6.1 Statistics For each situation construct a histogram and use it to answer the question. (For Exercises 1 5, histograms may vary.) 1. Blue Book Values A random sample of thirty-two cars parked at Emmett s Field last June found the following blue book values (in thousands of dollars) (use 6 classes of width 4 starting at 2.5): 15 5 8 17 18 17 8 9 5 13 6 9 5 19 8 13 20 9 7 13 25 16 5 25 25 17 24 10 23 14 15 4 5197 6012 1859 6207 3497 1097 4785 5950 3652 6279 3639 4553 6249 5999 6601 1193 5961 4625 5918 4289 2948 5043 1329 5483 968 5297 1423 5159 5527 4880 Based on your histogram, are more claims grouped at the lower end, in the middle, or at the upper end of the range? 4. Light Bulbs A random sample of thirty-six 75-watt light bulbs found the following lifetimes (number of hours until burn out): Based on your histogram, which two adjacent classes have the highest total number of cars? 2. Football Players A random sample of twenty-four defensive linemen in the East Coast Division II college conference found the following weights (in pounds) (use 8 classes of width 15 starting at 140.5): Based on your histogram, are the weights evenly distributed or more concentrated in one part of the range? 3. Insurance Claims A random sample of thirty accident repair claims received in December at the western division office of the Some States Insurance Company were for the following dollar amounts: 1070 1210 1280 1230 1340 1280 1200 1120 1310 1100 1170 1000 1290 1420 1190 1280 1160 1290 1250 1130 1180 1150 1110 1200 1230 1160 1240 1250 1230 1240 1000 1280 1180 1280 1270 1210 Based on your histogram, are the lifetimes roughly evenly distributed 173 171 211 169 258 213 143 236 or concentrated in one part of the range? 191 163 192 235 196 208 157 221 5. Dining Out A random sample of 174 200 195 198 228 152 188 201 forty dinner checks at the Lotus East Chinese Restaurant last Saturday evening showed the following total costs: 43 104 45 60 25 103 33 96 108 41 67 53 43 48 99 45 35 40 66 103 27 110 60 33 46 126 112 84 42 90 74 46 102 41 48 41 54 44 82 33 Based on your histogram, are the dinner checks grouped more at the lower or the upper end of the cost range?
6.2 Measures of Central Tendency Find the mode, median, and mean of each data set. 1. {10, 7, 9, 13, 11, 14, 15, 14, 8, 9} 2. {11, 19, 20, 11, 12, 18, 11, 20, 16, 18, 17, 10, 13, 11, 15} 3. {12, 5, 10, 6, 7, 6, 15, 7, 6, 5, 6, 5, 13, 10, 7, 13, 8, 11, 12} 4. {11, 8, 23, 21, 14, 16, 15, 11, 9, 22, 15, 11, 12, 11, 10, 15, 10, 25, 18, 6, 11} 75 60 50 55 90 53 65 71 33 81 60 74 71 75 83 53 66 75 47 36 77 54 67 65 8. Cereal Killers The calories per cup of several popular cold cereals, served with half a cup of milk and a peach or half a banana, are given below. Find the mode, the median, and the mean. Looking at the data, can you see why the mean is higher than the median? 5. {10, 13, 8, 15, 8, 12, 12, 12, 14, 8, 12, 11, 14, 9, 8, 14, 14, 14, 10, 10} APPLIED EXERCISES Find the mode, median, and mean for the data in each situation. 6. Car Sales A random sample of fifteen General Motors salesmen at Tri-State Dealers found the following earnings (in dollars) for the first week of September: 1200 1200 1100 1300 900 600 800 700 1200 500 1000 1200 600 900 900 7. Commuting Times A random sample of twenty-four employees at the Grover Smith Forge found the following morning commute times (in minutes) for last Monday: 9. Raising IQs When the mathematician Frederick Mosteller left Princeton for Harvard, he is reputed to have joked that he was raising the average IQ of both places. How is this possible? 10. New York Income New York State s mean income is the fourth highest in the nation, while its median income scores only 29th place, well below that of half of the states. What does this say about the incomes of New Yorkers?
