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probability statistics
The Practice Of Statistics For Business And Economics 3rd Edition David S. Moore, George P. McCabe, Layth C. Alwan, Bruce A. Craig, William M. Duckworth - Solutions
Tossing four coins. The distribution of the count X of heads in four tosses of a balanced coin was found in Example 4.10 to be Number of heads xi: 0 1 2 3 4 Probability pi : 0.0625 0.25 0.375 0.25 0.0625 Find the mean μx for this distribution. Then find the mean number of heads for a single coin
How many rooms? Which of the two distributions for room counts in Exercise 4.73 appears more spread out in the probability histograms? Why? Find the standard deviation for each distribution. The standard deviation provides a numerical measure of spread.
and 4.56. Compare the two distributions using probability histograms, means, and standard deviations. Write a brief comparison, using your calculations to back up your statements.
Households and families. In government data, a household consists of all occupants of a dwelling unit, while a family consists of two or more persons who live together and are related by blood or marriage. Here are the distributions of household size and of family size in the United States:Number
How many rooms? Furniture makers and others are interested in how many rooms housing units have, because more rooms can generate more sales. Here are the distributions of the number of rooms for owner-occupied units and renter-occupied units in the United States:18 Rooms 1 2 3 4 5 Owned 0.0001
Insurance policy value. Continue analyzing the insurance policy described in the previous exercise.(a) Calculate the mean value of the insurance policy.(b) To the policyholder, what does the mean value of the policy represent? Under what conditions will this number become meaningful to the
Insurance policy value. Consider an insurance policy that will pay the holder of the policy the market price of her house if the house is completely destroyed by fire. The market price of a particular house is set at $220,000. The probability of this house being completely destroyed by fire during
Exporting wood to Japan. Timber is a common raw material for the manufacture of many products such as building materials for homes, furniture, and paper products. The United States exports hundreds of thousands of metric tons of wood chips each year to Japan. The amount exported varies from year to
Online betting exchange. Betfair is an online betting exchange.Individuals can make bets and take bets at whatever odds someone is willing to offer. You might think of Betfair as a mixture of eBay and the stock exchange applied to sports betting. The company is a U.K. company and does not knowingly
How many transactions? Continue with the data of the previous exercise. You will need to have filled in the last row of the table to complete this exercise.(a) Make a probability histogram for the number of transactions per client. The vertical scale should measure the proportion of clients
How many transactions? A small investment firm is beginning to reach its limit for the number of clients it can handle without expanding operations significantly. Before making this decision, the firm collects data from its own records to obtain a better understanding of how much business it really
Managing new-product development process. Exercise 4.59 (page 250) gives the distributions of X, the number of weeks to complete the development of product specifications, and Y , the number of weeks to complete the design of the manufacturing process. You did some useful variance calculations in
Comparing sales. It is unlikely that the daily sales of Tamara and Derek in the previous problem are independent. They will both sell more during the Christmas season, for example. Suppose that the correlation between their sales is ρ = 0.4. What are now the mean and standard deviation of the
Comparing sales. Tamara and Derek are sales associates in a large electronics and appliance store. Their store tracks each associate’s daily sales in dollars. Tamara’s sales total X varies from day to day with mean and standard deviationμX = $1100 and σX = $100 Derek’s sales total Y also
Managing new-product development process. Exercise 4.59 (page 250) gives the distribution of time to complete two steps in the new-product development process.(a) Calculate the variance and the standard deviation of the number of weeks to complete the development of product specifications.(b)
Civilian sales. Example 4.16 also gives the distribution of Gain’s civilian sales Y . Find the variance σ2 Y and the standard deviation σY .4.62 Hard-drive sizes. Example 4.9 (page 240) gives the distribution of hard-drive sizes for sales of a particular model computer. You found the mean
Mutual funds. The addition rule for means extends to sums of any number of random variables. Let’s look at a portfolio containing three mutual funds.The monthly returns on Fidelity Magellan Fund, Fidelity Energy Fund, and Fidelity Japan Fund for the 60 months ending in August 2008 had
