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introduction to operations research
Introduction To The Practice Of Statistics 10th Edition David S. Moore, George P. McCabe, Bruce A. Craig - Solutions
4.31 Parents with alleles A and O. Andreona and Caleb both have alleles A and O.a. What blood types can their children have?b. What is the probability that their next child has each of these blood types?
4.30 Blood types of children. Emily and Michael both have alleles O and O.a. What blood types can their children have?b. What is the probability that their next child has each of these blood types?
4.29 Independence of complements. Show that if events A and B obey the multiplication rule, P(A and B)=P(A)P(B) , then A and the complement Bc of B also obey the multiplication rule, P(A and Bc)=P(A)P(Bc) .That is, if events A and B are independent, then A and Bc are also independent.(Hint: Start
4.28 Axioms of probability. Show that any assignment of probabilities to events that obeys Rules 2 and 3 on page 213 automatically obeys the complement rule (Rule 4). This implies that a mathematical treatment of probability can start from just Rules 1, 2, and 3. These rules are sometimes called
4.27 Universal blood donors. People with type O-negative blood are universal donors. That is, any patient can receive a transfusion of O-negative blood. Only 7% of the American population have Onegative blood. If six people appear at random to give blood, what is the probability that at least one
4.26 PINs. The personal identification numbers (PINs) for automatic teller machines usually consist of four digits. You notice that most of your PINs have at least one 0, and you wonder if the issuers use lots of 0s to make the numbers easy to remember. Suppose that PINs are assigned at random, so
4.25 Winning the lottery. A state lottery’s Pick 3 game asks players to choose a three-digit number, 000 to 999. The state chooses the winning three-digit number at random so that each number has probability 1/1000. You win if the winning number contains the digits in your number, in any order.a.
4.24 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
4.23 Rh blood types. Human blood is typed as O, A, B, or AB and also as Rh-positive or Rh-negative.ABO type and Rh-factor type are independent because they are governed by different genes. In the American population, 84% of people are Rh-positive. Use the information about ABO type in Exercise 4.17
4.22 Loaded dice. There are many ways to produce crooked dice. To load a die so that 6 comes up too often and 1 (which is opposite 6) comes up too seldom, add a bit of lead to the spot on the 1 face.Because the spot is solid plastic, this works even with transparent dice. If a die is loaded so that
4.21 Education levels of young adults. Choose a young adult (age 25 to 34 years) at random. The probability is 0.12 that the person chosen did not complete high school, 0.31 that the person has a high school diploma but no further education, and 0.29 that the person has at least a bachelor’s
4.20 French and English in Canada. 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:Language English French Other Probability 0.56 0.21 ?a. What probability should replace “?” in
4.19 Are the probabilities legitimate? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate—that is, whether it satisfies the rules of probability. If not, give specific reasons for your answer.a. Deal a card from a
4.18 Blood types in Ireland. The distribution of blood types in Ireland differs from the U.S. distribution given in the previous exercise:Blood type A B AB O Ireland probability 0.35 0.10 0.03 0.52 Choose a person from the United States and a person from Ireland at random, independently of each
4.17 Distribution of blood types. All human blood can be “ABO-typed” as one of O, A, B, or AB, but the distribution of the types varies a bit among groups of people. Here is the distribution of blood types for a randomly chosen person in the United States:Blood type A B AB O U.S. probability
4.16 Record the length of time spent on the page. Refer to the previous exercise. You also decide to measure the length of time a visitor spends on your page. Give the sample space for this measure.
4.15 Evaluating web page designs. You are a web page designer, and you set up a page with four different links. A user of the page can click on one of the links, or he or she can leave that page. Describe the sample space for the outcome of someone visiting your web page.
