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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

6.29 More on determining hypotheses. State the null hypothesis H0 and the alternative hypothesis Ha in each case. Be sure to identify the parameters that you use to state the hypotheses.a. A university gives credit in first-year calculus to students who pass a placement test. The mathematics
6.28 Determining hypotheses. State the appropriate null hypothesis H0 and alternative hypothesis Ha in each of the following cases.a. A 2019 study reported that 97.8% of college students own a cell phone. You plan to take an SRS of college students to see if the percent has increased.b. The
6.27 What’s wrong? For each of the following statements, explain what is wrong and why.a. A significance test rejected the null hypothesis that the sample mean is equal to 500.b. A test preparation company wants to test that the average score of its students on the ACT is better than the national
6.26 What’s wrong? For each of the following statements, explain what is wrong and why.a. A researcher tests the following null hypothesis:H0: x¯=23 .b. A random sample of size 30 is taken from a population that is assumed to have a standard deviation of 5. The standard deviation of the sample
6.25 More than one confidence interval. As we prepare to take a sample and compute a 95% confidence interval, we know that the probability that the interval we compute will cover the parameter is 0.95. That’s the meaning of 95% confidence. If we plan to use several such intervals, however, our
6.24 Accuracy of a laboratory scale. To assess the accuracy of a laboratory scale, a standard weight known to weigh 10 grams is weighed repeatedly. The scale readings are Normally distributed with unknown mean. (This mean is 10 grams if the scale has no bias.) The standard deviation of the scale
6.23 Radio poll. A National Public Radio (NPR) station invites listeners to enter a dispute about a proposed “pay as you throw” waste collection program. The station asks listeners to call in and state how much each 10-gallon bag of trash should cost. A total of 179 listeners call in. The
6.22 Adjusting required sample size for dropouts. Refer to the previous exercise. In similar previous studies, about 20% of the subjects drop out before the study is completed. Adjust your sample size requirement so that you will have enough subjects at the end of the study to meet the margin of
6.21 Required sample size for specified margin of error. A new bone study is being planned that will measure the biomarker TRAP described in Exercise 6.9. Using the value ofσ given there, 6.5 U/l, find the sample size required to provide an estimate of the mean TRAP with a margin of error of 1.5
6.20 How many “hits”? The Confidence Intervals applet lets you simulate large numbers of confidence intervals quickly. Select 95% confidence and then sample 50 intervals. Record the number of intervals that cover the true value. Repeat this process until you have 30 counts of hits. Make a
6.19 Outlook on life. Since 2008, the Gallup-Sharecare Well-Being Index tracks how people feel about their daily lives. In 2017, 56.3% of the respondents were classified as “thriving.” This classification is based on how a respondent rates his or her current and future lives. This is the
6.18 Average minutes per week listening to audio. Refer to the previous exercise.a. Give the sample mean and sample standard deviation in minutes.b. Calculate the 95% confidence interval in minutes from your answers to part (a).c. Explain how you could have directly calculated this interval from
6.17 Average hours per week listening to audio. An iHeartMedia-sponsored survey of 6016 consumers who listen at least once a week to an audio platform reported an average of 17.2 hours a week listening to audio, such as broadcast radio, streaming music services, and podcasts. Assume that the
6.16 Inference based on skewed data. The mean OC for the 31 subjects in Exercise 6.10 was 33.4 ng/ml. In our calculations, we assumed that the standard deviation was known to be 19.6 ng/ml. Use the 68–95–99.7 rule from Chapter 1 (page 51) to find the approximate bounds on the values of OC that
6.15 Determining sample size. Refer to the previous exercise. You really want to use a sample size such that about 95% of the averages fall within±5 minutes of the population mean μ .a. Based on your answer to part (b) in Exercise 6.14, should the sample size be larger or smaller than 175?
