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Stats Data And Models 4th Global Edition Richard D. De Veaux, Paul Velleman, David E. Bock - Solutions
Web visitors A website manager has noticed that during the evening hours, about 4 people per minute check out from their shopping cart and make an online purchase.She believes that each purchase is independent of the others and wants to model the number of purchases per minute.a) What model might
New bow, again The archer in Exercise 56 continues shooting arrows, ending up with 45 bull’s-eyes in 50 shots. Now are you convinced that the new bow is better?Explain.
Hotter hand The basketball player in Exercise 55 has new sneakers, which he thinks improve his game. Over his past 60 shots, he’s made 32—much better than the 50% he usually shoots. Do you think his chances of making a shot really increased? In other words, is making at least 32 of 60 shots
New bow The archer in Exercise 30 purchases a new bow, hoping that it will improve her success rate to more than 80% bull’s-eyes. She is delighted when she first tests her new bow and hits 6 consecutive bull’s-eyes. Do you think this is compelling evidence that the new bow is better? In other
Hot hand A basketball player who ordinarily makes about 50% of his free throw shots has made 6 in a row.Is this evidence that he has a “hot hand” tonight? That is, is this streak so unusual that it means the probability he makes a shot must have changed? Explain.
ESP Scientists wish to test the mind-reading ability of a person who claims to have ESP. They use five cards with different and distinctive symbols (square, circle, triangle, line, squiggle). Someone picks a card at random and thinks about the symbol. The “mind reader” must correctly identify
Rickets Vitamin D is essential for strong, healthy bones. Our bodies produce vitamin D naturally when sunlight falls upon the skin, or it can be taken as a dietary supplement. Although the bone disease rickets was largely eliminated in England during the 1950s, some people there are concerned that
Seatbelts II Police estimate that 80% of drivers now wear their seatbelts. They set up a safety roadblock, stopping cars to check for seatbelt use.a) How many cars do they expect to stop before finding a driver whose seatbelt is not buckled?b) What’s the probability that the first unbelted driver
Earthquakes Suppose the probability of a major earthquake on a given day is 1 out of 20,000.a) What’s the expected number of major earthquakes in the next 2000 days?b) Use the Poisson model to approximate the probability that there will be at least one major earthquake in the next 2000 days.
TB, again In Chapter 14 we saw that the probability of contracting TB is small, with p about 0.0005 for a new case in a given year. In a town of 6000 people:a) What’s the expected number of new cases?b) Use the Poisson model to approximate the probability that there will be at least one new case
Bank tellers I am the only bank teller on duty at my local bank. I need to run out for 10 minutes, but I don’t want to miss any customers. Suppose the arrival of customers can be modeled by a Poisson distribution with mean 2 customers per hour.a) What’s the probability that no one will arrive
Hurricanes, redux We first looked at the occurrences of hurricanes in Chapter 3 (Exercise 47). Suppose we find and the arrivals can be modeled by a Poisson distribution with mean 2.45.a) What’s the probability of no hurricanes next year?b) What’s the probability that during the next two years,
The euro Shortly after the introduction of the euro coin in Belgium, newspapers around the world published articles claiming the coin is biased. The stories were based on reports that someone had spun the coin 250 times and gotten 140 heads—that’s 56% heads. Do you think this is evidence that
Annoying phone calls A newly hired telemarketer is told he will probably make a sale on about 15% of his phone calls. The first week he called 150 people, but only made 20 sales. Should he suspect he was misled about the true success rate? Explain.
No-shows An airline, believing that 5% of passengers fail to show up for flights, overbooks (sells more tickets than there are seats). Suppose a plane will hold 265 passengers, and the airline sells 275 tickets. What’s the probability the airline will not have enough seats, so someone gets bumped?