4.14 What’s wrong? In each of the following scenarios, there is something wrong. Describe what is wrong and give a reason for your answer.a. If we select a digit at random, then the probability of selecting a 4 is 0.4.b. If the probability of A is 0.2, the probability of B is 0.3, and the
4.13 Equally likely events. For each of the following situations, explain why you think that the events are equally likely or not. Explain your answers.a. You roll a fair die and get an even number or an odd number.b. You are observing cars at an intersection. You classify the movement of each car
4.12 The multiplication rule for independent events. The probability that a randomly selected person prefers the vehicle color white is 0.25. Can you apply the multiplication rule for independent events in the situations described in parts (a) and (b)? If your answer is Yes, apply the rule.a. Two
4.11 Probability rules. For each of the following situations, state the probability rule or rules that you would use and apply it or them. Write a sentence explaining how the situation illustrates the use of the probability rules.a. A coin is tossed three times. The probability of zero heads is
4.10 What’s wrong? In each of the following scenarios, there is something wrong. Describe what is wrong and give a reason for your answer.a. If the probability of A is 0.6 and the probability of B is 0.7, the probability of both A and B happening is 1.3.b. If the probability of A is 0.55, then
4.9 Define the sample space. For each of the following questions, define a sample space for the associated random phenomenon. Explain your answers. Be sure to specify units if that is appropriate.a. How many people do you talk with in a typical day?b. How much did you spend on textbooks this
4.8 What is the sample space? You toss a fair coin five times.a. What is the sample space if you record the result of each toss (H or T)?b. What is the sample space if you record the number of heads?
4.7 A question about dice. Here is a question that a French gambler asked the mathematicians Fermat and Pascal at the very beginning of probability theory: What is the probability of getting at least one 6 in rolling four dice? The Law of Large Numbers applet allows you to roll several dice and
4.6 Use the Probability applet. 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
4.5 Are the trials independent? For each of the following situations, identify the trials as independent or not. Explain your answers.a. Throw a ball at a target 100 times. For the first throw, you use your right hand, and for the second you use your left hand. You continue alternating right and
4.4 Winning at craps. The game of craps starts with a “come-out” roll, in which the shooter rolls a pair of dice. If the total of the “spots” on the up-faces is 7 or 11, the shooter wins immediately. (There are ways that the shooter can win on later rolls if other numbers are rolled on the
4.3 Interpret the probabilities. Refer to Exercise 4.1. In each case, interpret the term probability for the phenomena that are random. For those that are not random, explain why the term probability does not apply.
4.2 Two possible outcomes. Explain why a random phenomenon with two possible outcomes does not necessarily have 0.5 as the probability for each of the outcomes.
4.1 Are these phenomena random? Identify each of the following phenomena as random or not. Give reasons for your answers.a. You roll a six-sided die that has one to six spots on each side.b. The last digit of your phone number.c. You draw a 10 from a well-shuffled deck of 52 cards.d. Your date of
3.80 Informed consent. The requirement that human subjects give their informed consent to participate in an experiment can greatly reduce the number of available subjects. For example, a study of new teaching methods asks the consent of parents for their children to be randomly assigned to be
3.79 Would the results be different for men and women? The drug that is the subject of the experiment in Exercise 3.78 may behave differently in men and women. How would you modify your experimental design to take this into account?
3.78 Compare two doses of a drug. A drug manufacturer is studying how a new drug behaves in patients. Investigators compare two doses: 5 milligrams(mg) and 10 mg. The drug can be administered by injection, by a skin patch, or by intravenous drip. Concentration in the blood after 30 minutes (the
3.77 Design a survey. You want to investigate the attitudes of students at your school about the faculty’s commitment to teaching. The student government will pay the costs of contacting about 500 students.a. Specify the exact population for your study. For example, will you include part-time
3.76 A cheating experiment. A psychologist conducts the following experiment. She measures the attitude of subjects toward cheating and then has them play a game rigged so that winning without cheating is impossible. The computer that organizes the game also records—unknown to the
3.75 A theft experiment. Students sign up to be subjects in a psychology experiment. When they arrive, they are told that interviews are running late and are taken to a waiting room. The experimenters then stage a theft of a valuable object left in the waiting room. Some subjects are alone with the
3.74 Should we ask for the consent of the parents? The Centers for Disease Control and Prevention, in a survey of teenagers, asked the subjects if they were sexually active. Those who said Yes were then asked, “How old were you when you had sexual intercourse for the first time?” Should consent
3.73 Use of data produced by the government. Data produced by the government are often available free or at low cost to private users. For example, satellite weather data produced by the U.S. National Weather Service are available free to TV stations for their weather reports and to anyone on the
3.72 Your institutional review board. 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 the
3.71 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
3.70 Design an experiment. The previous two exercises illustrate the use of statistically designed experiments to answer questions that arise in everyday life. Select a question of interest to you that an experiment might answer and carefully discuss the design of an appropriate experiment.