Managing new-product development process. Managers often have to oversee a series of related activities directed to a desired goal or output. As a new-product development manager, you are responsible for two sequential steps of the product development process, namely, the development of product
Customer orders. Each week your business receives orders from customers who wish to have air-conditioning units installed in their homes. From past data, you estimate the distribution of the number of units ordered in a week X to be Units ordered X: 0 1 2 3 4 5 Probability: 0.05 0.15 0.27 0.33 0.13
Hard-drive sizes. Example 4.9 (page 240) gives the distribution of customer choices of hard-drive size for a laptop computer model. Find the mean μ of this probability distribution. Explain in simple language what μ tells us about customer choices. Also explain why knowing μ is not very helpful
Household size. Choose an American household at random and let X be the number of persons living in the household. If we ignore the few households with more than seven inhabitants, the probability model for X is as follows:Household size X: 1 2 3 4 5 6 7 Probability: 0.25 0.32 0.17 0.15 0.07 0.03
Normal probabilities. Example 4.11 gives the Normal distribution N(50,000, 5500) for the tread life X of a type of tire (in miles). Calculate the following probabilities:(a) The probability that a tire lasts more than 50,000 miles.(b) P(X > 60,000)(c) P(X ≥ 60,000)
Uniform probabilities. Let Y be a random number between 0 and 1 produced by the idealized uniform random number generator with density curve pictured in Figure 4.12. Find the following probabilities:(a) P(0 ≤ Y ≤ 0.4)(b) P(0.4 ≤ Y ≤ 1)(c) P(0.3 ≤ Y ≤ 0.5)(d) P(0.3 < Y < 0.5)(e) P(0.226
How many cars? Choose an American household at random and let the random variable X be the number of cars (including SUVs and light trucks) they own. Here is the probability model if we ignore the few households that own more than 5 cars:Number of cars X: 0 1 2 3 4 5 Probability: 0.09 0.36 0.35
Discrete or continuous? For each exercise listed below, decide whether the random variable described is discrete or continuous and sketch the sample space on a number line.(a) Exercise 4.17(a), page 225(b) Exercise 4.17(c), page 225(c) Exercise 4.17(e), page 225
Discrete or continuous? For each exercise listed below, decide whether the random variable described is discrete or continuous and give a brief explanation for your response.(a) Exercise 4.16(b), page 225(b) Exercise 4.16(e), page 225(c) Exercise 4.26, page 234(d) Exercise 4.47, page 237
Miles per gallon for 2009 vehicles. Exercise 4.26 indicates that city gas mileage for 2009 model year vehicles is approximately Normal with mean 17.3 miles per gallon and standard deviation 4.2 miles per gallon.(a) What is the probability that a randomly selected 2009 vehicle has city mpg greater
SAT Math Reasoning Scores. Exercise 4.27 indicates that SAT Math Reasoning test scores are approximately Normal with mean 515 points and standard deviation 116 points.(a) What is the probability that a randomly selected student has a score greater than 550?(b) What is the probability that a
How large are households? Choose an American household at random and let X be the number of persons living in the household. If we ignore the fewhouseholds with more than seven inhabitants, the probability model for X is as follows:4.48 Random numbers. Many random number generators allow users to
Moving up. A study of class mobility in England looked at the economic class reached by the sons of lower-class fathers.Economic classes are numbered from 1 (low) to 5 (high). Take X to be the class of a randomly chosen son of a father in Class 1.The study found that the probability model for X
Consumer preference. Suppose that half of all computer users prefer the new version of your company’s best-selling software, and half prefer the old version. If you randomly choose three consumers and record their preferences, there are 8 possible outcomes. For example, NNO means the first two
Roulette. A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction. The wheel is carefully balanced so
How big are farms? Choose an American farm at random and measure its size in acres. The probabilities that the farm chosen falls in several acreage categories are shown below.Let A be the event that the farm is less than 50 acres in size, and let B be the event that it is 500 acres or more.(a) Find
Who goes to Paris? Abby, Deborah, Sam, Tonya, and Roberto work in a firm’s public relations office. Their employer must choose two of them to attend a conference in Paris. To avoid unfairness, the choice will be made by drawing two names from a hat. (This is an SRS of size 2.)(a) Write down all
Legitimate probabilities? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, satisfies the rules of probability. If not, give specific reasons for your answer.(a) When a coin is spun, P(H) = 0.55 and P(T) =
Almond M&M’s. Exercise 4.39 gives the probabilities that an M&M candy is each of blue, orange, green, brown, yellow, and red. If “Almond” M&M’s are equally likely to be any of these colors, what is the probability of any one color?
