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Name:_______________________________________ HW 1-1: Using the Empirical Rule For each problem set, label the normal curve with the appropriate values, and use the curve to answer the questions. 1. The mean score on the midterm was an 82 with a standard deviation of 5. Find the probability that a randomly selected person: a. scored between 77 and 87 b. scored between 82 and 87 c. scored between 72 and 87 d. scored higher than 92 e. scored less than 77 2. The mean SAT score is 490 with a standard deviation of 100. Find the probability that a randomly selected student: a. scored between 390 and 590 b. scored above 790 c. scored less than 490 d. scored between 290 and 490 3. The mean weight of college football players is 200 pounds with a standard deviation of 30. Find the probability that a randomly selected player: a. weighs between 170 and 260 b. weighs less than 170 c. weighs over 290 d. weighs less than 140 e. weighs between 140 and 230 4. The average life of a car tire is 28,000 miles with a standard deviation of 3000. Find the probability that a randomly selected tire will have a life of: a. between 19,000 and 37,000 miles b. less than 25,000 miles c. between 31,000 and 37,000 miles d. over 22,000 miles e. below 31,000 miles Reiland, NCSU 1 Using Z-Scores to Pick a Winner The decathlon is an event in track and field that consists of 10 events. To determine who wins, we have to know whether it is harder to jump an inch higher or run 5 seconds faster. We have to be able to compare two fundamentally different activities involving different units. Standard deviations to the rescue! If we knew the mean performance (by world-class athletes) in each event, and the standard deviation, we could compute how far each performance was from the mean in standard deviation units (that is, the z-scores). So consider the three athletes’ performances shown below in a three event competition. Note that each placed first, second, and third in an event. Competitor A B C mean standard deviation 100 m dash 10.1 sec 9.9 sec 10.3 sec 100 m dash 10 sec 0.2 sec Event Shot put 66’ 60’ 63’ Long Jump 26’ 27’ 27’ 3” Shot Put 60’ 3’ Long Jump 26’ 6” Who gets the gold medal? Who turned in the most remarkable performance of the competition? Explain your reasoning using mathematics. Reiland, NCSU 2 HW 1-2: Practice with Normal Distributions For each question, draw and shade a normal curve, then solve. 1. The percentage impurity of a chemical can be modelled by a normal distribution with a mean of 5.8 and a standard deviation of 0.5. Obtain the probability that a sample of the chemical has percentage impurity between 5 and 6. 2. Melons sold on a market stall have weights that are normally distributed with a mean of 2.18 kg and a standard deviation of 0.25 kg. For a melon chosen at random, find the probability that its weight lies between 2 kg and 2.5 kg. 3. A teacher travels from home to work by car each weekday by one of two routes, X or Y. For route X, her journey times are normally distributed with a mean of 30.4 minutes and a standard deviation of 3.6 minutes. Calculate the probability that her journey time on a particular day takes between 25 minutes and 35 minutes. 4. Soup is sold in tins which are filled by a machine. The actual weight of soup delivered to a tin by the filling machine is always normally distributed about the mean weight with a standard deviation of 8g. The machine is set originally to deliver a mean weight of 810g. (a) Determine the probability that the weight of soup in a tin, selected at random, is less than 800g. (b) Determine the probability that the weight of soup in a tin, selected at random, is between 795 g and 820 g. 5. The weight, X grams, of a particular variety of orange is normally distributed with mean 205 and standard deviation 25. A wholesaler decides to grade such oranges by weight. He decides that the smallest 30 per cent should be graded as small, the largest 20 per cent graded as large, and the remainder graded as medium. Determine, to one decimal place, the maximum weight of an orange graded as: (i) small (ii) medium. 6. Jars of bolognese sauce, sold by a supermarket, are stated to have contents of weight 500 g. The weights, in grams, of the actual contents of jars in a large batch are normally distributed with mean 506 and standard deviation 5. Find the weight which is exceeded by the contents of 99.9% of the jars in this batch. 