paul is 15 years old and 175 cm tall. (a) Find the z-score corresponding to Pauls height. Explain what this value means. (b) Pauls height puts him at the 75th percentile among 15-year-old males. Explain what this means to someone who knows no statistics.

Is Miranda taller? Miranda is 16 years old and 170 cm tall. (a) Find the z-score corresponding to Mirandas height. Explain what this value means. (b) Mirandas height puts her at the 88th percentile among 16-year-old females. Explain what this means to someone who knows no statistics. (c) Refer to Exercise 3.19. Who is taller relative to their peersMiranda or Paul? Justify your answer.

IQ test scores, I Between what values do the IQ scores of the middle 95% of all rural Midwest seventh-graders lie? Explain.

IQ test scores, II What percent of IQ scores for all rural Midwest seventh-graders are greater than 100? Explain. How does this compare with the percent in our sample?

IQ test scores, III What percent of all students have IQ scores of 144 or higher? Explain. None of the 74 students in our sample school had scores this high. Are you surprised at this? Why?

A Normal curve Estimate the mean and standard deviation of the Normal curve in Figure 3.16.

Horse pregnancies Bigger animals tend to carry their young longer before birth. Th e length of horse pregnancies from conception to birth varies according to a roughly Normal distribution with mean 336 days and standard deviation 3 days. Use the 689599.7 rule to answer the following questions. (a) Almost all (99.7%) horse pregnancies fall in what range of lengths? (b) What percent of horse pregnancies are longer than 339 days? Show your work.

Eggs A truck is loaded with cartons of eggs that weigh an average of 2 pounds each with a standard deviation of 0.1 pound. A histogram of these weights looks very much like a Normal distribution. Use the 689599.7 rule to answer the following questions. (a) What percent of the cartons weigh less than 2.1 pounds? Show your work. (b) What percent of the cartons weigh less than 1.8 pounds? Show your work. (c) What percent of the cartons weigh more than 1.9 pounds? Show your work.

Table A practice, I (a) z is less than 20.37 (b) z is greater than 20.37 (c) z is less than 2.15 (d) z is greater than 2.15

Table A practice, II (a) z is less than 21.58 (b) z is greater than 1.58 (c) z is greater than 20.46 (d) z is less than 0.93

Table A practice, III (a) z is between 21.33 and 1.65 (b) z is between 0.50 and 1.79

Table A practice, IV (a) z is between 22.05 and 0.78 (b) z is between 21.11 and 20.32 For Exercises 3.31 and 3.32, use Table A to fi nd the value z from the standard Normal distribution that satisfi es each of the following conditions. (Use the value of z from Table A that comes closest to satisfying the condition.) In each case, sketch a standard Normal curve with your value of z marked on the axis. Use the Normal Curve applet to check your answers.

Working backward, I (a) Th e 20th percentile of the standard Normal distribution. (b) 45% of all observations are greater than z.

Working backward, II (a) Th e 63rd percentile of the standard Normal distribution. (b) 75% of all observations are greater than z.

Tiger prowls Aft er hitting plenty of balls on the practice range, Tiger Woods heads out to the fi rst tee to begin a golf tournament. A large creek crosses the fairway 317 yards from the tee. Assume that the distance traveled by Tigers ball follows a Normal distribution with m  304 yards and s  8 yards, as in Example 3.10. Is Tiger safe hitting his driver? Show your work.

High IQ scores Scores on the Wechsler Adult Intelligence Scale for 20- to 34-year-olds are approximately Normally distributed with mean 110 and standard deviation 25. How high must a person score to be in the top 25% of all scores? Show your work.

SAT scores, I Th e average performance of females on the SAT, especially the Math section, is lower than that of males. Th e reasons for this gender gap are controversial. In 2007, female scores on the SAT Math test followed a Normal distribution with mean 500 and standard deviation 111. Male scores had a mean of 533 and a standard deviation of 118. What percent of females scored higher than the male mean? Show your work.

