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intro stats
Stats Data And Models 3rd Canadian Edition Richard De Veaux, Paul Velleman, David Bock, Augustin Vukov, Augustine Wong - Solutions
Loans Based on past experience, a bank believes that 7% of the people who receive loans will not make payments on time. The bank has recently approved 200 loans.a) What are the mean and standard deviation of the proportion of clients in this group who may not make timely payments?b) What
Mortgages In early 2009, rising unemployment and the house-price slump in Alberta led to a mortgage delinquency rate of 0.54% of all big bank mortgages (rising for the nineteenth straight month), the highest delinquency rate in Canada. Suppose a large bank holds 1731 mortgages.a) Can you apply
Vision It is generally believed that nearsightedness affects about 12% of all children. A school district has registered 170 incoming kindergarten children.a) Can you apply the Central Limit Theorem to describe the sampling distribution model for the sample proportion of children who are
Smoking pot The Canadian Drug Use and Monitoring Survey estimates that 33% of 15- to 24-year-olds used cannabis in 2008. Using the 68–95–99.7 Rule, describe the sampling distribution model for the proportion of cannabis users among a randomly selected group of fifty 15- to 24-year-olds. Be sure
Speeding Provincial police believe that 70% of the drivers travelling on a particular section of the Trans-Canada Highway exceed the speed limit. They plan to set up a radar trap and check the speeds of 80 cars.a) Using the 68–95–99.7 Rule, draw and label the distribution of the proportion of
Too many green ones? In a really large bag of M&Ms, the students in Exercise 18 found 500 candies, and 12% of them were green. Is this an unusually large proportion of green M&Ms? Explain.
Just (un)lucky? One of the students in the introductory Statistics class in Exercise 17 claims to have tossed her coin 200 times and found only 42% heads. What do you think of this claim? Explain.
Bigger bag Suppose the class in Exercise 16 buys bigger bags of candy, with 200 M&M’s each. Again the students calculate the proportion of green candies they find.a) Explain why it’s appropriate to use a Normal model to describe the distribution of the proportion of green M&M’s they might
More coins Suppose the class in Exercise 15 repeats the coin-tossing experiment.a) The students toss the coins 25 times each. Use the 68–95–99.7 Rule to describe the sampling distribution model.b) Confirm that you can use a Normal model here.c) They increase the number of tosses to 64 each.
M&M’s The candy company claims that 10% of the M&M’s it produces are green. Suppose that the candies are packaged at random in small bags containing about 50 M&M’s. A class of elementary school students learning about percents opens several bags, counts the various colours of the candies,
Coin tosses In a large class of introductory Statistics students, the professor has each student toss a coin 16 times and calculate the proportion of his or her tosses that were heads. The students then report their results, and the professor plots a histogram of these several proportions.a) What
More groceries Suppose the store in Exercise 12 had 312 customers this Sunday.a) Estimate the probability that the store’s revenues were at least $10,000.b) If, on a typical Sunday, the store serves 312 customers, how much does the store take in on the worst 10% of such days?
More tips The waiter in Exercise 11 usually waits on about 40 parties over a weekend of work.a) Estimate the probability that he will earn at most $500 in tips.b) How much does he earn on the best 10% of such weekends?