3.69 Bicycle gears. How does the time it takes a bicycle rider to travel 100 meters depend on which gear is used and how steep the course is? It may be, for example, that higher gears are faster on level ground, but lower gears are faster on steep inclines. Discuss the design of a two-factor
3.68 Compare the fries. Do consumers prefer the fries from Burger King or from McDonald’s? Design a blind test in which the source of the fries is not identified. Describe briefly an experiment with matched pairs design that you could use to investigate this question. How will you use
3.67 Choose the type of study. Give an example of a question about pets and their owners, their behavior, or their opinions that would best be answered bya. a sample survey.b. an observational study that is not a sample survey.c. an experiment.
3.66 Calcium and healthy bones. Adults need to eat foods or supplements that contain enough calcium to maintain healthy bones. Calcium intake is generally measured in milligrams per day (mg/d), and one measure of healthy bones is total body bone mineral density measured in grams per centimeter
3.65 Price promotions and consumer expectations. A researcher studying the effect of price promotions on consumer expectations makes up two different histories of the store price of a hypothetical brand of laundry detergent for the past year. Students in a marketing course view one price history or
3.64 Name the designs. What is the name for each of these study designs?a. A study to compare two methods of preserving wood started with boards of southern white pine. Each board was ripped from end to end to form two edge-matched specimens. One was assigned to Method A; the other was assigned to
3.63 Make it an experiment! In the following observational studies, describe changes that could be made to the data collection process that would result in an experiment rather than an observational study. Also, offer suggestions about unseen biases or lurking variables that may be present in the
3.62 Confidentiality at NORC. The National Opinion Research Center(NORC) conducts a large number of surveys and has established procedures for protecting the confidentiality of its survey participants. For its Survey of Consumer Finances, NORC provides a pledge to participants regarding 35
3.61 Online behavioral advertising. The Federal Trade Commission Staff Report “Self-Regulatory Principles for Online Behavioral Advertising” defines behavioral advertising as “the tracking of a consumer’s online activities over time—including the searches the consumer has conducted, the
3.60 Ethics and statistics news. Studies using statistical methods to analyze data and draw conclusions should be conducted in an ethical manner. Visit the website www.sciencedaily.com/news/science_society/ethics. Read one of the articles posted on this site and then write a short report
3.59 Informed consent to take blood samples. Researchers from Yale, working with medical teams in Tanzania, wanted to know how common infection with the AIDS virus is among pregnant women in that country. To do this, they planned to test blood samples drawn from pregnant women.Yale’s
3.58 Should poll results be made public? Some people think that the law should require that all political poll results be made public. Otherwise, the possessors of poll results can use the information to their own advantage. They can act on the information, release only selected parts of it, or
3.57 Political polls. The presidential election campaign is in full swing, and the candidates have hired polling organizations to take regular polls to find out what the voters think about the issues.a. What information should the pollsters be required to give out?b. What does the standard of
3.56 Use of stored blood. Long ago, doctors drew a blood specimen from you as part of treating minor anemia. Unknown to you, the sample was stored. Now researchers plan to use stored samples from you and many other people to look for genetic factors that may influence anemia. It is no longer
3.55 Anonymity and confidentiality in mail surveys. Some common practices may appear to offer anonymity while actually delivering only confidentiality. Market researchers often use mail surveys that do not ask the respondent’s identity but contain hidden codes on the questionnaire that identify
3.54 Anonymity and confidentiality in health screening. Texas A&M, like many other universities, offers free screening for HIV, the virus that causes AIDS. The announcement says, “Persons who sign up for the HIV Screening will be assigned a number so that they do not have to give their name.”
3.53 The General Social Survey. One of the most important nongovernment surveys in the United States is the National Opinion Research Center’s General Social Survey. The GSS regularly monitors public opinion on a wide variety of political and social issues. Interviews are conducted in person in
3.52 The Vytorin controversy. Vytorin is a combination pill designed to lower cholesterol. It consists of a relatively inexpensive and widely used drug, Zocor, and a newer drug called Zetia. Early study results suggested that Vytorin was no more effective than Zetia. Critics claimed that the makers
3.51 How should the samples have been analyzed? Refer to the ovarian cancer diagnostic test study in Example 3.50 (page 198).Describe how you would process the samples through the mass spectrometer.