6.14 Total sleep time of college students. In Example 5.4 (page 282), the total sleep time per night among college students was approximately Normally distributed with meanμ=7.13 hours and standard deviationσ=1.67 hours. Consider an SRS of size n=175 from this population.a. What is the standard
6.13 Consumption of sweet snacks. A study reported that the U.S. per capita consumption of sweet snacks among healthy-weight children aged 12 to 19 years is 251.2 kilocalories per day (kcal/d).This was based on 24-hour dietary recall records of n=2265 adolescents.a. Suppose that the population
6.12 Average starting salary. The National Association of Colleges and Employers (NACE)Summer Salary Survey shows that the current class of college graduates received an average startingsalary offer of $50,994. Your institution collected an SRS(n=200) of its recent graduates and obtained a 95%
6.11 Populations sampled and margins of error. Consider the following two scenarios. (A) Take a simple random sample (SRS) of 250 first-year students at your college or university. (B) Take an SRS of 250 students at your college or university. For each of these samples, you will record the amount
6.10 Mean OC in young women. Refer to the previous exercise. A biomarker for bone formation measured in the same study was osteocalcin (OC), measured in the blood. For the 31 subjects in the study, the mean was 33.4 nanograms per milliliter (ng/ml). Assume that the standard deviation is known to be
6.9 Mean TRAP in young women. For many important processes that occur in the body, direct measurement of characteristics of the process is not possible. In many cases, however, we can measure a biomarker, a biochemical substance that is relatively easy to measure and is associated with the process
6.8 Inference based on integer values. Refer to Exercise 6.7. The data for this study are integer values between 1 and 10. Explain why the confidence interval for the mean 8μ based on the Normal distribution should be a good approximation.
6.7 The state of stress in the United States. Since 2007, the American Psychological Association has supported an annual nationwide survey to examine stress across the United States. This year, a total of 3602 adults were asked to indicate their average stress level (on a 10-point scale) during the
6.6 More confidence interval mistakes and misunderstandings. Suppose that 100 randomly selected subscribers of Stingray Karaoke on YouTube asked how much time they typically spend on the site during the week.7 The sample mean x¯ is found to be 2.4 hours. Assume that the population standard
6.5 Confidence interval mistakes and misunderstandings. Suppose that 500 randomly selected alumni of the University of Okoboji were asked to rate the university’s academic advising services on a scale of 1–10. The sample mean x¯ was found to be 8.6.Assume that the population standard deviation
6.4 Changing the confidence level. Consider the setting of the previous two exercises. Suppose that the sample mean is still 1123, the sample size is 400, and the population standard deviation is 441.Make a diagram similar to Figure 6.6 (page 339) that illustrates the effect of the confidence level
6.3 Changing the sample size. Consider the setting of the previous exercise. Suppose that the sample mean is again 1123, and the population standard deviation is 441. Make a diagram similar to Figure 6.5 (page 338) that illustrates the effect of sample size on the width of a 95% interval. Use the
6.2 Margin of error and the confidence interval. A study of stress on the campus of your university using the College Undergraduate Stress Scale (CUSS) reported an average stress level of 1123 (a higher score indicating more stress) with a margin of error of 43 for 95% confidence. The study was
6.1 Student loan debt. The average student loan debt among college graduates is reported using the 95% confidence interval ($23,923, $34,447).a. Describe what this interval tells us about average student loan debt.b. What is the estimated average student loan debt among college graduates?c. What is
5.76 How large a sample is needed? The changing probabilities you found in Exercise 5.74 are due to the fact that the standard deviation of the sample proportion p^ gets smaller as the sample size n increases. If the population proportion is p=0.41 , how large a sample is needed to reduce the
5.75 Is the ESP result better than guessing? When the ESP study of Exercise 5.63 discovers a subject whose performance appears to be better than guessing, the study continues at greater length. The experimenter looks at many cards bearing one of five shapes (star, square, circle, wave, and cross)
5.74 The ideal number of children. “What do you think is the ideal number of children for a family to have?” A Gallup Poll asked this question of 1020 randomly chosen adults. Roughly 41% thought that a total of three or more children was ideal. Suppose that p=0.41 is exactly true for the