Colors of M&M’s. The colors of candies such as M&M’s are carefully chosen to match consumer preferences. The color of an M&M drawn at random from a bag has a probability distribution determined by the proportions of colors among all M&M’s of that type.(a) Here is the
Car colors. Choose a new car or light truck at random and note its color. Here are the probabilities of the most popular colors for cars purchased in China in 2006:What is the probability that the car you choose has any color other than the six listed? What is the probability that a randomly chosen
Land in Iowa. Choose an acre of land in Iowa at random.The probability is 0.92 that it is farmland and 0.01 that it is forest.(a) What is the probability that the acre chosen is not farmland?(b) What is the probability that it is either farmland or forest?(c) What is the probability that a randomly
Stock market movements. You watch the price of the Dow Jones Industrial Index for four days. Give a sample space for each of the following random phenomena.(a) You record the sequence of up-days and down-days.(b) You record the number of up-days.
Great Britain household size. In government data, a household consists of all occupants of a dwelling unit. Here is the distribution of household size in Great Britain according to the 2005 General Household Survey:Choose a Great Britain household at random and let the random variable Y be the
World Internet usage. Approximately 23.6% of the world’s population uses the Internet (as of December 2008).Furthermore, a randomly chosen Internet user has the following probabilities of being from the given region of the world:(a) What is the probability of a randomly chosen Internet user not
Modes of transportation. Governments (local and national)find it important to gather data on modes of transportation for commercial and workplace movement. Such information is useful for policy making as it pertains to infrastructure (like roads and railways), urban development, energy use, and
Spill or spell? Spell-checking software catches “nonword errors” that result in a string of letters that is not a word, as when “the” is typed as “teh.” When undergraduates are asked to write a 250-word essay (without spell-checking), the number X of nonword errors has the following
Online health information. Based on a random sample of 1010 adults (18 years or older), a Harris Poll (July 8 to 13, 2008) estimates that 150 million U.S. adults have gone online for health information. Such individuals have been labeled as “cyberchondriacs.”Cyberchondriacs in the sample were
Demographics—language. Canada has two official languages, English and French. Choose a Canadian at random and ask, “What is your mother tongue?” Here is the distribution of responses, combining many separate languages from the broad Asian/Pacific region:(a) What probability should replace
Confidence in institutions. A Gallup Poll (June 9 to 12, 2008) interviewed a random sample of 822 adults (18 years or older). The people in the sample were asked about their level of confidence in a variety of institutions in the United States. Here are the results for small and big businesses:(a)
Second Life around the world. Second Life(www.secondlife.com) is a free social network service based on an online 3D virtual world in which users, called “Residents,”create avatars to interact with each other. Active Second Life users are distributed worldwide:8(a) What probability should
SAT Math Reasoning scores. Scores on the SAT Math Reasoning test for 2008 high school graduates are approximately Normal with mean 515 points (out of a possible 800 points) and standard deviation 116 points. Let Y stand for the score of a randomly chosen student. Express each of the following
Miles per gallon for 2009 vehicles. The Normal distribution with mean μ =17.3 miles per gallon and standard deviation σ = 4.2 miles per gallon is an approximate model for the city gas mileage of 2009 model year vehicles. Figure 1.17 (page 41)pictures this distribution. Let X be the city miles per
Adding random numbers. Generate two random numbers between 0 and 1 and take T to be their sum. The sum T can take any value between 0 and 2. The density curve of T is the triangle shown in Figure 4.6.(a) Verify by geometry that the area under this curve is 1.(b) What is the probability that T is
Random numbers. Let Y be a random number between 0 and 1 produced by the idealized uniform random number generator described in Example 4.7 and Figure 4.5.Find the following probabilities:(a) P(0 ≤ Y ≤ 0.4)(b) P(0.4 ≤ Y ≤ 1)(c) P(0.3 ≤ Y ≤ 0.5)(d) P(0.3 < Y < 0.5)