3.50 What is wrong? Explain what is wrong in each of the following scenarios.a. Clinical trials are always ethical, as long as they randomly assign patients to the treatments.b. An institutional review board is done with its job when it decides to allow a study to be conducted.34c. A treatment that
3.49 Is the IRB responsible? An institutional review board (IRB)approved an experimental cancer vaccine for use in a clinical trial. The subjects were patients who had advanced disease and had received standard treatments with no success. Of the 94 subjects who received the vaccine, 26 died during
3.48 Are the data confidential? You have participated in a study, and the results were published in an article in a very prestigious journal. Only summary information was published. The policy of the journal requires that all data used in the articles it publishes be available to the public, and it
3.47 Did you give informed consent? You were asked to participate in a study by a friend who is recruiting subjects. You trust your friend and tell her that you are willing to do whatever is needed for the study. Have you given informed consent? Explain your answer.
3.46 Survey questions. Comment on each of the following as a potential sample survey question. Is the question clear? Is it slanted toward a desired response?a. “Some cell phone users have developed brain cancer. Should all cell phones come with a warning label explaining the danger of using cell
3.45 The sampling frame. The list of individuals from which a sample is actually selected is called the sampling frame. Ideally, the frame should list every individual in the population, but in practice this is often difficult. A common source of undercoverage is frames leaving out part of the
3.44 Stratified samples for accounting audits. Accountants use stratified samples during audits to verify a company’s records of such things as accounts receivable. The stratification is based on the dollar amount of the item and often includes 100% sampling of the largest items. One company
3.43 Select club members to go to a convention. A club has 30 student members and 10 faculty members. The students are Abel Fisher Huber Moran Reinmann Carson Golomb Jimenez Moskowitz Santos Chen Griswold Jones Neyman Shaw David Hein Kiefer O’Brien Thompson Deming Hernandez Klotz Pearl Utts
3.42 Random digit telephone dialing. An opinion poll in California uses random digit dialing to choose telephone numbers at random. Numbers are selected separately within each California area code. The size of the sample in each area code is proportional to the population living there.a. What is
3.41 Systematic random samples versus simple random samples. The previous exercise introduces systematic random samples. Explain carefully why a systematic random sample does give every individual the same chance to be chosen but is not a simple random sample.
3.40 Systematic random samples. Systematic random samples are often used to choose a sample of apartments in a large building or dwelling units in a block at the last stage of a multistage sample. An example will illustrate the idea of a systematic sample. Suppose that we must choose 5 addresses
3.39 Stratified samples for attitudes about alcohol. At a party, there are 30 students over age 21 and 20 students under age 21. You choose at random 6 of those over 21 and separately choose at random 4 of those under 21 to interview about attitudes toward alcohol. You have given every student at
3.38 Select another simple random sample. There are 38 active telephone area codes in California. You want to choose an SRS of 8 of these area codes for a study of available telephone numbers. Label the codes 01 to 38 and use the Simple Random Sample applet, Table B, or software to choose your
3.37 Select a simple random sample. After you have labeled the individuals in a population, the Simple Random Sample applet automates the task of choosing an SRS. Use the applet to choose the sample in the previous exercise.
3.36 Using GIS to identify mint field conditions. A geographic information system (GIS) is to be used to distinguish different conditions in mint fields. Ground observations will be used to classify regions of each field as either healthy mint, diseased mint, or weed-infested mint. The GIS divides
3.35 Interview residents of apartment complexes. You are planning a report on apartment living in a college town. You decide to select eight apartment complexes at random for in-depth interviews with residents. Select a simple random sample of eight of the following apartment complexes. If you use
3.34 Identify the populations. For each of the following sampling situations, identify the population as exactly as possible. That is, say what kind of individuals the population consists of and say exactly which individuals fall in the population. If the information given is not complete, complete
3.33 Importance of students as customers. A committee on community relations in a college town plans to survey local businesses about the importance of students as customers. From telephone book listings, the committee chooses 70 businesses at random. Of these, 26 return the questionnaire mailed by
3.32 What’s wrong? Explain what is wrong with each of the following random selection procedures and explain how you would do the randomization correctly.a. To determine the reading level of an introductory statistics text, you evaluate all the written material in the third chapter.b. You want to
3.31 What kind of sample? In each of the following situations, identify the sample as an SRS, a stratified random sample, a multistage random sample, or a voluntary response sample. Explain your answers.a. A student organization has 45 members. A table of random digits is used to select a sample of
3.30 What’s wrong? Explain what is wrong in each of the following scenarios.a. The population consists of all individuals selected in a simple random sample.b. In a poll of an SRS of residents in a local community, respondents are asked to indicate the level of their concern about the dangers of
3.29 Select the students to be interviewed. Refer to the previous exercise.a. Use software and Table B to select the students. Give details.b. Compare the use of Table B with software for selecting the students. Which do you prefer? Give reasons for your answer.
