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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
5.5 Constructing a sampling distribution. Refer to Example 5.1 (page 272). Suppose Student Monitor also reported that the median number of weekly hours per course spent outside of class was 2.5 hours.a. Explain why we’d expect the population median to be less than the population mean in this
5.4 Is it unbiased? A statistic has a sampling distribution that is somewhat skewed. The mean is 20.0, the median is 19.3, and the quartiles are 15.3 and 23.9.a. If the true parameter value is 19.3, is the estimator unbiased?b. If the true parameter value is 20.0, is the estimator unbiased?c. If
5.3 Describe the population and the sample. For each of the following situations, describe the population and the sample.a. A survey of 18,875 people aged 18 to 25 reported that 55.1% drank alcohol in the past month.b. In a study of work stress, 250 restaurant workers were asked about the impact of
5.2 What’s wrong? For each of the following statements, explain what is wrong and why.a. A parameter describes a sample.b. Bias and variability are two names for the same thing.c. Large samples are always better than small samples.d. A sampling distribution summarizes the values of a statistic
5.1 A change in the requirement rules? Thirty students from your liberal arts college are randomly selected to be on a committee to evaluate immediate changes in the quantitative competency requirement. There are 2600 students in your college. The current rules say that a statistics course is one
4.124 Odds bets at craps. Refer to the odds bets at craps in Exercise 4.116. Suppose that whenever the shooter has an initial roll of 4, 5, 6, 8, 9, or 10, you take the odds. Here are the probabilities for these initial rolls:19 Point 4 5 6 8 9 10 Probability 3/36 4/36 5/36 5/36 4/36 3/36 Draw a
4.123 Higher education at two-year and four-year institutions. The following table gives the counts of U.S. institutions of higher education classified as public or private and as two-year or four-year:Public Private Two-year 1000 721 Four-year 2774 672 Convert the counts to probabilities and
4.122 Wine tasting. In the setting of the previous exercise, Taster 1’s rating for a wine is 3. What is the conditional probability that Taster 2’s rating is higher than 3?
4.121 Wine tasters. Two wine tasters rate each wine they taste on a scale of 1 to 5. From data on their ratings of a large number of wines, we obtain the following probabilities for both tasters’ ratings of a randomly chosen wine:Taster 2 Taster 1 1 2 3 4 5 1 0.03 0.02 0.01 0.00 0.00 2 0.02 0.07
4.120 Find some conditional probabilities. Choose a point at random in the square with sides 0≤x≤1 and 0≤y≤1 . This means that the probability that the point falls in any region within the square is the area of that region. Let X be the x coordinate and Y the y coordinate of the point
Suppose that, in fact, the probability is 0.3 that a randomly chosen student has plagiarized a paper. Draw a tree diagram in which the first stage is tossing the coin and the second is the truth about plagiarism.The outcome at the end of each branch is the answer given to the randomized-response
4.119 Sample surveys for sensitive issues. It is difficult to conduct sample surveys on sensitive issues because many people will not answer questions if the answers might embarrass them.Randomized response is an effective way to guarantee anonymity while collecting information on topics such as
4.118 Lottery tickets. Michael buys a ticket in the Tri-State Pick 3 lottery every day, always betting on 491. He will win something if the winning number contains 4, 9, and 1 in any order. Each day, Michael has probability 0.006 of winning, and he wins (or not) independently of other days because
4.117 An interesting case of independence. Independent events are not always easy to identify. Toss two balanced coins independently. The four possible combinations of heads and tails in order each have probability 0.25. The events A=head on the first tossB=both tosses have the same outcome may
4.116 A fair bet at craps. Almost all bets made at gambling casinos favor the house. In other words, the difference between the amount bet and the mean of the distribution of the payoff is a positive number.An exception is “taking the odds” at the game of craps, a bet that a player can make
4.115 Some probability distributions. Here is a probability distribution for a random variable X:Value of X 4 5 6 Probability 0.3 0.4 0.3a. Find the mean and standard deviation for this distribution.b. Construct a different probability distribution with the same possible values, the same mean, and
4.114 Find the probabilities. Refer to the previous exercise. Find the probabilities for each event.
4.113 Roll a pair of dice two times. Consider rolling a pair of fair dice two times (see Exercise 4.48, page 234). For each of the following pairs of events, tell whether they are disjoint, independent, or neither.a. A={8 on the first roll}, B={8 or more on the first roll}.b. A={8 on the first
4.112 A different transformation. Refer to the previous exercise. Now let Y=4X2−2 .a. Find the distribution of Y.b. Find the mean and standard deviation for the distribution of Y.c. Explain why the rules that you used for part (b) of the previous exercise do not work for this transformation.