5.73 Binge drinking. The Centers for Disease Control and Prevention finds that 28% of people aged 18 to 24 years binge drank. Those who binge drank averaged 9.3 drinks per episode and 4.2 episodes per month. The study took a sample of over 18,000 people aged 18 to 24 years, so the population
5.72 Risks and insurance. The idea of insurance is that we all face risks that are unlikely but carry high cost. Think of a fire destroying your home. So we form a group to share the risk: we all pay a small amount, and the insurance policy pays a large amount to those few of us whose homes burn
5.71 Treatment and control groups. The previous exercise illustrates a common setting for statistical inference. This exercise gives the general form of the sampling distribution needed in this setting. We have a sample of n observations from a treatment group and an independent sample of m
5.70 Iron depletion without anemia and physical performance. Several studies have shown a link between iron depletion without anemia (IDNA) and physical performance. In one study, the physical performance of 24 female collegiate rowers with IDNA was compared with that of 24 female collegiate rowers
5.69 More on watching live television. Consider the settings of Exercises 5.50 and 5.52.a. Using the reported 30% from the survey, what is the largest number m out of n=20 undergraduates such that P(X≤m)
5.68 Income of working couples. A study of working couples measures the income X of the husband and the income Y of the wife in a large number of couples in which both partners are employed. Suppose that you knew the meansμX andμY and the variancesσX2 andσY2 of both variables in the
5.67 Summer employment of college students. Suppose (as is roughly true)that 88% of college men and 82% of college women were employed last summer. A sample survey interviews SRSs of 400 college men and 400 college women. The two samples are, of course, independent.a. What is the approximate
5.66 Learning a foreign language. Does delaying oral practice hinder learning a foreign language? Researchers randomly assigned 25 beginning students of Russian to begin speaking practice immediately and another 25 to delay speaking for four weeks.At the end of the semester both groups took a
5.65 A roulette payoff revisited. Refer to the previous exercise. In part (d), the central limit theorem was used to approximate the probability that Sam ends the year ahead.The estimate was about 0.10 too large. Let’s see if we can get closer using the Normal approximation to the binomial with
5.64 A roulette payoff. A $1 bet on a single number on a casino’s roulette wheel pays$35 if the ball ends up in the number slot you choose. Here is the distribution of the payoff X:Payoff X $0 $35 Probability 0.974 0.026 Each spin of the roulette wheel is independent of other spins.a. What are
5.63 A test for ESP. In a test for ESP (extrasensory perception), the experimenter looks at cards that are hidden from the subject. Each card contains either a star, a circle, a wave, or a square. As the experimenter looks at each of 20 cards in turn, the subject names the shape on the card.a. If a
5.62 A lottery payoff. A $1 bet in a state lottery’s Pick 3 game pays $500 if the threedigit number you choose exactly matches the winning number, which is drawn at random. Here is the distribution of the payoff X:Payoff X $0 $500 Probability 0.999 0.001 Each day’s drawing is independent of
5.61 Poisson distribution? Suppose you find in your spam folder an average of two spam emails every 10 minutes. Furthermore, you find that the rate of spam mail from midnight to 6 a.m. is twice the rate during other parts of the day. Explain whether or not the Poisson distribution is an appropriate
5.60 Wi-Fi interruptions. Suppose that the number of Wi-Fi interruptions on your home network follows the Poisson distribution, with an average of 1.6 Wi-Fi interruptions per day.a. Show that the probability of no interruptions on a given day is 0.2019.b. Treating each day as a trial in a binomial
5.59 The geometric distribution. Generalize your work in the previous exercise. You have independent trials, each resulting in a success or a failure. The probability of a success is p on each trial. The binomial distribution describes the count of successes in a fixed number of trials. Now the
5.58 Tossing a die. You are tossing a balanced die that has probability 1/6 of coming up 1 on each toss. Tosses are independent. We are interested in how long we must wait to get the first 1.a. The probability of a 1 on the first toss is 1/6. What is the probability that the first toss is not a 1
5.57 A random walk. A particle moves along the line in a random walk. That is, the particle starts at the origin (position 0) and moves either right or left in independent steps of length 1. If the particle moves to the right with probability 0.6, its movement at the ith step is a random variable