7. The distance, in kilometers, travelled to work by the employees of a city council may be modelled by a normal distribution with mean 7.5 and standard deviation 2.5. Find d such that 10% of the council’s employees travel less than d kilometers to work. 8. An airline operates a service between Manchester and Paris. The flight time may be modelled by a normal distribution with mean 85 minutes and standard deviation 8 minutes. In order to gain publicity for the service, the airline decides to refund fares when a flight time exceeds q minutes. Find the value of q such that the probability of fares being refunded on a particular flight is 0.001. Reiland, NCSU 3 HW 1-3: Biased versus Unbiased Questions and Sampling Methods Tell whether the question is potentially biased. Explain your answer. If the question is potentially biased, rewrite it so that it is not. 1. Do you think the city should risk an increase in pollution by allowing expansion of the Northern Industrial Park? 2. In a survey about Americans’ interest in soccer, the first 25 people admitted to a high school soccer game were asked, “How interested are you in the world’s most popular sport, soccer?” 3. Don’t you agree that the school needs a new baseball field more than a new science lab? 4. You want to determine whether to serve hamburgers or pizza at a soccer team party. a) Write a survey question that would likely produce biased results. b) Write a survey question that would likely produce unbiased results. 5. You want to find students’ opinions on the current attendance policy. Give two ways that your sample for the survey might be selected. The first must be an example of a biased sample and the second must be an example of an unbiased sample. Thoroughly explain your answers. 6. Two toothpaste manufacturers each claim that 4 out of every 5 dentists use their brand exclusively. Both manufacturers can support their claims with survey results. Explain how this is possible. For each situation, identify the sampling technique used. (simple random, cluster, stratified, convenience, voluntary response, or systematic) 7. Every fifth person boarding a plane is searched thoroughly. 8. At a local community College, five math classes are randomly selected out of 20 and all of the students from each class are interviewed. 9. A researcher randomly selects and interviews fifty male and fifty female teachers. 10. A researcher for an airline interviews all of the passengers on five randomly selected flights. 11. Based on 12,500 responses from 42,000 surveys sent to its alumni, a major university estimated that the annual salary of its alumni was 92,500. 12. A community college student interviews the first 100 students to enter the building to determine the percentage of students that own a car. 13. The names of 70 contestants are written on 70 cards, The cards are placed in a bag, and three names are picked from the bag. 14. Calling randomly generated telephone numbers, a study asked 855 U.S. adults which medical conditions could be prevented by their diet. Reiland, NCSU 4 HW 1-4: Margin of Error, Population Parameters and Sample Statistics Practice A survey of a sample population gathers information from a few people and then the results are used to reflect the opinions of a larger population. The reason that researchers and pollsters use sample population is that it is cheaper and easier to poll a few people rather than everybody. One key to successful surveys of sample populations is finding the appropriate size for the sample that will give accurate results without spending too much time or money. Suppose that 900 American teens were surveyed about their favorite ski category of the 2002 Winter Olympics in Park City, Utah. Ski jumping was the favorite for 20% of those surveyed. This result can be used to predict how many of all 31 million American teens favor ski jumping. How? To determine how accurately the results of surveying 900 American teens truly reflect the results of surveying all 31 million American teens, a margin of error should be given. When pollsters report the margin or error for their surveys, they are stating their confidence mathematically in the data they have collected. The margin of 1 error can be calculated by using the formula 𝑛, where n is the number in a sample size. √ 1 1 For the above sample, the margin of error would be = 30 = 0.03, 𝑜𝑟 3%,. Since the actual statistic could √900 be larger or smaller than the true amount, the margin of error can be expressed as ± 3%. 1. Find the margin of error for a survey of 100 American teens. 2. Compare that to the margin of error for a survey of 90,000 teens. 3. Draw a conclusion about the margin of error based on the size of the sample. Why do you think this is so? 4. If you want to cut your margin of error in half, what would you have to do to the sample size? Why? 5. Find the sample size needed to create a margin of error of 2%. 6. For each statement, identify whether the numbers underlined are statistics or parameters. a. Of all U.S. kindergarten teachers, 32% say that knowing the alphabet is an essential skill. b. Of the 800 U.S. kindergarten teachers polled, 34% say that knowing the alphabet is an essential skill. Reiland, NCSU 5 7. Of the U.S. adult population, 36% has an allergy. A sample of 1200 randomly selected adults resulted in 33.2% reporting an allergy. a. Describe the population. b. What is the sample? c. Describe the variable. d. Identify the statistic and give its value. e. Identify the parameter and give its value. 8. In your own words, explain why the parameter is fixed and the statistic varies. 