3.28 Interview some students. You are a teaching assistant for an introductory statistics class. The instructor would like you to interview some of the students in the class to find out their opinions regarding the new interactive activities that she has introduced to the course. There are 33
3.27 Response rate. A survey designed to assess satisfaction with food items sold at a college’s football games was sent to 100 fans who had season tickets. The total number of fans who have season tickets is 6874. Responses to the survey were received from 62 fans.a. Describe the population for
3.26 How many text messages? You would like to know something about how many text messages you will receive in the next 50 days. Counting the number for each of the 50 days would take more time than you would like to spend on this project, so you randomly select 5 days from the 50 to count.a.
3.25 Measuring water quality in streams and lakes. Water quality of streams and lakes is an issue of concern to the public. Although trained professionals typically are used to take reliable measurements, many volunteer groups are gathering and distributing information based on data that they
3.24 A different design for calcium and the bones of young girls. Refer to the previous exercise.a. Outline a matched pairs design in which each girl receives both levels of calcium (with a “washout period” in which no calcium supplementation was given between the two treatment periods).b. What
3.23 Calcium and the bones of young girls. Calcium is important to the bone development of young girls. To study how the bodies of young girls process calcium, investigators used the setting of a summer camp. Calcium was given in punch at either a high level or a low level. The camp diet was
3.22 Random digits. Table B is a table of random digits. Which of the following statements are true of a table of random digits, and which are false? Explain your answers.a. Each triple of digits has 1/1000 chance of being 333.b. The digits 1234 can never appear as a group because this pattern is
3.21 Health benefits of bee pollen. “Bee pollen is effective for combating fatigue, depression, cancer, and colon disorders.” So says a website that offers the pollen for sale. We wonder if bee pollen really does prevent colon disorders. Here are two ways to study this question. Explain why the
3.20 Use the Simple Random Sample applet. The Simple Random Sample applet can demonstrate how randomization works to create similar groups for comparative experiments. Suppose that (unknown to the experimenters) the first 20 students among the 40 subjects for the smartphone study in Example 3.18
3.19 Use the Simple Random Sample applet. The Simple Random Sample applet allows you to randomly assign experimental units to more than two groups without difficulty. Consider a randomized comparative experiment in which 90 students are randomly assigned to three groups of 30.a. Use the applet to
3.18 Use the Simple Random Sample applet. You can use the Simple Random Sample applet to choose a group at random once you have labeled the subjects. Example 3.20 (page 168) uses Excel to choose 5 students from a group of 10 to receive a treatment in an experiment. The remaining 5 students will
3.17 Calcium and vitamin D. The body needs vitamin D in order to use calcium. An experiment is designed to study the effects of calcium and vitamin D supplements on the bones of first-year college students. The outcome measure is the total body bone mineral content (TBBMC), a measure of bone
3.16 What is wrong? Explain what is wrong with each of the following randomization procedures and describe how you would do the randomization correctly.a. Twenty subjects are to be assigned to two treatments, 10 to each. For each subject, a coin is tossed.If the coin comes up heads, the subject is
3.15 Evaluate a new orientation program. Your company runs a two-day orientation program for new employees on Monday and Tuesday each week. A new program is to be compared with the current one.Set up an experiment to compare the new program with the old. Be sure to provide details regarding
3.14 The Madden curse. Some people believe that individuals who appear on the cover of the football game Madden NFL will soon have a serious injury. Can you evaluate this belief with an experiment?Explain your answer.
3.13 What is needed? Explain what is deficient in each of the following proposed experiments and explain how you would improve the experiment.a. A study compares three marketing campaigns to encourage individuals to eat more fruits and vegetables. The first campaign is launched in Florida at the
3.12 Randomize the web pages for the running shoes. Refer to Exercise 3.10. Use software or Table B to randomize the treatments. Give a step-by-step detailed description of how you performed the randomization.
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