Groceries A grocery store’s receipts show that Sunday customer purchases have a skewed distribution with a mean of $30 and a standard deviation of $20.a) Explain why you cannot determine the probability that the next Sunday customer will spend at least $40.b) Can you estimate the probability that
Tips A waiter believes the distribution of his tips has a model that is slightly skewed to the right, with a mean of $9.60 and a standard deviation of $5.40.a) Explain why you cannot determine the probability that a given party will tip him at most $20.b) Can you estimate the probability that the
Soup A machine is supposed to fill cans with 16 oz of soup. Of course there, will be some variation in the amount actually dispensed, and measurement errors are often approximately normally distributed. The manager would like to understand the variability of the variances of the samples, so he
Sample maximum The distribution of scores on a Statistics test for a particular class is skewed to the left. The professor wants to predict the maximum score and so wants to understand the distribution of the sample maximum. She simulates the distribution of the maximum of the test for 30 different
Character recognition, again The automatic character recognition device discussed in Exercise 4 successfully reads about 85% of handwritten credit card applications. In Exercise 4, you looked at the histograms showing distributions of sample proportions from 1000 simulated samples of size 20, 50,
Send more money The philanthropic organization in Exercise 3 expects about a 5% success rate when they send fundraising letters to the people on their mailing list. In Exercise 3, you looked at the histograms showing distribution of sample proportions from 1000 simulated mailings for samples of
Campus sample For her final project, Stacy plans on surveying a random sample of 50 students on whether they plan to go to Florida for Spring Break. From past years, she guesses that about 10% of the class goes. Is it reasonable for her to use a Normal model for the sampling distribution of the
Marriage According to a Pew Research survey, 27% of American adults are pessimistic about the future of marriage and the family. That is based on a random sample of about 1500 people. Is it reasonable for Pew Research to use a Normal model for the sampling distribution of the sample proportion?
Character recognition An automatic character recognition device can successfully read about 85% of handwritten credit card applications. To estimate what might happen when this device reads a stack of applications, the company did a simulation using samples of size 20, 50, 75, and 100.For each
Send money When it sends out its fundraising letter, a philanthropic organization typically gets a return from about 5% of the people on their mailing list. To see what the response rate might be for future appeals, it did a simulation using samples of size 20, 50, 100, and 200.For each sample
Marketing The proportion of adult women in Canada is approximately 51%. A marketing survey company telephones 400 people at random.a) What proportion of the sample of 400 would you expect to be women?b) What would the standard deviation of the sampling distribution be?c) How many women, on
Website An investment website can tell what devices are used to access the site. The site managers wonder whether they should enhance the facilities for trading via smartphones so they want to estimate the proportion of users who access the site that way (even if they also use their computers
Door prize You are among 100 people attending a charity fundraiser at which a large-screen television will be given away as a door prize. To determine who wins, 99 white balls and 1 red ball have been placed in a box and thoroughly mixed. The guests will line up and, one at a time, pick a ball
Pregnant? Suppose that 70% of the women who suspect they may be pregnant and purchase an in-home pregnancy test are actually pregnant. Further suppose that the test is 98% accurate. What’s the probability that a woman whose test indicates that she is pregnant actually is?
Recalls In a car rental company’s fleet, 70% of the cars are American brands, 20% are Japanese, and the rest are German. The company notes that manufacturers’ recalls seem to affect 2% of the American cars, but only 1% of the others.a) What’s the probability that a randomly chosen car is
The Drake equation Not the latest hit by the Toronto rapper, but an equation developed in 1961 by astronomer Frank Drake. He was trying to estimate the number of extraterrestrial civilizations in our galaxy that might be able to communicate with us via radio transmissions. Now largely accepted by
Coins A coin is tossed 36 times.a) What are the mean and standard deviation of the number of heads?b) Suppose the resulting number of heads is unusual, two standard deviations above the mean. How many “extra” heads were observed?c) If the coin were tossed 100 times, would you still consider
Socks In your sock drawer you have four blue socks, five grey socks, and three black ones. Half asleep one morning, you grab two socks at random and put them on. Find the probability you end up wearinga) two blue socks.b) no grey socks.c) at least one black sock.d) a green sock.e) matching socks.