9. Suppose a 12 year old asked you to explain the difference between a sample and a population, how would you explain it to him/her? How might you explain why you would want to take a sample, rather than surveying every member of the population? 10. In your own words, explain the difference between a statistic and a parameter. 11. Determine whether the numerical value is a parameter or a statistic (and explain): a) A recent survey by the alumni of a major university indicated that the average salary of 10,000 of its 300,000 graduates was 125,000. b) The average salary of all assembly-line employees at a certain car manufacturer is $33,000. c) The average late fee for 360 credit card holders was found to be $56.75. 12. Identify each of the following as a parameter or a statistic. If you need to make an assumption about who or what the population is, explain your assumption. a. The proportion of voters who voted for President Bush in the 2004 election. b. The proportion of voters surveyed by CNN who voted for John Kerry in the 2004 election. c. The proportion of voters among our school's faculty who voted for Ralph Nader in the 2004 election. d. The average number of points scored in a Super Bowl game. Reiland, NCSU 6 HW 1-5: Simulation Practice with Random Number Table Random digit table Line 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 19223 73676 45467 52711 95592 68417 82739 60940 36009 38448 81486 59636 62568 45149 61041 14459 38167 73190 95857 35476 71487 13873 54580 71035 96746 95034 47150 71709 38889 94007 35013 57890 72024 19365 48789 69487 88804 70206 32992 77684 26056 98532 32533 07118 55972 09984 81598 81507 09001 12149 05756 99400 77558 93074 69971 15529 20807 17868 15412 18338 60513 04634 40325 75730 94322 31424 62183 04470 87664 39421 29077 95052 27102 43367 37823 28713 01927 00095 60227 91481 72765 47511 24943 39638 24697 09297 71197 03699 66280 24709 80371 70632 29669 92099 65850 14863 90908 56027 49497 71868 96409 27754 32863 40011 60779 85089 81676 61790 85453 39364 00412 19352 71080 03819 73698 65103 23417 84407 58806 04266 61683 73592 55892 72719 18442 12531 42648 29485 85848 53791 57067 55300 90656 46816 42006 71238 73089 22553 56202 14526 62253 26185 90785 66979 35435 47052 75186 33063 96758 35119 42544 82425 82226 48767 17297 50211 94383 87964 83485 76688 27649 84898 11486 02938 31893 50490 41448 65956 98624 43742 62224 87136 41842 27611 62103 82853 36290 90056 52573 59335 47487 14893 18883 41979 08708 39950 45785 11776 70915 32592 61181 75532 86382 84826 11937 51025 95761 81868 91596 39244 1. A club contains 33 students and 10 faculty members. The students are: Aisen Albrecht Bhagava Bonini Chakra Ding DuFour Dwivedi Gartner Hollensburg Huang Joseph The faculty members are: Beck Burse Brown Diamente Kittaka Kuhn Lee Lipp Lundberg Marshall May MacDonald Neukam Patel Pham Ranaweera Rokop Sommer Stuy Terrell Thomas Thompson Laumeyer Mitchener Nesbitt Sohalski Twining Wernke Thyen Wang Yerant The club can send four students and two faculty members to a convention and decides to choose those who will go by random selection. Reiland, NCSU 7 a. Use the random digit table above to choose a sample of 4 students. Start on line 109. List the numbers and the names below. b. Use the random digit table to choose a sample of 2 faculty members. Start on line 112. List the numbers and the names below. 2. You are a marketing executive for a clothing company. Choose a SRS of 10 of the 440 retail outlets in New York that sell your company's products. Describe how you would label the retail outlets and select your sample, starting on line 102. 3. Your school will send a delegation of 35 seniors to a student life convention. 200 girls and 150 boys are eligible to be chosen. If a sample of 20 girls and a separate sample of 15 boys are each selected randomly, it gives each senior the same chance to be chosen to attend the convention. Beginning at line 105 in the random digits table below, select the first four senior girls to be in the sample. Explain your procedures clearly. 4. Five boxes, each containing 24 cartons of strawberries, are delivered in a shipment to a grocery store. The produce manager always selects a few cartons randomly to inspect. He knows better than to just look at some of the cartons on the top or only in one box, because sometimes the rotten ones are on the bottom. Today he wishes to select a total of 6 cartons to inspect. He has the boxes arranged in order and has a set way to count the cartons inside each box. Explain the process used to make the random selection using a random digit table. 