4.111 Work with a transformation. Here is a probability distribution for a random variable X:Value of X 3 7 Probability 0.4 0.6a. Find the mean and the standard deviation of this distribution.b. Let Y=4X−2 . Use the rules for means and variances to find the mean and the standard deviation of the
4.110 Repeat the experiment many times and take the mean. Here is a probability distribution for a random variable X:Value of X −6 7 Probability 0.5 0.5 A single experiment generates a random value from this distribution. If the experiment is repeated many times, what will be the approximate
4.109 Repeat the experiment many times. Here is a probability distribution for a random variable X:Value of X −5 4 Probability 0.4 0.6 A single experiment generates a random value from this distribution. If the experiment is repeated many times, what will be the approximate proportion of times
4.108 Muscular dystrophy. Muscular dystrophy is an incurable muscle-wasting disease. The most common and serious type, called DMD, is caused by a sex-linked recessive mutation. Specifically, women can be carriers but do not get the disease; a son of a carrier has probability 0.5 of having DMD; a
4.107 Find some conditional probabilities. Beth knows the probabilities for her genetic types from part(c) of the previous exercise. She marries Bob, who is albino. Bob’s genetic type must be aa.a. What is the conditional probability that a child of Beth and Bob is non-albino if Beth has type
4.106 Albinism. People with albinism have little pigment in their skin, hair, and eyes. The gene that governs albinism has two forms (called alleles), which we denote a and A. Each person has a pair of these genes, one inherited from each parent. A child inherits one of each parent’s two alleles
4.105 Find a conditional probability. If Emily is offered the federal job, what is the conditional probability that she is also offered the New Jersey job? If Emily is offered the New Jersey job, what is the conditional probability that she is also offered the federal job?Genetic counseling.
4.104 Find the probability of another event. What is the probability that Emily is offered both the Connecticut and New Jersey jobs but not the federal job?
4.103 Find the probability of at least one offer. What is the probability that Emily is offered at least one of the three jobs?
4.102 Job offers. Emily is graduating from college. She has studied biology, chemistry, and computing and hopes to work as a forensic scientist applying her science background to crime investigation. Late one night she thinks about some jobs she has applied for. Let A, B, and C be the events Emily
4.101 Conditional probabilities and independence. Using the information in Exercise 4.100, answer these questions.a. Given that a vehicle is imported, what is the conditional probability that it is a light truck?b. Are the events “vehicle is a light truck” and “vehicle is imported”
4.100 Sales of cars and light trucks. Motor vehicles sold to individuals are classified as either cars or light trucks (including SUVs) and as either domestic or imported. In a recent year, 72% of vehicles sold were light trucks, 68% were domestic, and 54% were domestic light trucks. Let A be the
4.99 Draw a Venn diagram. Draw a Venn diagram that shows the relation between the events A and B in Exercise 4.97. Indicate each of the following events on your diagram and use the information in Exercise 4.97 to calculate the probability of each event. Finally, describe in words what each event
4.98 Find a conditional probability. In the setting of the previous exercise, what is the conditional probability that a household is prosperous, given that it is educated? Explain why your result shows that events A and B are not independent.
4.97 Education and income. Call a household prosperous if its income exceeds $100,000. Call the household educated if the householder completed college. Select an American household at random and let A be the event that the selected household is prosperous and B the event that it is educated.
4.96 Draw a different tree diagram for the same setting. Refer to the previous two exercises. Draw a tree diagram to illustrate the probabilities in a situation where you first identify the gender of the student and then identify the type of institution attended. Explain why the probabilities in
4.95 Draw a tree diagram. Refer to the previous exercise. Draw a tree diagram to illustrate the probabilities in a situation where you first identify the type of institution attended and then identify the gender of the student.