Lefties, again A lecture hall has 170 seats with folding arm tablets, 27 of which are designed for left-handers.The average size of classes that meet there is 150, and we can assume that about 12% of students are left-handed.What’s the probability that a right-handed student in one of these
Frogs, part II Based on concerns raised by his preliminary research, the biologist in Exercise 38 decides to collect and examine 150 frogs.a) Assuming the frequency of the trait is still 1 in 8, determine the mean and standard deviation of the number of frogs with the trait he should expect to find
Apples An orchard owner knows that he’ll have to use about 5% of the apples he harvests for cider because they will have bruises or blemishes. He expects a tree to produce about 200 apples.a) Describe an appropriate model for the number of cider apples that may come from that tree. Justify your
More arrows The archer in Exercise 30 will be shooting 200 arrows in a large competition.a) What are the mean and standard deviation of the number of bull’s-eyes she might get?b) Is a Normal model appropriate here? Explain.c) Use the 68–95–99.7 Rule to describe the distribution of the number
And more tennis Suppose the tennis player in Exercise 37 serves 80 times in a match.a) What are the mean and standard deviation of the number of good first serves expected?b) Verify that you can use a Normal model to approximate the distribution of the number of good first serves.c) Use the
Frogs A wildlife biologist examines frogs for a genetic trait he suspects may be linked to sensitivity to industrial toxins in the environment. Previous research had established that this trait is usually found in 1 of every 8 frogs.He collects and examines a dozen frogs. If the frequency of the
Tennis, anyone? A certain tennis player makes a successful first serve 70% of the time. Assume that each serve is independent of the others. If she serves 4 times, what’s the probability she getsa) all 4 serves in?b) exactly 3 serves in?c) at least 2 serves in?d) no more than 3 serves in?
International students At a certain college, 6% of all students come from outside the United States. Incoming students there are assigned at random to freshman dorms, where students live in residential clusters of 55 freshmen sharing a common lounge area. How many international students would you
Vision It is generally believed that nearsightedness affects about 14% of all children. A school district tests the vision of 156 incoming kindergarten children. How many would you expect to be nearsighted? With what standard deviation?
Still more arrows Suppose the archer from Exercise 30 shoots 10 arrows.a) Find the mean and standard deviation of the number of bull’s-eyes she may get.b) What’s the probability that she never misses?c) What’s the probability that there are no more than 8 bull’s-eyes?d) What’s the
Still more lefties Suppose we choose 10 people instead of the 5 chosen in Exercise 29.a) Find the mean and standard deviation of the number of right-handers in the group.b) What’s the probability that they’re not all right-handed?c) What’s the probability that there are no more than 10
More arrows Consider our archer from Exercise 30.a) How many bull’s-eyes do you expect her to get?b) With what standard deviation?c) If she keeps shooting arrows until she hits the bull’seye, how long do you expect it will take?
Lefties, redux Consider our group of 5 people from Exercise 29.a) How many lefties do you expect?b) With what standard deviation?c) If we keep picking people until we find a lefty, how long do you expect it will take?
Arrows An Olympic archer is able to hit the bull’s-eye 80% of the time. Assume each shot is independent of the others. If she shoots 6 arrows, what’s the probability of each of the following results?a) Her first bull’s-eye comes on the third arrow.b) She misses the bull’s-eye at least
Lefties Assume that 12% of people are left-handed. If we select 4 people at random, find the probability of each outcome.a) The first lefty is the fourth person chosen.b) There are some lefties among the 4 people.c) The first lefty is the third or fourth person.d) There are exactly 3 lefties in the
Roulette and intuition An American roulette wheel has 38 slots, of which 18 are red, 18 are black, and 2 are green (0 and 00). If you spin the wheel 38 times,a) Intuitively, how many times would you expect the ball to wind up in a green slot?b) Use the formula for expected value to verify your
Coins and intuition If you flip a fair coin 160 times,a) Intuitively, how many heads do you expect?b) Use the formula for expected value to verify your intuition.
Color blindness About 8% of males are color-blind. A researcher needs some color-blind subjects for an experiment and begins checking potential subjects.a) On average, how many men should the researcher expect to check to find one who is color-blind?b) What’s the probability that she won’t find
Blood Only 4% of people have Type AB blood.a) On average, how many donors must be checked to find someone with Type AB blood?b) What’s the probability that there is a Type AB donor among the first 5 people checked?c) What’s the probability that the first Type AB donor will be found among the
Cold calls Justine works for an organization committed to raising money for Alzheimer’s research. From past experience, the organization knows that about 20% of all potential donors will agree to give something if contacted by phone. They also know that of all people donating, about 5% will give
Customer center operator Raaj works at the customer service call center of a major credit card bank. Cardholders call for a variety of reasons, but regardless of their reason for calling, if they hold a platinum card, Raaj is instructed to offer them a double-miles promotion. About 14% of all
Chips ahoy For the computer chips described in Exercise 20, how many do you expect to test before finding a bad one?
More hoops For the basketball player in Exercise 19, what’s the expected number of shots until he misses?
Chips Suppose a computer chip manufacturer rejects 2%of the chips produced because they fail presale testing.a) What’s the probability that the fifth chip you test is the first bad one you find?b) What’s the probability you find a bad one within the first 9 you examine?