SAT scores, II Refer to the previous exercise. (a) Find the 85th percentile of the SAT Math score distribution for males. Show your work. (b) To what percentile in the female score distribution does your answer to (a) correspond? Show your work.

Potatoes Bags of potatoes in a shipment averaged 10 pounds with a standard deviation of 0.5 pounds. A histogram of these weights followed a Normal curve quite closely. (a) What percent of the bags weighed less than 10.25 pounds? Show your work. (b) What percent weighed between 9.5 and 10.25 pounds? Show your work.

Brush your teeth Th e amount of time Ricardo spends brushing his teeth follows a Normal distribution with unknown mean m and standard deviation s 5 20 seconds. Ricardo spends less than 60 seconds brushing his teeth about 40% of the time. Use this information to determine the mean of this distribution. Show your method clearly.

Heights of young women Th e distribution of the heights of young women aged 18 to 24 is approximately Normal with mean 65 inches and standard deviation 2.5 inches. Sketch a picture of a Normal curve and then use the 689599.7 rule to show what the rule states about these womens heights.

Normal curve properties, I Figure 3.26 is a Normal density curve. Estimate the mean and the standard deviation of this distribution.

Normal curve properties, II Explain why the point that is one standard deviation below the mean in a Normal distribution is always the 16th percentile. Explain why the point that is two standard deviations above the mean is the 97.5th percentile.

More Table A practice, I Use Table A to fi nd the proportion of observations from a standard Normal distribution that satisfi es each of the following statements. In each case, sketch a standard Normal curve and shade the area under the curve that is the answer to the question. Use an applet or your calculator to check your answers. (a) z . 21.81 (b) z , 2.29 (c) 21.81 # z # 20.47 (d) 21.02 # z # 0.65

More Table A practice, II Use Table A to fi nd the value z from a standard Normal distribution that satisfi es each of the following conditions. In each case, sketch a standard Normal curve with your value of z marked on the axis. Use an applet or your calculator to check your answers. (a) Th e point z with 32% of the observations falling to its left . (b) Th e point z with 40% of the observations falling to its right. (c) Th e 10th percentile of the standard Normal distribution. (d) Th e 83rd percentile of the standard Normal distribution.

NCAA rules for athletes The National Collegiate Athletic Association (NCAA) requires Division I athletes to score at least 820 on the combined Mathematics and Critical Reading parts of the SAT exam in order to compete in their first college year. (Higher scores are required for students with poor high school grades.) In 2007, the combined scores of the millions of students taking the SATs were approximately Normal with mean 1017 and standard deviation 211. What percent of all students had scores less than 820? Show your work.

Are we getting smarter? When the Stanford-Binet IQ test came into use in 1932, it was adjusted so that scores for each age group of children followed roughly the Normal distribution with mean 100 and standard deviation 15. Th e test is readjusted from time to time to keep the mean at 100. If present-day American children took the 1932 Stanford-Binet test, their mean score would be about 120. Th e reasons for the increase in IQ over time are not known but probably include better childhood nutrition and more experience in taking tests.5 (a) IQ scores above 130 are oft en called very superior. What percent of children had very superior scores in 1932? (b) If present-day children took the 1932 test, what percent would have very superior scores? (Assume that the standard deviation 15 does not change.)

Get smart Refer to the previous exercise. What IQ score was at the 98th percentile in 1932? Show your work.

Textbook costs Students taking a college introductory statistics class reported spending an average of $305 on textbooks that quarter with a standard deviation of $90. Heres a rough sketch of a Normal density curve that fi tted the histogram well: (a) Approximately what percent of the students spent between $215 and $395 on textbooks that quarter? Explain how you got this answer. (b) One student spent $287 on textbooks. What was her standard score? What percent of the students spent less than she did on textbooks that quarter?