Volcanoes Almost every year, there is some incidence of volcanic activity on the island of Japan. In 2005, there were five volcanic episodes, defined as either eruptions or sizable seismic activity. Suppose the mean number of episodes is 2.4 per year. Let X be the number of episodes in the two-year
Plan B Here’s another attempt at developing a good strategy for the dice game in Exercise 33.Instead of stopping after a certain number of rolls, you could decide to stop when your score reaches a certain number of points.a) How many points would you expect a roll to add to your score?b) In
Dogs A census by the municipal dog control officer found that 18% of homes kept one dog as a pet, 4% had two dogs, and 1% had three or more. If a salesperson visits two homes selected at random, what’s the probability he encountersa) no dogs?b) some dogs?c) dogs in each home?d) more than one dog
Technology on campus 2005 Every five years, the Conference Board of the Mathematical Sciences surveys college math departments. In 2005, the board reported that 51% of all undergraduates taking Calculus I were in classes that used graphing calculators and 21% were in classes that used computer
Plan B Here’s another attempt at developing a good strategy for the dice game in Exercise 33.Instead of stopping after a certain number of rolls, you could decide to stop when your score reaches a certain number of points.a) How many points would you expect a roll to add to your score?b) In
When to stop? You play a game that involves rolling a die. You can roll as many times as you want, and your score is the total for all the rolls. But . . . if you roll a 6, your score is 0 and your turn is over. What might be a good strategy for a game like this? You attempted to devise a good
Jerseys A Statistics professor comes home to find that all four of his children got white team shirts from soccer camp this year. He concludes that this year, unlike other years, the camp must not be using a variety of colours. But then he finds out that in each child’s age group there are four
Who’s the boss 2011? According to Industry Canada, 17% of all small businesses were owned by women at the end of 2011. You call some small business firms doing business locally, assuming that the national percentage is true in your area.a) What’s the probability that the first three you call
Buying melons The first store in Exercise 28 sells watermelons for 32 cents a pound. The second store is having a sale on watermelons—only 25 cents a pound. Find the mean and standard deviation of the difference in the price you may pay for melons randomly selected at each store.
Home sweet home 2011 According to Statistics Canada, 67% of Canadian households own the home they live in. A mayoral candidate conducts a survey of 820 randomly selected homes in your city and finds only 523 owned by the current residents. The candidate then attacks the incumbent mayor, saying
Picking melons Two stores sell watermelons. At the first store, the melons weigh an average of 22 pounds, with a standard deviation of 2.5 pounds. At the second store, the melons are smaller, with a mean of 18 pounds and a standard deviation of 2 pounds. You select a melon at random at each
Travel to Kyrgyzstan Your pocket copy of Kyrgyzstan on 4237 ± 360 Soms a Day claims that you can expect to spend about 4237 soms each day with a standard deviation of 360 soms. How well can you estimate your expenses for the trip?a) Your budget allows you to spend 90 000 soms. To the nearest day,
Meals A university student on a seven-day meal plan reports that the amount of money he spends daily on food varies with a mean of $13.50 and a standard deviation of $7.a) What are the mean and standard deviation of the amount he might spend in two consecutive days?b) What assumption did you make
Youth survey According to a recent Gallup survey, 93% of teens use the Internet, but there are differences in how teen boys and girls say they use computers. The telephone poll found that 77% of boys had played computer games in the past week, compared with 65% of girls. On the other hand, 76% of
Merger Explain why the facts you know about variances of independent random variables might encourage two small insurance companies to merge. (Hint: Think about the expected amount and potential variability in payouts for the separate and the merged companies.)
Random variables Given independent random variables with means and standard deviations as shown, find the mean and standard deviation of each of the variables below. The variables X1 and X2 are independent with the same distribution as X.a) X + 50b) 10Yc) X + 0.5Yd) X - Ye) X1 + X2 Mean SD X 50 8 Y
Teen smoking II Suppose that, as reported by the Centers for Disease Control, about 23% of high school students smoke tobacco. You randomly select 120 high school students to survey them on their attitudes toward scenes of smoking in the movies.a) What’s the expected number of smokers?b) What’s
Passing stats Molly’s university offers two sections of Statistics 101.From what she has heard about the two professors listed, Molly estimates that her chances of passing the course are 0.80 if she gets Professor Scedastic and 0.60 if she gets Professor Kurtosis. The registrar uses a lottery to
Teen smoking The Centers for Disease Control say that about 23% of high-school students smoke tobacco (down from a high of 38% in 1997). Suppose you randomly select high school students to survey them on their attitudes toward scenes of smoking in the movies. What’s the probability thata) none
Insurance A 65-year-old woman takes out a $100 000 term life insurance policy. The company charges an annual premium of $520. Estimate the company’s expected profit on such policies if mortality tables indicate that only 2.6% of women age 65 die within a year.