Hoops A basketball player has made 60% of his foul shots during the season. Assuming the shots are independent, find the probability that in tonight’s game hea) misses for the first time on his fifth attempt.b) makes his first basket on his fourth shot.c) makes his first basket on one of his
Lost luggage A Department of Transportation report about air travel found that airlines misplace about 5 bags per 1000 passengers. Suppose you are traveling with a group of people who have checked 22 pieces of luggage on your flight. Can you consider the fate of these bags to be Bernoulli trials?
On time A Department of Transportation report about air travel found that, nationwide, 76% of all flights are on time. Suppose you are at the airport and your flight is one of 50 scheduled to take off in the next two hours. Can you consider these departures to be Bernoulli trials? Explain.
Hope, again Let’s take one last look at the Hope Solo picture search. You know her picture is in 20% of the cereal boxes. You buy five boxes to see how many pictures of Hope you might get.a) Describe how you would simulate the number of pictures of Hope you might find in five boxes of cereal.b)
Simulation II You are one space short of winning a child’s board game and must roll a 1 on a die to claim victory. You want to know how many rolls it might take.a) Describe how you would simulate rolling the die until you get a 1.b) Run at least 30 trials.c) Based on your simulation, estimate the
Simulating the model Think about the Hope Solo picture search again. You are opening boxes of cereal one at a time looking for her picture, which is in 20% of the boxes. You want to know how many boxes you might have to open in order to find Hope.a) Describe how you would simulate the search for
Website sales Suppose occurrences of sales on a small company’s website are well modeled by a Poisson model with l = 5>hour.a) If a sale just occurred, what is the expected waiting time until the next sale?b) What is the probability that the next sale will happen in the next 6 minutes?
Component lifetimes Lifetimes of electronic components can often be modeled by an exponential model.Suppose quality control engineers want to model the lifetime of a hard drive to have a mean lifetime of 3 years.a) What value of l should they use?b) With this model, what would the probability be
Serial numbers In an effort to check the quality of their cell phones, a manufacturing manager decides to take a random sample of 10 cell phones from yesterday’s production run, which produced cell phones with serial numbers ranging (according to when they were produced)from 43005000 to 43005999.
Telephone numbers A cable provider wants to contact customers in a particular telephone exchange to see how satisfied they are with the new digital TV service the company has provided. All numbers are in the 452 exchange, so there are 10,000 possible numbers from 452-0000 to 452-9999. If they
Passing on A large hospital has an average of 7 fatalities in a week. Using the Poisson model, what is the probability that this week it has 10 fatalities?
Soccer A soccer team estimates that they will score on 8% of the corner kicks. In next week’s game, the team hopes to kick 15 corner kicks. What are the chances that they will score on 2 of those opportunities?
Toasters A manufacturer ships toasters in cartons of 20.In each carton, they estimate a 5% chance that one of the toasters will need to be sent back for minor repairs. What is the probability that in a carton, there will be exactly 3 toasters that need repair?
Bernoulli 2 Do these situations involve Bernoulli trials?Explain.a) You are rolling 5 dice and need to get at least two 6’s to win the game.b) We record the distribution of eye colors found in a group of 500 people.c) A manufacturer recalls a doll because about 3% have buttons that are not
Bernoulli Do these situations involve Bernoulli trials?Explain.a) We roll 50 dice to find the distribution of the number of spots on the faces.b) How likely is it that in a group of 120 the majority may have Type A blood, given that Type A is found in 43% of the population?c) We deal 7 cards from a
Some LCD panels have stuck or “dead” pixels that have defective transistors and are permanently unlit. If a panel has too many dead pixels, it must be rejected. A manufacturer knows that, when the production line is working correctly, the probability of rejecting a panel is 0.07.Questions:1.
