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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

7.17 Rudeness and its effect on onlookers. Many believe that an uncivil environment has a negative effect on people. A pair of researchers performed a series of experiments to test whether witnessing rudeness and disrespect affects task performance. In one study, 34 participants met in small groups
7.16 Using an app or the Web on a smartphone. The Nielsen Company reports that U.S. residents aged 18 to 34 years spend an average of 2.83 hours per day using an app or the Web on a smartphone.You wonder if this it true for students at your large university because you rarely see students not using
7.15 Beer before wine, and you’ll feel fine? Some researchers recently performed a study to investigate the order of beer and wine on hangover intensity. The study was run with each participant undergoing both orders of beer and wine with at least a week between sessions. Each session involved
7.14 Number of friends on Facebook. To mark Facebook’s 10th birthday, Pew Research surveyed people using Facebook to see what they like and dislike about the site. The survey found that among adult Facebook users, the average number of friends is 338. This distribution takes only integer values,
7.13 Uber driver earnings. On its blog back in 2014, Uber posted a scatterplot using a sample of several thousand drivers in New York City. The plot showed each driver’s average net earnings per hour versus the number of hours worked. Here is a sample of earnings (dollars) for 27 drivers working
7.12 Testing the sticker information. Refer to the previous exercise. The vehicle sticker information for this model stated a city average of 19 mpg. Are these mpg values consistent with the vehicle sticker?Perform a significance test using the 0.05 significance level. Be sure to specify the
7.11 Fuel economy. Although the Environmental Protection Agency (EPA) establishes the tests to determine the fuel economy of new cars, it often does not perform them. Instead, the test protocols are given to the car companies, and the companies perform the tests themselves. To keep the industry
7.10 Business bankruptcies in Canada. Business bankruptcies in Canada are monitored by the Office of the Superintendent of Bankruptcy Canada (OSB). Included in each report are the assets and liabilities the company declared at the time of the bankruptcy filing. A study is based on a random sample
7.9 Margin of error. For each of the following situations, compute the 95% margin of error for the population mean μ .a. An SRS of n=25 gives x¯=23 and SEx¯=0.5 .b. An SRS of n=16 gives x¯=32 and s=4 .c. An SRS of n=25 gives x¯=44 and s2=25 .
7.8 A final one-sample t test. The one-sample t statistic for testing H0: μ=20Ha: μ
7.7 Another one-sample t test. The one-sample t statistic for testing H0: μ=40Ha: μ≠40 from a sample of n=13 observations has the value t=2.78 .a. What are the degrees of freedom for t?b. Locate the two critical values t* from Table D that bracket t.c. Between what two values does the P-value
7.6 A one-sample t test. The one-sample t statistic for testing H0: μ=8Ha: μ>8 from a sample of n=22 observations has the value t=2.24 .a. What are the degrees of freedom for this statistic?b. Give the two critical values t* from Table D that bracket t.c. Between what two values does the P-value
7.5 More on the distribution of the t statistic. Repeat the previous exercise for the two situations where the alternative is one-sided.
7.4 Distribution of the t statistic. Assume a sample size of n=20 . Draw a picture of the distribution of the t statistic under the null hypothesis. Use Table D and your picture to illustrate the values of the test statistic that would lead to rejection of the null hypothesis at the 1% level for a
7.3 Finding the critical value t* . What critical value t* from Table D should be used to calculate the margin of error for a confidence interval for the mean of the population in each of the following situations?a. A 95% confidence interval based on n=18 observations.b. A 95% confidence interval
7.2 What’s wrong? For each of the following statements, explain what is wrong and why.a. As the degrees of freedom k decrease, the t distribution density curve gets closer to the N(0, 1)curve.b. The standard error of the sample mean is s2/n .c. A researcher wants to test H0: x¯=30 versus the
7.1 What’s wrong? For each of the following statements, explain what is wrong and why.a. The degrees of freedom for the one-sample t statistic is n−2 .b. The margin of error for the population meanμ is m=t*s .c. The P-value in a one-sample equivalence test quantifies the evidence provided by
6.112 Where do you buy? Consumers can purchase nonprescription medications at food stores, mass merchandise stores such as Target and Walmart, or pharmacies. About 45% of consumers make such purchases at pharmacies. What accounts for the popularity of pharmacies, which often charge higher prices?A
6.111 Odor threshold of future wine experts. Many food products contain small quantities of substances that would give an undesirable taste or smell if they are present in large amounts. An example is the “off-odors” caused by sulfur compounds in wine. Oenologists (wine experts) have determined
6.110 Cellulose content in alfalfa hay. An agronomist examines the cellulose content of a variety of alfalfa hay. Suppose that the cellulose content in the population has standard deviationσ=8 milligrams per gram(mg/g). A sample of 15 cuttings has mean cellulose content x¯=145 mg/g .a. Give a 90%
6.109 Blood phosphorus level in dialysis patients. Patients with chronic kidney failure may be treated by dialysis, in which a machine removes toxic wastes from the blood, a function normally performed by the kidneys. Kidney failure and dialysis can cause other changes, such as retention of
6.108 Simulation study of power. Explain why the result from the previous exercise approximates the power of detecting a difference of four units(H0: μ=24 versus μ=20) in this setting.PUTTING IT ALL TOGETHER
6.107 Another simulation study of a test of significance. Use the same procedure for generating data as in the previous exercise. Now test the null hypothesis thatμ=24 . Explain your results.