6.3 Measures of Variation Find the range of each data set. 1. {26, 35, 47, 59, 86, 92, 116} 2. {2.0, 3.8, 5.2, 1.6, 4.3, 2.3, 5.4, 3.5, 2.5, 3.1} 3. {121, 115, 147, 163, 171, 116, 167, 147, 169, 131} Make a five-point summary and draw the box-and-whisker plot for each data set. 4. {5, 8, 11, 13, 16, 20, 26} 5. {2, 3, 5, 9, 10, 12, 15, 17, 20} 6. {12, 8, 19, 17, 20, 8, 13, 14, 23, 5, 6, 9, 23, 12, 9} 7. {21, 13, 18, 14, 24, 8, 24, 24, 9, 12, 25, 20, 16, 17, 12} 8. {14, 17, 17, 13, 11, 6, 10, 21, 20, 18, 14, 11, 19, 14, 14} 9. {18, 7, 8, 18, 6, 15, 12, 15, 18, 24, 6, 18, 16, 17, 24} 10. {16, 9, 14, 12, 18, 15, 21, 19, 14, 18, 8, 4, 17, 15, 21, 14, 23, 13, 20, 17} 11. {19, 18, 7, 16, 5, 8, 9, 16, 18, 11, 20, 16, 17, 17, 8, 10, 15, 10, 8, 20} Find the sample standard deviation of each data set. 12. {8, 13, 14, 17, 13} 13. {18, 11, 18, 1, 5, 19} 14. {14, 11, 11, 9, 2, 13} APPLIED EXERCISES In Exercises 15 and 16, analyze the data by finding the range, the five-point summary, and the sample standard deviation and then drawing the box-and-whisker plot. 15. Kitchen Cabinets A random sample of fifteen electric bills at Rick s Always Cooking cabinet factory found the following monthly electricity consumptions (in thousands of kilowatt hours): 10 12 6 15 13 15 12 9 14 6 5 8 9 12 9 16. Malpractice Settlements A random sample of twenty-four malpractice claims against a midwestern health maintenance organization (HMO) that were settled out of court found the following settlement amounts (in tens of thousands of dollars): 16 22 16 18 29 22 30 18 19 28 18 37 17 32 24 31 19 16 16 6 14 15 8 18 17. Income and Education The following five-point summary is based on a survey of annual incomes (in thousands of dollars) for those whose education includes an advanced degree (beyond the bachelor s). Draw the box-andwhisker plot and interpret the results in comparison with the plots in Example 3 on pages 235-236 for people without an advanced degree. 17.4, 30.2, 40.7, 54.4, 89.3 18. Home Run Records In 1998, a baseball record that had stood for 37 years, Roger Maris's record of 61 home runs in a season, was broken
by Sammy Sosa (with 66) and by Mark McGwire (with 70). To compare their records, make a fivepoint summary and draw the boxand-whisker plot for the following lists of home runs per season for each player during their major league careers. Interpret the results. 20. Process Control Suppose that ten samples of size 3 taken from a manufacturing process provided the following data. 19. Fuel Efficiency The following tables list the fuel economy in miles per gallon for model year 2003 family sedans and for sport utility vehicles (SUVs). For each, make a five-point summary and draw the box-and-whisker plot. Give an interpretation of the results. a. Find the upper and lower control limits for the sample mean (use k 5 2). b. If a later sample were {5, 6, 7}, would the process be under control? c. If a later sample were {7, 7, 6}, would the process be under control? 21. Process Control Suppose that ten samples of size 3 taken from a manufacturing process provided the following data. a. Find the upper and lower control limits for the sample mean (use k 5 2). b. If a later sample were {4, 4, 6}, would the process be under control? c. If a later sample were {5, 6, 5}, would the process be under control? Source: Consumer Reports
6.4 Normal Distributions and Binomial Approximation Let X be a normal random variable with mean µ = 12 and standard deviation σ = 3. Find each probability as an area under the normal curve. 1. P(6 # X # 18) 2. P(0 # X # 24) 3. P(9 # X # 12) 4. P(15 # X # 16) 5. P(7 # X # 10) 6. P( 6 1 X 11 2 ) 7. P( 10 1 X 14 2 ) Find the z-score corresponding to each x- value. 8. x = 20 with µ= 15 and σ= 5 9. x = 10 with µ= 10 and σ= 3 10. x = 6 with µ= 6 and σ= 5 11. x = 6 with µ= 8 and σ= 2 12. x = 15 with µ= 20 and σ= 1 13. x = 6 with µ= 8 and σ= 1 14. x = 130 with µ= 100 and σ= 15 15. x = 90 with µ= 100 and σ= 10 Use the normal approximation to the binomial random variable X with n = 20 and p = 0.7 to find each probability. 16. P(X 5 12) 17. P(11 # X # 12) 18. P(8 # X # 10) 19. P(3 # X # 8) 20. P(13 # X # 19) 21. P(2 # X # 7) 22. P(0 # X # 20) 23. P(14 # X # 20) APPLIED EXERCISES Answer each question using an appropriate normal probability. 24. Advertising The percentage of early afternoon television viewers who watch episodes of As the World Spins is a normal random variable with mean 22% and standard deviation 3%. The producers guarantee advertisers that the viewer percentage will be better than 20% or else the advertisers will receive free air time. What is the probability that the producers will have to give free air time after any particular episode? 25. SAT Scores The scores at Centerville High School on last year s mathematics SAT test were approximately normally distributed with mean 490 and standard deviation 140. What proportion of the scores were between 550 and 750? 26. Coin Tosses A fair coin is tossed fifty times. What is the probability that it comes up heads at least thirty times? 27. Actuarial Exams The probability of