5.56 Use the Probability applet. The Probability applet simulates tosses of a coin. You can choose the number of tosses n and the probability p of a head. You can therefore use the applet to simulate binomial random variables.The count of misclassified sales records in Example 5.20 has the binomial
5.55 Use the Normal approximation. Suppose that we toss a fair coin. Use the Normal approximation to find the probability that the sample proportion of heads isa. between 0.45 and 0.55 when n=100.b. between 0.48 and 0.52 when n=625.c. Use these results to describe the relationship between the
5.54 Common last names. The U.S. Census Bureau says that the 10 most common names in the United States are (in order) Smith, Johnson, Williams, Brown, Jones, Garcia, Miller, Davis, Rodriguez, and Martinez. These names account for 4.9% of all U.S. residents. Out of curiosity, you look at the authors
5.53 Marks per round in cricket. Cricket is a dart game that uses the numbers 15 to 20 and the bull’s-eye. Each time you hit one of these regions, you score either 0, 1, 2, or 3 marks. Thus, in a round of three throws, a person can score 0 to 9 marks. Lex plans to play 20 games. Her distribution
5.52 Watching live television, continued. Refer to Exercise 5.50. You think that the undergraduate rate of those who watch live television every day at your university is 15%.a. Using this rate, what is the expected number of students in your sample who say that they watch live television every
5.51 Leaking gas tanks. Leakage from underground gasoline tanks at service stations can damage the environment. It is estimated that 25% of these tanks leak. You examine 15 tanks chosen at random, independently of each other.a. What is the mean number of leaking tanks in such a sample of 15?b. What
5.50 Watching live television. A survey of 442 people aged 18 to 29 revealed that 30%watch live television every day. You take a random sample of 20 undergraduates from your university and ask them whether they watch live TV every day. If their rate matches the 30% rate:a. What is the distribution
5.49 Benford’s law. It is a striking fact that the first digits of numbers in legitimate records often follow a distribution known as Benford’s law. Here it is:First digit 1 2 3 4 5 6 7 8 9 Proportion 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046 Fake records usually have fewer first
5.48 Attitudes toward drinking and studies of behavior. Some of the methods in this chapter are based on approximations rather than exact probability results. We have given rules of thumb for safe use of these approximations.a. You are interested in attitudes toward drinking among the 75 members of
5.47 Monitoring the emerald ash borer. The emerald ash borer is a beetle that poses a serious threat to ash trees. Purple traps are often used to detect or monitor populations of this pest. In the counties of your state where the beetle is present, thousands of traps are used to monitor the
5.46 The effect of sample size on the standard deviation. Assume that the standard deviation in a very large population is 100.a. Calculate the standard deviation for the sample mean for samples of size 1, 4, 25, 100, 250, 500, 1000, and 5000.b. Graph your results with the sample size on the x axis
5.45 Dust in coal mines. A laboratory weighs filters from a coal mine to measure the amount of dust in the mine atmosphere. Repeated measurements of the weight of dust on the same filter vary Normally, with standard deviationσ=0.11 milligram (mg) because the weighing is not perfectly precise. The
5.44 The cost of Internet access. In Canada, households spent an average of $54.17 CDN monthly for high-speed Internet access. Assume that the standard deviation is$17.83. If you ask an SRS of 500 Canadian households with high-speed Internet how much they pay, what is the probability that the
5.43 Metal fatigue. Metal fatigue refers to the gradual weakening and eventual failure of metal that undergoes cyclic loads. The wings of an aircraft, for example, are subject to cyclic loads when in the air, and cracks can form. It is thought that these cracks start at large particles found in the
5.42 Number of colony-forming units. In microbiology, colony-forming units (CFUs) are used to measure the number of microorganisms present in a sample. To determine the number of CFUs, the sample is prepared, spread uniformly on an agar plate, and then incubated at some suitable temperature.Suppose
5.41 English Premier League goals. The total number of goals scored per soccer match in the English Premier League (EPL) often follows the Poisson distribution. In one recent season, the average number of goals scored per match (over 380 games played) was 2.821. Compute the following
5.40 Show that these facts are true. Use the definition of binomial coefficients to show that each of the following facts is true. Then restate each fact in words in terms of the number of ways that k successes can be distributed among n observations.a. (nn)=1 for any whole number n≥1 .b.
