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intro stats
Stats Data And Models 5th Global Edition Richard De Veaux, Paul Velleman, David Bock - Solutions
Graduation It’s believed that as many as 30% of adults over 50 never graduated from high school. We wish to see if this percentage is the same among the 25 to 30 age group.a) How many of this younger age group must we survey in order to estimate the proportion of non-grads to within 6%with 90%
Quality control, part 2 The cell phone manufacturer in Exercise 60 wants to model the time between events. The mean number of defective cell phones is 2 per day.a) What model would you use to model the time between events?b) What would the probability be that the time to the next failure is 1 day
Web visitors, part 2 The website manager in Exercise 59 wants to model the time between purchases. Recall that the mean number of purchases in the evening is 3 per minute.a) What model would you use to model the time between events?b) What is the mean time between purchases?c) What is the
Quality control In an effort to improve the quality of their cell phones, a manufacturing manager records the number of faulty phones in each day’s production run. The manager notices that the number of faulty cell phones in a production run of cell phones is usually small and that the quality of
Web visitors A website manager has noticed that during the evening hours, about 3 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
Are you now convinced that the new gun is better? Explain.
New gun, again The marksman in Exercise 56 continues shooting, ending up with 90 hits out of a
Ice hockey, again The ice hockey player in Exercise 55 has new blades, which he thinks will improve his game. Off his past 60 shots, he has scored 30, much better than the 33% he usually scores. Do you think his chances of making a shot really increased?In other words, is making at least 30 out of
Different gun The marksman in Exercise 30 uses a different gun, hoping that it will improve the success rate to more than 70%. The marksman is delighted when, in their first test, they hit 10 consecutive plates. Do you think this is compelling evidence that the new gun is better? In other words, is
Ice hockey An ice hockey player usually scores 33% of his shots on goal. He has scored 3 in a row. Is this evidence that he is having a good game tonight? In other words, is this streak so unusual that it means the probability that he scores a shot on goal has changed? Explain.
True–false A true–false test consists of 50 questions. How many does a student have to get right to convince you that he is not merely guessing? 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 10,000.a) What’s the expected number of major earthquakes in the next 1000 days?b) Use the Poisson model to approximate the probability that there will be at least one major earthquake in the next 1000 days.
TB Worldwide, tuberculosis is the number-one cause of death from an infectious disease. But the majority of cases are in developing countries. In the U.S., the probability of contracting TB is small, with p about 0.0005 for a new case in a given year.In a town of 8000 people:a) What’s the
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 have looked at various facts about hurricanes in previous chapters. Suppose we find that the arrivals of hurricanes 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
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 12% of his phone calls. The first week he called 200 people, but only made 10 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?
Cold calling, again The sales department intends to do 1000 cold calls in a week and stocks 300 items. On average, 900 cold calls are answered, and we can assume that 30% of these will result in a sale. What’s the probability that there are more sales than what is available in stock?
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 6% of the apples he harvests for cider because they will have bruises or blemishes. He expects a tree to produce about 300 apples.a) Describe an appropriate model for the number of cider apples that may come from that tree. Justify your
More shooting The marksman in Exercise 30 will be shooting 100 plates in a competition.a) What are the mean and standard deviation of the number of times they hit?b) Is a normal model appropriate here? Explain.c) Use the 68-95-99.7 Rule to describe the distribution of the number of hits the
More electric scooters Suppose the company in Exercise 37 expands and now produces 120 scooters in one day.a) What are the mean and standard deviation of the number of electric scooters that pass first inspection?b) Verify that you can use a normal model to approximate the distribution of 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
Electric scooters In a company, 90% of electric scooters produced pass first inspection. Assume that the quality of each electric scooter is independent of the other. If 10 electric scooters are made in one day, what is the probability that:a) all electric scooters pass first inspection?b) exactly
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 40 freshmen sharing a common lounge area. How many international students would you
Vision It is generally believed that nearsightedness affects about 12% of all children. A school district tests the vision of 169 incoming kindergarten children. How many would you expect to be nearsighted? With what standard deviation?
Still more shooting Suppose the marksman from Exercise 30 shoots 8 times.a) Find the mean and standard deviation of the number of hits.b) What is the probability that the marksman never misses?c) What is the probability that there are no more than 6 hits?d) What is the probability that there are
More cold calling Suppose we choose 10 people instead of 6 in Exercise 29.a) Find the mean and standard deviation of the number of calls that don’t lead to sales.b) What’s the probability that not all calls end without a sale?c) What is the probability that not more than 3 calls end without a
More shooting Consider our marksman from Exercise 30.a) How many plates is the marksman expected to hit?b) With what standard deviation?c) If the marksman keeps shooting until she hits a plate, how many tries do you expect it will take?
Cold calling, redux Consider our group of 6 people in Exercise 29.a) How many sales do you expect?b) What is the standard deviation on the number of sales?c) If we keep calling until we get a sale, how many calls do we expect to make?
Shooting range A marksman in a shooting range is able to hit a plate 70% of the time. Assume each shot is independent of the others. If the marksman shoots 5 times, what is the probability of each of the following results?a) The marksman’s first hit comes on the second shot.b) The marksman misses
Cold calling A sales department knows that 30% of cold calls for a particular product end up in a sale. If 6 potential customers are selected at random, find the probability of each of the following outcomes.a) The first customer to buy is the third one.b) At least one sale happens.c) The first
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 100 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 10% of all
Appliances ahoy For the appliances described in Exercise 20, how many does the department store expect to sell before selling a faulty one?