Job satisfaction. We can use the results of a 2008 poll on job satisfaction to give a probability model for the job satisfaction rating of a randomly chosen employed(full-time or part-time) American.7 Here is the model:(a) What is the probability of a randomly selected employed American being
U.S. power plant energy consumption. Drawa U.S. power plant at random and record the primary source of energy that the power plant relies on to produce electricity and/or heat for the public. “At random” means that we give every such power plant the same chance to be the one we choose. That is,
Rolling a die. Figure 4.4 displays several assignments of probabilities to the six faces of a die. We can learn which assignment is actually accurate for a particular die only by rolling the die many times. However, some of the assignments are not legitimate assignments of probability. That is,
Grading Canadian health care. Annually, the Canadian Medical Association uses the marketing research firm Ipsos Canada to measure public opinion with respect to the Canadian health care system. Between June 10 and June 12 of 2008, Ipsos Canada interviewed a random sample of 1002 adults.5 The people
Causes of death. Government data on job-related deaths assign a single occupation for each such death that occurs in the United States. The data on occupational deaths in 2007 show that the probability is 0.105 that a randomly chosen death was agriculture related and 0.072 that it was manufacturing
Moving up. An economist studying economic class mobility finds that the probability that the son of a lower-class father remains in the lower class is 0.46. What is the probability that the son moves to one of the higher classes?
Describing sample spaces. In each of the following situations, describe a sample space S for the random phenomenon. In some cases you have some freedom in specifying S, especially in setting the largest and the smallest value in S.(a) Choose a student in your class at random. Ask how much time that
Describing sample spaces. In each of the following situations, describe a sample space S for the random phenomenon. In some cases, you have some freedom in your choice of S.(a) A new business is started. After two years, it is either still in business or it has closed.(b) A rust prevention
Customer returns. Shoppers return about 6% of everything they buy. Use your statistical software to complete this exercise.(a) Set the probability for Bernoulli trials to 0.06. Generate 100 Bernoulli trials. How many of the 100 trials resulted in a “success”?What percent of the trials does this
Financial fraud. Continue with the Experian-Gallup Personal Credit Index survey described in Exercise 4.3. In Exercise 4.3, a fair die was used to simulate the selection of a fraud victim.The die provided a 1-in-6 (or 16.66667%) chance of a fraud victim’s knowing the perpetrator of the fraud. The
More efficient simulation. Continue the exploration begun in Exercise 4.12. Software allows you to simulate many independent Yes/No trials more quickly if all you want to save is the count of Yes outcomes. The key word “Binomial” simulates n independent Bernoulli trials, each with probability p
Simulating an opinion poll. A recent opinion poll showed that about 59% of the American public approve of labor unions.3 Suppose that this is exactly true. Choosing a person at random then has probability 0.59 of getting one who approves of labor unions. Use the Probability applet or your
Simulating consumer behavior. About half of the customers entering a local computer store will purchase a newcomputer before leaving the store. Use the Probability applet or your statistical software to simulate 100 customers independently entering the store with each having a probability of 0.5 of
What probability doesn’t say. The idea of probability is that the proportion of heads in many tosses of a balanced coin eventually gets close to 0.5. But does the actual count of heads get close to one-half the number of tosses? Let’s find out. Set the “Probability of heads” in the
Thinking about probability statements. Probability is a measure of how likely an event is to occur. Match one of the probabilities that follow with each statement of likelihood given.(The probability is usually a more exact measure of likelihood than is the verbal statement.)0 0.01 0.3 0.6 0.99
Three of a kind. You read in a book on poker that the probability of being dealt three of a kind in a five-card poker hand is 1/50. Explain in simple language what this means.