Stock strategy Many investment advisors argue that after stocks have declined in value for two consecutive years, people should invest heavily because the market rarely declines three years in a row.a) Since the stock market began in 1872, there have been two consecutive losing years eight times.
Multiple choice A multiple choice test has 50 questions, with 4 answer choices each. You must get at least 30 correct to pass the test, and the questions are very difficult.a) Are you likely to be able to pass by guessing on every question? Explain.b) Suppose, after studying for a while, you
Stocks Since the stock market began in 1872, stock prices have risen in about 73% of the years. Assuming that market performance is independent from year to year, what’s the probability thata) the market will rise for three consecutive years?b) the market will rise three out of the next five
Beanstalks In some cities, tall people who want to meet and socialize with other tall people can join Beanstalk Clubs. To qualify, a man must be over 6’2” tall and a woman over 5’10.” According to the National Health Survey, heights of adults may have a Normal model with mean heights of
Play again If you land in a “penalty zone” on the game board described in Exercise 12, your move will be determined by subtracting the roll of the die from the result on the spinner. Now what are the mean and standard deviation of the number of spots you may move?
Language Neurological research has shown that in about 80% of people, language abilities reside in the brain’s left side. Another 10% display right-brain language centres, and the remaining 10% have two-sided language control. (The latter two groups are mainly left-handers.)1a) Assume that a
Child’s play In a board game, you determine the number of spaces you may move by spinning a spinner and rolling a die. The spinner has three regions: Half of the spinner is marked “5,” and the other half is equally divided between “10”and “20.” The six faces of the die show 0, 0, 1,
Friend me? One hundred fifty-eight teenagers are standing in line for a big movie premier night. Refer to Exercise 7.a) What are the mean and standard deviation of the number of Facebook users we might expect?b) Can we use a Normal model in this situation?c) What’s the probability that no more
At fault The car insurance company in Exercise 8 believes that about 0.5% of drivers have an at-fault accident during a given year. Suppose the company insures 1355 drivers in that city.a) What are the mean and standard deviation of the number who may have at-fault accidents?b) Can you describe
More Facebook Using the percentages from Exercise 7, suppose there is a group of five teenagers. What’s the probability thata) all will be on Facebook?b) exactly one will be on Facebook?c) at least three will be on Facebook?
Deductible A car owner may buy insurance that will pay the full price of repairing the car after an at-fault accident, or save $12 a year by getting a policy with a $500 deductible. Her insurance company says that about 0.5% of drivers in her area have an at-fault auto accident during any given
Facebook According to Pew Research, 50% of adults and 75% of teenagers were using a social networking site in early 2012. Most of that activity was on Facebook, so let’s assume these probabilities apply strictly to Facebook. Among a group of 10 people, what’s the probability thata) at least one
Emergency switch Safety engineers must determine whether industrial workers can operate a machine’s emergency shutoff device. Among a group of test subjects, 66% were successful with their left hands, 82% with their right hands, and 51% with both hands.a) What percent of these workers could not
A game To play a game, you must pay $5 for each play. There is a 10% chance you will win $5, a 40% chance you will win $7, and a 50% chance you will win only $3.a) What are the mean and standard deviation of your net winnings?b) You play twice. Assuming the plays are independent events, what are
Autism Psychiatrists estimate that about 1 in 100 adults has autism. What’s the probability that in a city of 20 000, there are more than 300 people with this condition? Be sure to verify that a Normal model can be used here.