Standardized test scores as percentiles Joey received a report that he scored in the 97th percentile on a national standardized math test but in the 72nd percentile on the reading portion of the test. 138 CHAPTER 3 Modeling Distributions of Data (a) Explain to Joeys grandmother, who knows little statistics, what these numbers mean. (b) Can we determine Joeys z-scores for his math and reading performance? Why or why not?

Finding areas Use Table A to fi nd the proportion of observations from a standard Normal distribution that falls in each of the following regions. In each case, sketch a standard Normal curve and shade the area representing the region. Use an applet or your calculator to check your answers. (a) z $ 22.25 (b) 22.25 , z , 1.77 (c) z , 2.89 (d) 1.44 , z , 2.89

Finding z-scores Use Table A to fi nd the value z from a standard Normal distribution that satisfi es each of the following conditions. In each case, sketch a standard Normal curve with your value of z marked on the axis. Use an applet or your calculator to check your answers. (a) Th e point z with 70% of the observations falling below it. (b) Th e point z such that 90% of all observations are greater than z. (c) Th e 46th percentile of the standard Normal distribution.

Low-birth-weight babies Researchers in Norway analyzed data on the birth weights of 400,000 newborns over a 6-year period. Th e distribution of birth weights is approximately Normal with a mean of 3668 grams and a standard deviation of 511 grams.6 Babies that weigh less than 2500 grams at birth are classifi ed as low birth weight. (a) What percent of babies will be identifi ed as low birth weight? Show your work. (b) Find the quartiles of the birth weight distribution. Show your work.

Bacteria in milk A study of bacterial contamination in milk counted the number of coliform bacteria per milliliter in 100 specimens of milk purchased in East Coast grocery stores. Th e U.S. Public Health Service recommends no more than 10 coliform bacteria per milliliter. Here are the data: 5 8 6 7 8 3 2 4 7 8 6 4 4 8 8 8 6 10 6 5 6 6 6 6 4 3 7 7 5 7 4 5 6 7 4 4 4 3 5 7 7 5 8 3 9 7 3 4 6 6 8 7 4 8 5 7 9 4 4 7 8 8 7 5 4 10 7 6 6 7 8 6 6 6 0 4 5 10 4 5 7 9 8 9 5 6 3 6 3 7 1 6 9 6 8 5 2 8 5 3 (a) Enter the data into your calculator and plot a histogram. Does the distribution of coliform counts appear to be approximately Normal? (b) Calculate the mean and standard deviation for the distribution. (c) What percent of the observations fall within one, two, and three standard deviations of the mean?

Density curves Figure 3.27 (on the facing page) shows density curves of two diff erent shapes. Briefl y describe the overall shape of each distribution. Two or three points are marked on each curve. Th e mean and the median are among these points. For each curve, which point is the median and which is the mean?

Helmet sizes Th e army reports that the distribution of head circumference among soldiers is approximately Normal with mean 22.8 inches and standard deviation 1.1 inches. Helmets are mass-produced for all except the smallest 5% and the largest 5% of head sizes. Soldiers in the smallest or largest 5% get custom-made helmets. What head sizes get custom-made helmets? Show your work.

Th e stock market Th e annual rate of return on stock indexes (which combine many individual stocks) is very roughly Normal. Since 1945, the Standard & Poors 500 index has had a mean yearly return of 12%, with a standard deviation of 16.5%. Take this Normal distribution to be the distribution of yearly returns over a long period. (a) In what range do the middle 95% of all yearly returns lie? Explain. (b) Th e market is down for the year if the return on the index is less than zero. In what proportion of years is the market down? Show your work. (c) In what proportion of years does the index gain between 15% and 25%? Show your work.

Do you sudoku? In the chapter opening story (page 101), one of the authors played an online game of sudoku. At the end of his game, this graph was displayed. Th e density curve shown was constructed from a histogram of times from 4,000,000 games played in one week at this Web site. (a) How would you describe the shape of the density curve? Explain why this shape makes sense in this setting. (b) Use what you have learned in this chapter to describe the authors performance in a few sentences.