Tossing a thumbtack. Toss a thumbtack on a hard surface 100 times. How many times did it land with the point up? What is the approximate probability of landing point up?
Random digits. As discussed in Chapter 3, generation of random numbers is one approach for obtaining a simple random sample (SRS). If we were to look at the random generation of digits, the mechanism should give each digit probability 0.1.Consider the digit “0” in particular.(a) The table of
Credit monitoring. In a recent study of consumers, 25%reported purchasing a credit-monitoring product that alerts them to any activity on their credit report. To study this situation with a physical device providing the randomness as opposed to a computer simulation, we will need to be a bit more
Financial fraud. In December 2006, the Experian-Gallup Personal Credit Index survey found that 17% of fraud victims knew the perpetrator as a friend or acquaintance. Financial fraud includes crimes like unauthorized credit card charges, withdrawal of money from a savings or checking account, and
Nickels falling over. You may feel that it is obvious that the probability of a head in a coin experiment is about 1/2 because the coin has two faces. Such opinions are not always correct. The previous exercise asked you to spin a nickel rather than toss it—that changes the probability of a head.
Nickels spinning. Hold a nickel upright on its edge under your forefinger on a hard surface, then snap it with your other forefinger so that it spins for some time before falling. Based on 50 spins, estimate the probability of heads.
Your institutional review board (optional). Your college or university has an institutional review board that screens all studies that use human subjects. Get a copy of the document that describes this board (you can probably find it online).(a) According to this document, what are the duties of
Two ways to ask sensitive questions. Sample survey questions are usually read from a computer screen. In a Computer Aided Personal Interview (CAPI), the interviewer reads the questions and enters the responses. In a Computer Aided Self Interview (CASI), the interviewer stands aside and the
Changing the sample size. Draw 100 samples of each of the sizes n = 60, n = 200, and n = 800 from a population with p = 0.6. Make a histogram of the ˆp-values for each simulation, using the same horizontal and vertical scales so that the three graphs can be compared easily. How does increasing the
Changing the population. Draw 100 samples of size n = 60 from populations with p = 0.1, p = 0.3, and p = 0.5.Make a stemplot of the 100 values of ˆp obtained in each simulation.Compare your three stemplots. Do they show about the same variability? How does changing the parameter p affect the
Randomization for testing a breakfast food. To demonstrate how randomization reduces confounding, return to the breakfast food testing experiment described in Example 3.20(page 172). Label the 30 rats 01 to 30. Suppose that, unknown to the experimenter, the 10 rats labeled 01 to 10 have a genetic
Design your own experiment. The previous two exercises illustrate the use of statistically designed experiments to answer questions of interest to consumers as well as to businesses.Select a question of interest to you that an experiment might answer and briefly discuss the design of an appropriate
Mac versus PC. Many people hold very strong opinions about the superiority of the computer that they use. Design an experiment to compare customer satisfaction with the Mac versus the PC. Consider whether or not you will include individuals who routinely use both types of computers and whether or
Quality of service. Statistical studies can often help service providers assess the quality of their service. The United States Postal Service is one such provider of services.Wewonder if the number of days a letter takes to reach another city is affected by the time of day it is mailed and whether
The product should not be discolored. Fewpeoplewant to eat discolored french fries. Potatoes are kept refrigerated before being cut for french fries to prevent spoiling and preserve flavor.But immediate processing of cold potatoes causes discoloring due to complex chemical reactions. The potatoes
Stocks go down on Monday. Puzzling but true: stocks tend to go down on Mondays, both in the United States and in overseas markets. There is no convincing explanation for this fact. A recent study looked at this “Monday effect” in more detail, using data on the daily returns of stocks on several