The communications monitoring company Postini has reported that 91% of e-mail messages are spam. Recently, you installed a spam filter. You observe that over the past week it okayed only 151 of 1422 e-mails you received, classifying the rest as junk. Should you worry that the filtering is too
The communications monitoring company Postini has reported that 91% of e-mail messages are spam. Suppose your inbox contains 25 messages.Questions: What are the mean and standard deviation of the number of real messages you should expect to find in your inbox? What’s the probability that you’ll
People with O-negative blood are called “universal donors” because O-negative blood can be given to anyone else, regardless of the recipient’s blood type. Only about 6% of people have O-negative blood.Questions:1. If donors line up at random for a blood drive, how many do you expect to
Postini is a global company specializing in communications security. The company monitors over 1 billion Internet messages per day and recently reported that 91%of e-mails are spam!Let’s assume that your e-mail is typical—91% spam. We’ll also assume you aren’t using a spam filter, so every
Weightlifting The Atlas BodyBuilding Company(ABC) sells “starter sets” of barbells that consist of one bar, two 20-pound weights, and four 5-pound weights.The bars weigh an average of 10 pounds with a standard deviation of 0.25 pounds. The weights average the specified amounts, but the standard
Coffee and doughnuts At a certain coffee shop, all the customers buy a cup of coffee; some also buy a doughnut.The shop owner believes that the number of cups he sells each day is normally distributed with a mean of 320 cups and a standard deviation of 20 cups. He also believes that the number of
Bike sale The bicycle shop in Exercise 50 will be offering 2 specially priced children’s models at a sidewalk sale. The basic model will sell for $120 and the deluxe model for $150. Past experience indicates that sales of the basic model will have a mean of 5.4 bikes with a standard deviation of
Farmers’ market A farmer has 100 lb of apples and 50 lb of potatoes for sale. The market price for apples(per pound) each day is a random variable with a mean of 0.5 dollars and a standard deviation of 0.2 dollars.Similarly, for a pound of potatoes, the mean price is 0.3 dollars and the standard
Bikes Bicycles arrive at a bike shop in boxes. Before they can be sold, they must be unpacked, assembled, and tuned (lubricated, adjusted, etc.). Based on past experience, the shop manager makes the following assumptions about how long this may take:• The times for each setup phase are
Medley In the 4 * 100 medley relay event, four swimmers swim 100 yards, each using a different stroke. A college team preparing for the conference championship looks at the times their swimmers have posted and creates a model based on the following assumptions:• The swimmers’ performances are
More pets You’re thinking about getting two dogs and a cat. Assume that annual veterinary expenses are independent and have a Normal model with the means and standard deviations described in Exercise 46.a) Define appropriate variables and express the total annual veterinary costs you may have.b)
More cereal In Exercise 45 we poured a large and a small bowl of cereal from a box. Suppose the amount of cereal that the manufacturer puts in the boxes is a random variable with mean 16.2 ounces and standard deviation 0.1 ounces.a) Find the expected amount of cereal left in the box.b) What’s the
Pets The American Veterinary Association claims that the annual cost of medical care for dogs averages $100, with a standard deviation of $30, and for cats averages $120, with a standard deviation of $35.a) What’s the expected difference in the cost of medical care for dogs and cats?b) What’s
Cereal The amount of cereal that can be poured into a small bowl varies with a mean of 1.5 ounces and a standard deviation of 0.3 ounces. A large bowl holds a mean of 2.5 ounces with a standard deviation of 0.4 ounces.You open a new box of cereal and pour one large and one small bowl.a) How much
Casino A casino knows that people play the slot machines in hopes of hitting the jackpot but that most of them lose their dollar. Suppose a certain machine pays out an average of $0.92, with a standard deviation of $120.a) Why is the standard deviation so large?b) If you play 5 times, what are the
Fire! An insurance company estimates that it should make an annual profit of $200 on each homeowner’s policy written, with a standard deviation of $8000.a) Why is the standard deviation so large?b) If it writes only two of these policies, what are the mean and standard deviation of the annual
Donations Organizers of a televised fundraiser know from past experience that most people donate small amounts($10–$25), some donate larger amounts ($50–$100), and a few people make very generous donations of $250, $500, or more. Historically, pledges average about $32 with a standard deviation
Tickets A delivery company’s trucks occasionally get parking tickets, and based on past experience, the company plans that each truck will average 1.4 tickets a month, with a standard deviation of 0.5 tickets.a) If they have 25 trucks, what are the mean and standard deviation of the total number
Stop! Find the mean and standard deviation of the number of red lights the commuter in Exercise 24 should expect to hit on her way to work during a 5-day work week.
Repair calls Find the mean and standard deviation of the number of repair calls the appliance shop in Exercise 23 should expect during an 8-hour day.
Garden A company selling vegetable seeds in packets of 20 estimates that the mean number of seeds that will actually grow is 18, with a standard deviation of 1.2 seeds.You buy 5 different seed packets.a) How many bad (non-growing) seeds do you expect to get?b) What’s the standard deviation?c)
Eggs A grocery supplier believes that in a dozen eggs, the mean number of broken ones is 0.7 with a standard deviation of 0.5 eggs. You buy 3 dozen eggs without checking them.a) How many broken eggs do you expect to get?b) What’s the standard deviation?c) What assumptions did you have to make
Random variables Given independent random variables with means and standard deviations as shown, find the mean and standard deviation o a) X 20 b) 0.5Y c) X + Y Mean SD X 80 12 d) X-Y Y 12 3 e) Y + Y
Batteries In a group of 10 batteries, 6 are dead. You choose 2 batteries at random.a) Create a probability model for the number of good batteries you get.b) What’s the expected number of good ones you get?c) What’s the standard deviation?