5.39 Shooting free throws. Since the mid-1960s, the overall free-throw percent at all college levels, for both men and women, has remained pretty consistent. For men, players have been successful on roughly 69% of free throws, with the season percent never falling below 67% or above 70%. Assume
5.38 Mishandled bags. In the airline industry, the term mishandled refers to a bag that was lost, delayed, damaged, or stolen. The latest report by SITA states that the mishandled baggage rate has plateaued at 5.7 per 1000 passengers. Suppose that this national rate holds for your airport, and your
5.37 Is the binomial distribution a reasonable approximation? In each of the following situations, is it reasonable to use a binomial distribution to approximate the sampling distribution of X? Give reasons for your answer in each case.a. According to the Red Cross, 0.5% of the Asian ethnic group
5.36 Online learning. The U.S. Department of Education released a report on online learning stating that blended instruction, a combination of conventional face-to-face and online instruction, appears to be more effective in terms of student performance than conventional teaching. You decide to
5.35 Cyberbullying, continued. Refer to Exercise 5.33.a. What is the expected number of undergraduates in your sample who say that they have received hurtful comments online in the past 30 days? What is the expected number of undergraduates who say that they have not received hurtful comments
5.34 Genetics of peas. According to genetic theory, the blossom color in the second generation of a certain cross of sweet peas should be red or white in a 3:1 ratio. That is, each plant has probability 3/4 of having red blossoms, and the blossom colors of separate plants are independent.a. What is
5.33 Cyberbullying. A survey of 4972 U.S. students aged 12 to 17 years reveals that 25% have received mean or hurtful comments online in the past 30 days. You take a random sample of 15 undergraduates and ask them whether they have received mean or hurtful comments online in the past 30 days. If
5.32 Admitting students to college. A selective college would like to have an entering class of 900 students. Because not all students who are offered admission accept, the college admits more than 900 students. Past experience shows that about 78% of the students admitted will accept. The college
5.31 Random digits. Each entry in a table of random digits like Table B has probability 0.1 of being a 0, and digits are independent of each other.a. What is the probability that a group of five digits from the table will contain at least one digit greater than 4?b. What is the mean number of
5.30 Should you use the binomial distribution? In each of the following situations, is it reasonable to use a binomial distribution for the random variable X? Give reasons for your answer in each case.a. In a random sample of students in a fitness study, X is the mean daily exercise time of the
5.29 Should you use the binomial distribution? In each of the following situations, is it reasonable to use a binomial distribution for the random variable X? Give reasons for your answer in each case. If a binomial distribution applies, give the values of n and p.a. A poll of 200 college students
5.28 What’s wrong? For each of the following statements, explain what is wrong and why.a. In the binomial setting, X is a proportion.b. The variance for a binomial count is p(1-p)/n .c. The Normal approximation to the binomial distribution is always accurate when n is very large.d. The binomial
5.27 What’s wrong? For each of the following statements, explain what is wrong and why.a. If you toss a fair coin four times and a head appears each time, then the next toss is more likely to be a head than a tail.b. If you toss a fair coin four times and observe the pattern THTH, then the next
5.26 Investments in two funds. Jennifer invests her money in a portfolio that consists of 65%Fidelity 500 Index Fund and 35% Fidelity Tax-Free Bond Fund. Suppose that, in the long run, the annual real return X on the Index Fund has mean 10% and standard deviation 12%, the annual real return Y on
5.25 Weights of airline passengers. In 2019, the Federal Aviation Administration (FAA) updated its standard average passenger weight to be based on data from U.S. government health agency surveys. It specified this average weight, which includes clothing, as 189 pounds in the summer (195 in the
5.24 Grades in a math course. Indiana University posts the grade distributions for its courses online.In one spring semester, students in Math 118 received 16.1% A’s, 34.3% B’s, 29.2% C’s, 9.6% D’s, and 9.8% F’s.a. Using the common scale A=4 , 11 B=3 , C=2 , D=1 , F=0 , take X to be the
5.23 Cholesterol levels of teenagers. A study of the health of teenagers plans to measure the blood cholesterol level of an SRS of 13- to 16-year-olds. The researchers will report the mean x¯ from their sample as an estimate of the mean cholesterol levelμ in this population.a. Explain to someone