Treating drunk drivers. Once a person has been convicted of drunk driving, one purpose of court-mandated treatment or punishment is to prevent future offenses of the same kind. Suggest three different treatments that a court might require. Then outline the design of an experiment to compare their
Student attitudes concerning labor practices. Youwant to investigate the attitudes of students at your school about the labor practices of factories that make college-brand apparel.You have a grant that will pay the costs of contacting about 500 students.(a) Specify the exact population for your
Attitudes toward collective bargaining. A labor organization wants to study the attitudes of college faculty members toward collective bargaining. These attitudes appear to be different depending on the type of college. The American Association of University Professors classifies colleges as
How much do students earn? A university’s financial aid office wants to know how much it can expect students to earn from summer employment. This information will be used to set the level of financial aid. The population contains 3478 students who have completed at least one year of study but
Learning about markets. Your economics professor wonders if playing market games online will help students understand how markets set prices. You suggest an experiment: have some students use the online games, while others discuss markets in recitation sections. The course has two lectures, at 8:30
Will regulation make the product safer? Canada requires that cars be equipped with “daytime running lights,” headlights that automatically come on at a low level when the car is started. Some manufacturers are now equipping cars sold in the United States with running lights. Will running lights
Questions about attitudes. Write two questions about an attitude that concerns you for use in a sample survey. Make the first question so that it is biased in one direction and make the second question biased in the opposite direction. Explain why your questions are biased and then write a third
Does the new product taste better? Before a newvariety of frozen muffins is put on the market, it is subjected to extensive taste testing. People are asked to taste the new muffin and a competing brand and to say which they prefer. (Both muffins are unidentified in the test.) Is this an
Executives and exercise. A study of the relationship between physical fitness and leadership uses as subjects middleaged executives who have volunteered for an exercise program.The executives are divided into a low-fitness group and a highfitness group on the basis of a physical examination. All
Employee meditation. You see a news report of an experiment that claims to show that a meditation technique increased job satisfaction of employees. The experimenter interviewed the employees and assessed their levels of job satisfaction. The subjects then learned how to meditate and did so
A hot fund. A large mutual funds group assigns a young securities analyst to manage its small biotechnology stock fund.The fund’s share value increases an impressive 43% during the first year under the new manager. Explain why this performance does not necessarily establish the manager’s
Marketing a dietary supplement. Your company produces a dietary supplement that contains a significant amount of calcium as one of its ingredients. The company would like to be able to market this fact successfully to one of the target groups for the supplement: men with high blood pressure. To
Can you remember how many? An opinion poll calls 1800 randomly chosen residential telephone numbers, then asks to speak with an adult member of the household. The interviewer asks, “How many movies have you watched in a movie theater in the past 12 months?”(a) What population do you think the
Coupons and customer expectations. A researcher studying the effect of coupons on consumers’ expectations makes up two different series of ads for a hypothetical brand of cola for the past year. Students in a family science course view one or the other sequence of ads on a computer. Some students
Compare the fries. Do consumers prefer the fries from Burger King or from McDonald’s? Design a blind test in which neither source of the fries is identified. Describe briefly the design of a matched pairs experiment to investigate this question.How will you use randomization?
Choose the type of study. Give an example of a question about your customers, their behavior, or their opinions thatwould best be answered by(a) a sample survey.(b) an observational study that is not a sample survey.(c) an experiment.
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