4.94 Attendance at two-year and four-year colleges. In a large national population of college students, 59% attend four-year institutions, and the rest attend two-year institutions. Males make up 44%of the students in the four-year institutions and 40% of the students in the two-year
4.93 Find another probability for lying to a teacher. Refer to the previous exercise. Suppose that you select a student from the subpopulation of those who would admit to lying to a teacher during the past year. What is the probability that the student is female? Be sure to show your work and
4.92 Find a probability for lying to a teacher. Suppose that 46% of high school students would admit to lying at least once to a teacher during the past year and that 28% of students are male and would admit to lying at least once to a teacher during the past year. Assume that 44% of the students
4.91 Venn diagram for exercise and sleep. Refer to the previous exercise. Draw a Venn diagram showing the probabilities for exercise and sleep.
4.90 Exercise and sleep. Suppose that 46% of adults get enough sleep, 40% get enough exercise, and 27% do both. Find the probabilities of the following events:a. Enough sleep and not enough exercise.b. Not enough sleep and enough exercise.c. Not enough sleep and not enough exercise.d. For each of
4.89 Are the events independent? Refer to Exercises 4.84 and 4.87. Are the age of the child and whether or not the child has adequate calcium intake independent? Calculate the probabilities that you need to answer this question and write a short summary of your conclusion.
4.88 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. P(A or B) is always equal to the sum of P(A) and P(B).b. The probability of an event minus the probability of its complement is always equal to 1.17c. Two
4.87 Use Bayes’s rule. Refer to Exercise 4.84. Use Bayes’s rule to find the probability that a child from this population who has inadequate intake is 5 to 10 years old.
4.86 Another probability. Refer to the previous exercise. What is the probability of the event that B occurs and A does not?
4.85 Find a probability. Suppose that P(A)=0.2 , P(B)=0.7 , and P(B|A)=0.1 . Find the probability that both A and B occur and use a Venn diagram to explain your calculation.
4.84 Is the calcium intake adequate? In the population of young children eligible to participate in a study of whether or not their calcium intake is adequate, 52% are 5 to 10 years of age and 48% are 11 to 13 years of age. For those who are 5 to 10 years of age, 18% have inadequate calcium intake.
4.83 Find the probability. Suppose that the probability that A occurs is 0.6 and the probability that A and B occur is 0.3. Find the probability that B occurs, given that A occurs and illustrate your calculations using a Venn diagram.
4.82 The complement. Refer to the previous exercise. Find the probability of the complement of the union of A, B, and C.
4.81 Unions. Assume that P(A)=0.3 , P(B)=0.2 , and P(C)=0.1 . If the events A, B, and C are disjoint, find the probability that the union of these events occurs. Draw a Venn diagram to illustrate your answer.
4.80 Why not? Suppose that P(B)=0.5 . Explain why P(A and B) cannot be 0.6.
4.79 Probability rules.a. Explain why a probability cannot be less than zero or greater than one.b. What is the probability of the event that A or Ac occurs. Explain your answer.c. If A and B are disjoint, what is the probability of A or B?d. What is the probability of the complement of A in terms
4.78 Find and explain some probabilities.a. Suppose P(A)=0.2 and P(B)=0.6 .Explain what it means for A and B to be disjoint. Assuming that they are disjoint, find the probability that A or B occurs.b. Explain in your own words the meaning of the rule P(S)=1 .c. Consider an event A. What is the name
4.77 Mean and standard deviation for 5 policies and for 20 policies. In fact, the insurance company in the previous exercise sees that in the entire population of homeowners, the mean loss from fire isμ=$300 , and the standard deviation of the loss isσ=$400 . What are the mean and standard
4.76 Fire insurance. An insurance company looks at the records for millions of homeowners and sees that the mean loss from fire in a year isμ=$300 per person. (Most of us have no loss, but a few lose their homes. The $300 is the average loss.) The company plans to sell fire insurance for $300 plus
4.75 Transform the distribution of heights from centimeters to inches. A report of the National Center for Health Statistics says that the heights of 20-year-old men have mean 176.8 centimeters (cm)and standard deviation 7.2 cm. There are 2.54 centimeters in an inch. What are the mean and standard