5.22 Number of friends on Facebook. In Australia, young people aged 18 to 29 have an average of 394 Facebook friends. This population distribution takes only integer values, so it is certainly not Normal.It is also highly skewed to the right. Suppose thatσ=280 and you take an SRS of 70 Facebook
5.21 Can volumes. Averages are less variable than individual observations. It is reasonable to assume that the can volumes in Exercise 5.19 vary according to a Normal distribution. In that case, the mean x¯ of an SRS of cans also has a Normal distribution.a. Make a sketch of the Normal curve for a
5.20 Average movie length on Netflix. Refer to Exercise 5.18. Suppose that the true mean movie length is 98.6 minutes, and you plan to take an SRS of n=50 movies.a. Explain why it may be reasonable to assume that the average x¯ is approximately Normal even though the population distribution is
5.19 Bottling an energy drink. A bottling company uses a filling machine to fill cans with an energy drink. The cans are supposed to contain 250 milliliters (ml) each. The machine, however, has some variability, so the standard deviation of the volume isσ=0.27 ml. A sample of five cans is
5.18 Length of a movie on Netflix. Flixable reports that Netflix’s U.S. catalog contains almost 4000 movies. You are interested in determining the average length of these movies. Previous studies have suggested the standard deviation for this population is 34 minutes.a. What is the standard
5.17 Determining sample size. Refer to the previous exercise. You want to use a sample size such that about 95% of the averages fall within±10 minutes (0.17 hour) of the true mean μ=7.13 .a. Based on your answer to part (b) in Exercise 5.16, should the sample size be larger or smaller than 60?
5.16 Sleep duration of college students. In Example 5.4, the daily sleep duration among college students was approximately Normally distributed with meanμ=7.13 hours and standard deviation σ=1.67 hours.You plan to take an SRS of size n=60 and compute the average total sleep time.a. What is the
5.15 Generating a sampling distribution. Let’s illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the 10 scholarship players currently on your women’s basketball team. For convenience, the 10 players have been
5.14 What’s wrong? For each of the following statements, explain what is wrong and why.a. The central limit theorem states that for large n, the population meanμ is approximately Normal.b. For large n, the distribution of observed values will be approximately Normal.c. For sufficiently large n,
5.13 What’s wrong? For each of the following statements, explain what is wrong and why.a. If the population standard deviation is 20, then the standard deviation of x¯ for an SRS of 10 observations is 20/10=2 .b. When taking SRSs from a population, larger sample sizes will result in larger
5.12 Twitter polls. Twitter provides the option for users to weigh in on questions posed by other Twitter users. A Twitter poll can remain open for a minimum of five minutes and maximum of one week after it is posted. Can you apply the ideas about populations and samples to these polls? Explain why
5.11 Sampling distributions and sample size. The software JMP includes some applets, one of which is called “Sampling Distribution of Sample Proportions.” This applet does sampling very quickly, especially if “Animate Illustration?” is set to No.a. In Exercise 5.7, you obtained 50 draws of
5.10 Use the Simple Random Sample applet, continued. Refer to the previous exercise.a. Suppose instead that a sample size of n=10 was used. Based on what you know about the effect of the sample size on the sampling distribution, which sampling distribution should have the smaller variability?b.
5.9 Use the Simple Random Sample applet. The Simple Random Sample applet can illustrate the idea of a sampling distribution. Form a population labeled 1 to 100. We will choose an SRS of 20 of these numbers. That is, in this exercise, the numbers themselves are the population, not just labels for
5.8 Comparing sampling distributions. Refer to the previous exercise.a. How do the centers of your two histograms reflect the differing truths about the two populations?b. Describe any differences in the shapes of the two histograms. Is one more skewed than the other?c. Compare the spreads of the
5.7 Constructing sampling distributions. The Probability applet simulates tossing a coin, with the advantage that you can choose the true long-term proportion, or probability, of a head. Suppose that we have a population in which proportion p=0.4 (the parameter) plan to vote in the next election.
5.6 Bias and variability. FIGURE 5.5 shows histograms of four sampling distributions of statistics intended to estimate the same parameter. Label each distribution relative to the others as high or low bias and as high or low variability.FIGURE 5.5 Determine which of these sampling distributions

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