More pings For the computer in Exercise 19, what is the expected number of pings until a server is found unavailable?
Appliances Suppose 5% stock of a particular appliance of a particular brand sold at a large department store will have a fault of some kind.a) What is the probability that the sixth appliance sold is the first faulty one?b) What is the probability that it takes more than twelve sales to sell a
Ping A computer sends a periodical ping to a server, and the server is available 90%. Assuming each ping trial is independent from the others, find the probability that, on any particular day,a) the computer finds the server unavailable on its fourth ping.b) the computer finds the server available
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
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.
Seatbelts Suppose 75% of all drivers always wear their seatbelts.Let’s investigate how many of the drivers might be belted among five cars waiting at a traffic light.a) Describe how you would simulate the number of seatbeltwearing drivers among the five cars.b) Run at least 30 trials.c) Based on
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?CHAPTER
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 select
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?SECTION 15.7
Sell! A car dealership sells an average of 5 cars in a day. Using the Poisson model, what is the probability that the dealer sells 3 cars tomorrow?
Soccer again If this team has 200 corner kicks over the season, what are the chances that they score more than 22 times?SECTION 15.4
Toasters again In a batch of 10,000 toasters, what are the chances that fewer than 450 need to be returned?
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?SECTION 15.3
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?
Toasters A manufacturer ships toasters in cartons of
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
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
He also believes that the number of desserts he sells each day is independent of the lunch deal sales and is normally distributed with a mean of 120 and a standard deviation of 10.a) The shop is open 5 days a week on weekdays. Assuming day-to-day sales are independent, what is the probability that
Lunch deals At a certain food outlet, all customers buy a lunch deal, and some also get dessert. The shop owner believes that the number of lunch deals he sells each day is normally distributed with mean 200 and standard deviation
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
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
Homework A survey in a particular school indicates that girls dedicate, on average, 4.1 hours a week on homework, with a standard deviation of 2.37, while boys dedicate, on average, 3.68 hours a week, with a standard deviation of 2.54.a) What’s the expected difference in hours per week on
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
Insurance For a particular insurance policy, individual claims are, on average, 100, while the standard deviation is 1000.Policy holders with zero claims are also included in calculating this.a) Why is the standard deviation so large?b) If 5 policy holders are picked at random, what are the mean
Fire! An insurance company estimates that it should make an annual profit of $150 on each homeowner’s policy written, with a standard deviation of $6000.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 1.3 tickets a month, with a standard deviation of 0.7 tickets.a) If they have 18 trucks, what are the mean and standard deviation of the total number
Bus again Find the mean and standard deviation of the number of stops the bus in Exercise 24 is expected to make in the next 20 identical journeys.
Reservation again Find the mean and standard deviation of the number of people the restaurant in Exercise 23 should expect in the next ten reservations.
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 0.6 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
Kittens In a litter of seven kittens, three are female. You pick two kittens at random.a) Create a probability model for the number of male kittens you get.b) What’s the expected number of males?c) What’s the standard deviation?
Batteries In a group of 10 batteries, 3 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?
Bus A bus must pass through five stations before reaching its destination, and if there are any people at the station, it will have to stop. Given past history, the probability model for the number of stops the bus has to make during its journey is shown below.X = # of stops 0 1 2 3 4 5 P (X = x)
Reservation The probability model below describes the number of people in table reservation in a restaurant.Number of people 1 2 3 4 Probability 0.1 0.2 0.2 0.5a) How many people should the restaurant expect on the next reservation?b) What is the standard deviation?
Game show, again Find the standard deviation of the net points scored in Exercise 14.
Driving license, again Find the standard deviation of the number of attempts for a person trying to obtain a driver’s license in Exercise 13.
Assets, again Find the standard deviation of the price of the asset in Exercise 12.
Shares again Find the standard deviation of the gains/losses 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.
Racehorse A man buys a racehorse for $20,000 and enters it in two races. He plans to sell the horse afterward, hoping to make a profit. If the horse wins both races, its value will jump to $100,000. If it wins one of the races, it will be worth$50,000. If it loses both races, it will be worth only
Software A small software company bids on two contracts and knows it can only get one of them. It anticipates a profit of$50,000 if it gets the larger contract and a profit of $20,000 on the smaller contract. The company estimates there’s a 30%chance it will get the larger contract and a 60%
Game show In a game show on television, a participant has up to 3 attempts to find the box that contains 100 points from among 4 boxes. Each attempt costs him 10 points.a) Create a probability model for the net points scored.b) Find the expected number of attempts.c) Find the expected net points
Driving license In a particular country, the chances of obtaining a driver’s license on any single attempt is 0.9; however, the law does not allow for any one person to sit for a driver’s license exam more than three times.a) Create a probability model for the number of attempts for a person
There is a 50/50 chance that its price the next day will either go up or down by 10.a) Create a probability model for the price of the asset in two days.b) Find the expected price of the model in two days.c) Do you expect to gain, lose, or stay the same, on average?
Asset The price of a particular asset today is
Shares For a particular share on the stock market with value of $1(per share), there is a 40% chance of retaining the same value the next day, while outcomes $0.8, $0.9, and $1.2 are equally likely.a) Create a probability model for gains/losses on the value of one share.b) Find the expected
Ketchup An automatic filling machine in a factory fills bottles of ketchup with a mean of 16.1 oz and a standard deviation of 0.05 oz with a distribution that can be well modeled by a Normal model. What is the probability that your bottle of ketchup contains less than 16 oz?
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