Airfares Each year a company must send three officials to a meeting in China and five officials to a meeting in France. Airline ticket prices vary from time to time, but the company purchases all tickets for a country at the same price. Past experience has shown that tickets to China have a mean
Workers A company’s human resources officer reports a breakdown of employees by job type and sex as shown in the table.Sex Male Female Job Type Management 7 6 Supervision 8 12 Production 45 72a) What’s the probability that a worker selected at random is i. female? ii. female or a production
Quality control A consumer organization estimates that 29% of new cars have a cosmetic defect, such as a scratch or a dent, when they are delivered to car dealers. This same organization believes that 7% have a functional defect—something that does not work properly— and that 2% of new cars
Correlated RVs Investors often purchase negatively cor- related investment vehicles to minimize risk. Suppose you purchase two investments. Based on past experi- ence, you believe that each has a mean gain of 5% with standard deviation of 2% (over some time frame). Let X the percentage return from
Geometric model again Suppose we inspect objects produced by a process in which 10% on average are defective. We will keep sampling and testing objects one at a time until we find a defective one.a) Are these Bernoulli trials?b) Can the Binomial model be applied here? Explain.c) Find the
Geometric model Suppose we keep flipping a loonie until we see a head. Let X = the flip on which we observe that first head. For example, if we get a tail, then a head, X = 2.Assume this is a typical loonie, flipped high and randomly.a) Are these Bernoulli trials?b) Can the Binomial model be
Bike sale The bicycle shop in Exercise 94 will be offering two 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
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
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 national 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 90.a) Define appropriate variables and express the total annual veterinary costs you might
More cereal In Exercise 89, 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 is
Pets The Canadian Veterinary Medical 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 is the expected difference in the cost of medical care for dogs and cats?b) What
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
Continuous models Suppose that X is a continuous random variable with the following density curve model:a) Find P(0 b) Find P(1 ≤ X ≤ 3) and shade in the relevant area. 0.5+ 0.0 0 +2
Continuous uniform model Suppose that X is continuous and uniformly distributed over the outcomes 0 to 360.(For example, you might be spinning a bottle and observing the final angle.)a) Draw a picture of the flat density curve to model this random variable. What is the correct height for the curve
New bow, again Kateri Vrakking in Exercise 84 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 Our basketball player in Exercise 83 has new sneakers, which he thinks improve his game. Over his past 40 shots, he’s made 32—much better than the 55% he usually shoots. Do you think his chances of making a shot really increased? In other words, is making at least 32 of 40 shots
New bow Kateri Vrakking, the archer in Exercise 62, 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 six consecutive bull’s-eyes. Do you think this is compelling evidence that the new bow
Hot hand A basketball player who ordinarily makes about 55% of his free throw shots has made four 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.
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
Rickets Vitamin D is essential for strong, healthy bones. Our bodies produce vitamin D naturally when the skin is exposed to sunlight, 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
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) What is the probability that the first 10 drivers are all wearing their seatbelts?b) If police stop 30 cars during the first hour, find the mean and
Lotto 6/49 scandal Earlier in the chapter, we mentioned Professor Jeff Rosenthal’s work on the Lotto 6/49 scandal. He estimated—conservatively—that for the 5713 major-prize lottery wins over the period 1996–2006, the expected number of wins by retailers was 123.2. In fact, retailers won
HIV It is given that the probability of contracting HIV is small, with p about 0.0005 for a new case in a given year. In a town of 8000 people,a) What is the expected number of new cases?b) Use the Poisson model to approximate the probability that there will be at least one new case of HIV next
Bank tellers You are the only bank teller on duty at your local bank. You need to run out for 10 minutes, but you don’t want to miss any customers. Suppose the arrival of customers can be modelled by a Poisson distribution with a mean of two customers per hour.a) What is the probability that no
Hurricanes 2015, redux We first looked at the occurrences of hurricanes in Chapter 3 exercises and found that they arrive with a mean of 2.65 per year. Suppose the number of hurricanes can be modelled by a Poisson distribution with this mean.a) What is the probability of no hurricanes next year?b)
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 is the probability that the airline will not have enough seats, so someone gets
Lefties again A lecture hall has 200 seats with folding arm tablets, 30 of which are designed for left-handers. The average size of classes that meet there is 188, and we can assume that about 13% of students are left-handed. What is 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 66 decides to collect and examine 150 frogs.a) Assuming the frequency of the trait is still one in eight, determine the mean and standard deviation of the number of frogs with the trait that he will 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
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