4.74 Will you assume independence? In which of the following games of chance would you be willing to assume independence of X and Y in making a probability model? Explain your answer in each case.a. In blackjack, you are dealt two cards and examine the total points X on the cards (face cards count
4.73 What happens when the correlation is 1? We know that variances add if the random variables involved are uncorrelated(ρ=0) but not otherwise.The opposite extreme is perfect positive correlation(ρ=1) . Show by using the general addition rule for variances that, in this case, the standard
4.72 Means and variances of sums. The rules for means and variances allow you to find the mean and variance of a sum of random variables without first finding the distribution of the sum, which is usually much harder to do.a. A single toss of a balanced coin has either 0 or 1 head, each with
4.71 Toss a four-sided die twice. Role-playing games like Dungeons & Dragons use many different types of dice. Suppose that a four-sided die has faces marked 1, 2, 3, and 4. The intelligence of a character is determined by rolling this die twice and adding 1 to the sum of the spots. The faces are
4.70 Calcium supplements and calcium in the diet. Refer to Example 4.52 (page 249). Suppose that people who have high intakes of calcium in their diets are more compliant than those who have low intakes. What effect would this have on the calculation of the standard deviation for the total calcium
4.69 Find the mean of the sum. Figure 4.12 (page 235) displays the density curve of the sum Y=X1+X2 of two independent random numbers, each uniformly distributed between 0 and 1.a. The mean of a continuous random variable is the balance point of its density curve. Use this fact to find the mean of
4.68 Suppose that the correlation is zero. Refer to Example 4.51 (page 248).a. Recompute the standard deviation for the total of the natural-gas bill and the electricity bill, assuming that the correlation is zero.b. Is this standard deviation larger or smaller than the standard deviation computed
4.67 What happens if the correlation is not zero? Suppose that X is a random variable with mean 20 and standard deviation 3. Also suppose that Y is a random variable with mean 60 and standard deviation 2. Assume that the correlation between X and Y is 0.4. Find the variance and the standard
4.66 Standard deviation for fruits and vegetables. Refer to Exercise 4.58. Find the variance and the standard deviation for the distribution of the number of servings of fruits and vegetables.
2. Assume that the correlation between X and Y is zero. Find the variance and the standard deviation of the random variable Z for each of the following cases. Be sure to show your work.a. Z=33−8X .b. Z=11X−6 .c. Z=X+Y .d. Z=X−Y .e. Z=−2X+2Y .
4.65 Find some variances and standard deviations. Suppose that X is a random variable with mean 20 and standard deviation 3. Also suppose that Y is a random variable with mean 60 and standard deviation
4.64 Standard deviation of the number of aces. Refer to Exercise 4.62. Find the standard deviation of the number of aces.
4.63 Find the variance and the standard deviation. A random variable X has the following distribution:X −2 −1 0 1 Probability 0.1 0.2 0.4 0.3 Find the variance and the standard deviation for this random variable. Show your work.
4.62 Mean of the distribution for the number of aces. In Exercise 4.47 (page 234) you examined the probability distribution for the number of aces when you are dealt two cards in the game Texas hold ’em.Let X represent the number of aces in a randomly selected deal of two cards in this game. Here
4.61 Find some means. Suppose that X is a random variable with mean 20 and standard deviation 2.Also suppose that Y is a random variable with mean 40 and standard deviation 7. Assume that the correlation between X and Y is zero. Find the mean of the random variable Z for each of the following
4.60 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 you toss a fair coin three times and get heads all three times, then the probability of getting a tail on the next toss is much greater than one-half.b.
4.59 Explain what happens when the sample size gets large. Consider the following scenarios: (1)You take a sample of two observations on a random variable and compute the sample mean, (2) you take a sample of 100 observations on the same random variable and compute the sample mean, (3) you take a
4.58 Servings of fruits and vegetables. The following table gives the distribution of the number of servings of fruits and vegetables consumed per day in a population:Number of servings X 0 1 2 3 4 5 Probability 0.4 0.1 0.1 0.2 0.1 0.1 Find the mean for this random variable.
4.57 Find the mean of the random variable. A random variable X has the following distribution:X −2 −1 0 1 Probability 0.1 0.2 0.4 0.3 Find the mean for this random variable. Show your work.