passing the first actuarial examination is 60%. If 940 college students take the exam in March, what is the probability that the number that pass is between 575 and 725? 28. Flu Shots Although getting a flu shot in October is a good way of planning ahead for the winter flu season, approximately 1% of those getting the shots develop side effects. If 800 students at Micheles College get flu shots, what is the probability that between 6 and 12 of them will develop side effects? 29. Real Estate Nationwide, 40% of the sales force at the Greener Pastures real estate franchises are college graduates. If 200 of these sales personnel are chosen at random, what is the probability that between 85 and 100 of them are college graduates? 30. Old Age If 3% of the population lives to age 90, what is the probability that between 10 and 15 of the 280 business majors at Henderson College will live to be that old?
Chapter 6 Review Exercises 6.1 Random Samples and Data Organization Construct a bar chart of the data in each situation. 1. Homicides A random sample of twenty homicides investigated last year by the Macon county Coroner's Office lists the causes as 12 by handguns, 3 by firearms other than handguns, 2 by knives, 2 by blunt instruments, and 1 by use of hands. Construct a histogram of the data in each situation. 2. Checking Accounts A random sample of thirty checking account balances on March 2 at the Lobsterman Trust Company branch in Bangor found that the following amounts (rounded to the nearest dollar):
6.2 Measures of Central Tendency Find the mode, median, and the mean of each data set. 3. {5, 6, 15, 13, 15, 6, 11, 13, 13, 6, 13, 6, 8} 4. {12, 13, 13, 5, 9, 14, 13, 8, 9, 14} 5. Stock Earnings The third-quarter earnings per share for twelve randomly selected textile stocks on the New York Stock Exchange were (in dollars): 12 8 8 11 4 7 8 7 5 13 10 9 6. Personal Calls A random sample of the telephone records for fifteen office clerks at the southern regional office of the Marston Glassware Products Company counted the following numbers of personal calls last week: 4 15 23 26 2 22 25 4 27 18 30 28 11 20 6
6.3 Measures of Variation 7. Find the range of the data {38, 28, 32, 43, 41, 25, 35, 37, 43, 36}. 8. Find the five-point summary and draw the box-and-whisker plot for the data {4, 5, 7, 9, 10, 13, 14, 15, 20}. 9. Find the sample standard deviation of the data {5, 7, 9, 10, 13, 16}. 10. Mortgages A random sample of twenty mortgages filed last month at the Seaford Town Clerk office had the following values (in the thousands of dollars): 132 141 154 143 115 141 125 132 123 132 119 126 123 114 126 132 128 131 136 131
6.4 Normal Distributions and Binomial Approximation Let X be a normal random variable with mean µ = 50 and the standard deviation σ = 10. Find each probability. 11. P(55 X 75) Find the z-score corresponding to each x- value. 12. x = 18 with µ = 24 and σ = 1 Use the normal approximation to the binomial random variable X with n = 25 and p = 0.35 to find each probability. 13. PX= ( 14) For Exercises 14-16, answer each question using an approximate normal distribution. 14. Heights Women's heights are approximately normally distributed with mean 63.2 inches and standard deviation 2.6 inches. Find the proportion of women with heights between 62 inches and 69 inches. 15. Lost Luggage During the Christmas rush, the chance that an airline will lose your suitcase increases to 1%. What is the probability that between 4 and 8 of the 690 suitcases checked in at the Bellemeade Regional Airport this holiday season will be lost? 16. Heart Disease The leading cause of death in the United States is heart disease, which causes 32% of all deaths. In a random sample of 150 death certificates, what is the probability that between 50 and 60 list heart disease as the cause of death?