Contracts Your company bids for two contracts. You believe the probability you get contract #1 is 0.8. If you get contract #1, the probability you also get contract #2 will be 0.2, and if you do not get #1, the probability you get#2 will be 0.3.a) Are the two contracts independent? Explain.b) Find
Contest You play two games against the same opponent.The probability you win the first game is 0.7. If you win the first game, the probability you also win the second is 0.5. If you lose the first game, the probability that you win the second is 0.4.a) Are the two games independent? Explain.b)
Day trading An option to buy a stock is priced at $300.If the stock closes above 30 on May 15, the option will be worth $700. If it closes below 20, the option will be worth nothing, and if it closes between 20 and 30 (inclusively), the option will be worth $300. A trader thinks there is a 40%
Cancelled flights Mary is deciding whether to book the cheaper flight home from college after her final exams, but she’s unsure when her last exam will be. She thinks there is only a 20% chance that the exam will be scheduled after the last day she can get a seat on the cheaper flight. If it is
Insurance An insurance policy costs $110 and will pay policyholders $11,000 if they suffer a major injury(resulting in hospitalization) or $3000 if they suffer a minor injury (resulting in lost time from work). The company estimates that each year 1 in every 2046 policyholders may have a major
Defects A consumer organization inspecting new cars found that many had appearance defects (dents, scratches, paint chips, etc.). While none had more than three of these defects, 8% had three, 11% two, and 21%one defect. Find the expected number of appearance defects in a new car and the standard
Red lights A commuter must pass through five traffic lights on her way to work and will have to stop at each one that is red. She estimates the probability model for the number of red lights she hits, as shown belowa) How many red lights should she expect to hit each day?b) What’s the standard
Darts Find the standard deviation of your winnings throwing darts in Exercise 14.
Kids, again Find the standard deviation of the number of children the couple in Exercise 13 may have.
Pick another card Find the standard deviation of the amount you might win drawing a card in Exercise 11.
Variation 2 Find the standard deviations of the random variables in Exercise 10.
Variation 1 Find the standard deviations of the random variables in Exercise 9.
Software A small software company bids on two contracts and knows it can only get one of them. It anticipates a profit of $45,000 if it gets the larger contract and a profit of $20,000 on the smaller contract. The company estimates there’s a 31% chance it will get the larger contract and a 62%
Carnival A carnival game offers a $100 cash prize for anyone who can break a balloon by throwing a dart at it.It costs $5 to play, and you’re willing to spend up to $20 trying to win. You estimate that you have about a 10%chance of hitting the balloon on any throw.a) Create a probability model
Kids A couple plans to have children until they get a girl, but they agree that they will not have more than three children even if all are boys. Assume that the probability of having a girl is 48.00%.a) Create a probability model for the number of children they might have.b) Find the expected
Golf scores A golfer keeps track of his score for playing nine holes of golf (half a normal golf round). His mean score is 85 with a standard deviation of 11. Assuming that the second 9 has the same mean and standard deviation, what is the mean and standard deviation of his total score if he plays
Caffeinated A coffee shop tracks sales and has observed the distribution in the following table. What is the average daily sales that it can expect? # of Sales Probability 145 150 155 160 170 0.15 0.22 0.37 0.19 0.07
Oranges A citrus farmer has observed the following distribution for the number of oranges per tree. How many oranges does he expect on average? Oranges 25 30 35 40 Probability 0.10 0.40 0.30 0.20
You’re planning to spend next year wandering through the mountains of Kyrgyzstan.You plan to sell your used SUV so you can purchase an off-road Honda motor scooter when you get there. Used SUVs of the year and mileage of yours are selling for a mean of $6940 with a standard deviation of $250.Your
The Quiet Nook’s Lucky Lovers promotion offers couples discounts averaging$5.83 with a standard deviation of $8.62. The restaurant owner is planning on serving 40 couples on Valentine’s Day.Question: What’s the expected total of the discounts the owner will give? With what standard deviation?
The Lucky Lovers discount at the Quiet Nook averages $5.83 with a standard deviation of $8.62. Just up the street, the Wise Fool restaurant has a competing Lottery of Love promotion. There a couple can select a specially prepared chocolate from a large bowl and unwrap it to learn the size of their
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