5. Five of the employees at the Stellar Boutique are going to be selected to go to training in Las Vegas for four days. Everyone wants to go of course, so the owner has decided to make the selection randomly. She has decided to send two managers and three sales representatives. The employees' names are listed in the table below. Managers Angela Barbara Elise Gigi Malena Rosie Tammie Veronica Sales Representatives Alfie Betty Carrie Cathy Darcy Fred Heidu Sales Representatives Irma Joe Katarina Lynn Marcie Nancy Orville Sales Representatives Ray Sandy Shirley Suzi Tawny Wendy Zoe Explain the process she can follow to use a random digit table to select the employees who will get to go to the training. Select the managers first, then select the sales representatives using the random number table starting on line 116. Reiland, NCSU 8 7. Use a random digit table, starting on line 112, to select an SRS of five of the fifty U.S. States. Explain your process thoroughly and report the five states that you chose. Repeat this a second time, but begin on a different line on the random digit table. Compare your lists to another classmate's lists. Did you end up with any of the same states in your samples? Alabama California Florida Illinois Kentucky Massachusetts Missouri New Hampshire North Carolina Oregon South Dakota Vermont Wisconsin Alaska Colorado Georgia Indiana Louisiana Michigan Montana New Jersey North Dakota Pennsylvania Tennessee Virginia Wyoming Arizona Connecticut Hawaii Iowa Maine Minnesota Nebraska New Mexico Ohio Rhode Island Texas Washington Arkansas Delaware Idaho Kansas Maryland Mississippi Nevada New York Oklahoma South Carolina Utah West Virginia 8. Washington High School has had some recent problems with students using steroids. The district decides that it will randomly test student athletes for steroids and other drugs. The boy's hockey team is to be tested. There are 13 players on the varsity team and 21 players on the junior varsity team. Use a table of random digits starting at line 122, to choose a stratified random sample of 3 varsity players and 5 junior varsity players to be tested. Remember to clearly describe your process. 10. At the start of this season, Major League Baseball fans were asked which American League Central team would be most likely to win the division this year. The table below gives the results of the poll. Most Likely to Win AL Central Team Chicago Cleveland Detroit Kansas City Minnesota Probability 0.14 0.23 0.33 0.02 0.28 Using the random digit table, starting on line 117, simulate the results when asking 10 fans who they think will win the AL Central. Reiland, NCSU 9 HW 1-6: Expected Value and Fair Games 1. The outcomes of a game of chance are determined by flipping three coins. The payouts for each trial are below. Option #1 Outcome All 3 coins are heads exactly 2 coins are heads any other result Payout Win $60 Win $5 lose $20 Option #2 Outcome All 3 coins are heads exactly 2 coins are heads any other result Payout Win $45 Win $20 lose $25 Option #3 Outcome All 3 coins are heads exactly 2 coins are heads any other result Payout Win $30 Win $10 lose $15 You must play this game 40 times. Choose the option you think will win the most money. State your option and explain why you are choosing this option below. I am choosing option # __________ Reason: 2. You roll a 6-sided die and win points or lose points as follows: Die shows a 1, 2 or 3: Die shows a 4 or 5: Die shows a 6: +10 points - 13 points - 1 point a. How many points can you expect to end up with on average (expected value)? b. Is this a fair game? Why or why not? 3. Two thousand tickets were sold for $25 each to benefit the Lake Zurich Baseball Association. The grand prize was a $5000 family trip to Florida. 2nd prize was $2500 cash. 3rd prize was $1000 cash. Find the expected value. Reiland, NCSU 10 4. If the sum of two rolled dice is 8 or more, you win $2; if not you lose $2. a. Is this a fair game? Why or why not? b. To have a fair game, the $2 winnings should instead be what amount? 5. An 18-year old student must decide whether to spend $160 for one year’s car collision damage insurance. The insurance carries a $100 deductible, which means that when the student files a damage claim, the student must pay $100 of the damage amount, with the insurance company paying the rest (up to the value of the car). Because the car is only worth $1500, the student consults with an insurance agent who presented him with the following table of damage amounts and probabilities based on the driving records for 18-year olds in the region. Event Payoff Probability Accident costing $1500 $1400 0.05 Accident costing $1000 $900 0.02 Accident costing $500 $400 0.03 No Accident $0 0.90 a. What is the expected value of this insurance? What does this mean? b. Should the student buy the insurance? 6. Throw a die. If you win $2 when the number is even and lose $1 when the number is odd, what is the expected value? If you pay $1 to play the game, will you win in the long run? 7. A company has a choice of three marketing strategies. The first will cost $150,000 and has a 40% chance of $1,500,000 in profits and a 60% chance of $500,000 in profits. The second strategy will cost $50,000 and has a 20% chance of $1,000,000 in profits and an 80% chance of $600,000 in profits. The third strategy will cost $80,000 and has a 50% chance of $1,000,000 in profits and a 50% chance of $400,000 in profits. Which is the best strategy? Reiland, NCSU 11