4.56 Different kinds of means. Explain the difference between the mean of a random variable and the mean of a sample.
4.55 Normal approximation for a sample proportion. A sample survey contacted an SRS of 700 registered voters in Oregon shortly after an election and asked respondents whether they had voted. Voter records show that 56% of registered voters had actually voted. We will see in the next chapter that,
4.54 How many close friends? How many close friends do you have? Suppose that the number of close friends adults claim to have varies from person to person with meanμ=8 and standard deviationσ=3 . An opinion poll asks this question of an SRS of 500 adults. We will see in the next chapter that, in
4.53 The sum of two uniform random numbers. Generate two random numbers between 0 and 1 and take Y to be their sum. Then Y is a continuous random variable that can take any value between 0 and 2. The density curve of Y is the triangle shown in FIGURE 4.12.a. Verify by geometry that the area under
4.52 Uniform numbers between 0 and 4. Many random number generators allow users to specify the range of the random numbers to be produced. Suppose that you specify that the range is to be all numbers between 0 and 4. Call the random number generated Y. Then the density curve of the random variable
4.51 Find the probabilities. Let the random variable X be a random number with the uniform density curve in Figure 4.9 (page 229). Find the following probabilities:a. P(X≥0.40) .b. P(X=0.40) .c. P(0.40
4.50 Spell-checking software. Spell-checking software catches “nonword errors,” which are strings of letters that are not words, as when “the” is typed as “eth.” When undergraduates are asked to write a 250-word essay (without spell-checking), the number X of nonword errors has the
4.49 Nonstandard dice. Nonstandard dice can produce interesting distributions of outcomes.You have two balanced, six-sided dice. One is a standard die, with faces having one, two, three, four, five, and six spots. The other die has three faces with two spots and three faces with five spots. Find
4.48 Tossing two dice. Some games of chance rely on tossing two dice. Each die has six faces, marked with one, two, . . . , six spots called pips. The dice used in casinos are carefully balanced so that each face is equally likely to come up. When two dice are tossed, each of the 36 possible pairs
4.47 Texas hold ’em. The game Texas hold ’em starts with each player receiving two cards. Here is the probability distribution for the number of aces in two-card hands:Number of aces 0 1 2 Probability 0.8507 0.1448 0.0045a. Verify that this assignment of probabilities satisfies the requirement
4.46 Discrete or continuous. In each of the following situations, decide whether the random variable is discrete or continuous and give a reason for your answer.a. Your web page has five different links, and a user can click on one of the links or can leave the page. You record the length of time
4.45 Households and families in government data. 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. So all families form households, but some households are not
4.44 Probabilities for Twitter. Refer to the Exercise 4.42. Find the probabilities for the number of Twitter users in a sample of size 2.
4.43 Use the Normal distribution. Suppose X is a Normal random variable with mean 20 and standard deviation 4. Find the following probabilities.a. The probability that X greater than or equal to 22.b. The probability that X less than 22.11c. The probability that X greater than 22 and less than
4.42 Use of Twitter. Suppose that the population proportion of Internet users who say that they use Twitter or another service to post updates about themselves or to see updates about others is 19%.Think about selecting random samples from a population in which 19% are Twitter users.a. Describe the
4.41 Use the uniform distribution. Suppose that a random variable X follows the uniform distribution described in Example 4.29 (page 229). For each of the following events, find the probability and illustrate your calculations with a sketch of the density curve similar to the ones in Figure 4.9
4.40 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. The possible values for a discrete random variable can’t be negative.b. A continuous random variable can take any value between 0 and 1.c. Normal
4.39 Find more probabilities. Refer to Exercise 4.35.a. Find the probability that a randomly selected student takes two or fewer courses.b. Find the probability that a randomly selected student takes three or four courses.c. Find the probability that a randomly selected student takes seven courses.
4.38 Find some probabilities. Refer to Exercise 4.36.a. Find the probability that a randomly selected student earns more than 18 credits.b. Find the probability that a randomly selected student earns 6 or fewer credits.c. Find the probability that a randomly selected student earns 15 credits or
4.37 Make a graphical display. Refer to Exercise 4.35. Use a probability histogram to provide a graphical description of the distribution of X.
4.36 A new random variable. Refer to the previous exercise. Suppose that a student earns three credits for each course taken. Let Y equal the number of credits a student would earn if they complete the course.a. What is the distribution of Y?b. Use a probability histogram to describe the
4.35 How many courses? At a small liberal arts college, students can register for one to six courses. Let X be the number of courses taken in the fall by a randomly selected student from this college. In a typical fall semester, 6% take one course, 6% take two courses, 12% take three courses, 20%
4.34 A random variable? You toss two coins and record the outcome as HH, HT, TH, or TT. Is the outcome a random variable? Explain your answer.
4.33 Three children. Anna has alleles B and O. Nathan has alleles A and O.a. What is the probability that a child of these parents has blood type O?b. If Anna and Nathan have three children, what is the probability that all three have blood type O?What is the probability that the first child has
4.32 Two children. Samantha has alleles B and O. Dylan has alleles A and B. They have two children.What is the probability that both children have blood type A? What is the probability that both children have the same blood type?
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