6.106 Simulation study of a test of significance. Use a computer to generate n=15 observations from a Normal distribution with mean 20 and standard deviation 5: N(20, 5). Test the null hypothesis thatμ=20 using a two-sided significance test. Repeat this process 100 times and then count the number
6.105 Simulation study of the confidence interval. Use a computer to generate n=15 observations from a Normal distribution with mean 20 and standard deviation 5: N(20, 5). Find the 95% confidence interval for μ . Repeat this process 100 times and then count the number of times that the confidence
6.104 What’s wrong? For each of the following statements, explain what is wrong and why.a. The z statistic for a two-sided test is−3.18 . The researcher concludes that the null hypothesis is not rejected forα=0.05 because−3.18
6.103 More on the meaning of “statistically significant.” Another student, when asked why statistical significance appears so often in research reports, says, “Because saying that results are significant tells us that they cannot easily be explained by chance variation alone.” Do you think
6.102 Meaning of “statistically significant.” When asked to explain the meaning of “statistically significant at the α=0.01 level,” a student says, “This means there is only probability 0.01 that the null hypothesis is true.” Is this an essentially correct explanation of statistical
6.101 CEO pay. A study of the pay of corporate chief executive officers (CEOs) examined the increase in cash compensation of the CEOs of 104 companies, adjusted for inflation, in a recent year. The mean increase in real compensation was x¯=6.9% , and the standard deviation of the increases was
6.100 Reporting margins of error. An AP News article from July 17, 2014, reported Commerce Department estimates of changes in the construction industry:Construction fell 9.3 percent last month to a seasonally adjusted annual rate of 893,000 homes.If we turn to the original Commerce Department
6.99 Survey response and margin of error. Suppose that a business conducts a marketing survey. As is often done, the survey is conducted by telephone. As it turns out, the business was only able to elicit responses from fewer than 10% of the randomly chosen customers. The low response rate is
6.98 Effect of sample size on significance. You are testing the null hypothesis thatμ=0 versus the alternativeμ>0 usingα=0.05 . Assume thatσ=16 . Suppose that x¯=8 and n=10 . Calculate the test statistic and its P-value. Repeat assuming the same value of x¯ but with n=20 . Do the same for
6.97 Coverage percent of 90% confidence interval. Refer to the previous exercise.Do the simulations and report the results for 90% confidence.29
6.96 Coverage percent of 95% confidence interval. For this exercise, you will use the Confidence Intervals applet. Set the confidence level at 95% and the number of sample to 10 to simulate 10 confidence intervals. Record the percent hit. Simulate another 10 intervals by resampling.Record the
6.95 Food selection by children in school cafeterias. A group of researchers examined whether children’s food selection in a school cafeteria met the standards set by the School Meals Initiative. They measured food selection and food intake of 2049 fourth- through sixth-grade students in 33
6.94 Roulette. A roulette wheel has 18 red slots among its 38 slots. You observe many spins and record the number of times that red occurs. Now you want to use these data to test whether the probability of a red has the value that is correct for a fair roulette wheel. State the hypotheses H0 and Ha
6.93 Exercise and statistics exams. A study examined whether light exercise performed an hour before the final exam in statistics affects how students perform on the exam. The P-value was given as 0.13.a. State null and alternative hypotheses that could be used for this study. (Note: There is more
6.92 Telemarketing wages. An advertisement in the student newspaper asks you to consider working for a telemarketing company. The ad states, “Earn between $500 and $1000 per week.” Do you think that the ad is describing a confidence interval? Explain your answer.
6.91 Why not a 100% confidence interval? The most common confidence levels are 90%, 95%, and 99%. If you were to consider a 100% confidence interval for the meanμ , what would it look like?Explain your reasoning and why such an interval is of no practical use.
6.90 Make a recommendation. Your manager has asked you to review a research proposal that includes a section on sample size justification. A careful reading of this section indicates that the power is 18% for detecting an effect that would be considered important. Write a short report for your
6.89 Other changes and the effect on power. Refer to the previous exercise. For each of the following changes, explain what happens to the power for each alternativeμ .a. Change to the two-sided alternative.b. Decrease σ to 0.5.c. Increase n from 10 to 30.
6.88 Effect of changing the alternative μ on power. The Statistical Power applet can be used to study power. Open the applet and set the null hypothesis toμ=0 , the alternative toμ>0 , the sample size to n=10 , the standard deviation to σ=1 , the significance level toα=0.05 , and the
6.87 Computer-assisted career guidance systems. A wide variety of computer-assisted career guidance systems have been developed over the past decade. These programs use factors such as student interests, aptitude, skills, personality, and family history to recommend a career path. For simplicity,
6.86 More on choosing the appropriate distribution. Refer to Exercise 6.80. Suppose that instead of a single observation X, you plan to obtain n observations and use the decision rule to reject H0 when x¯≤k .a. What are the means of x¯ under p0 and p1 ?b. Based on your answers in part (a), what
6.85 Power for a different alternative. For the ball bearing example (page 376), the power is 0.92 when μ=22.015 .a. Would the power for the alternativeμ=22.030 be larger than, smaller than, or equal to 0.92? Sketch a plot like Figure 6.18 to explain your answer.b. Use the Statistical Power
6.84 Computing the power in the ball bearing study. Recall Example 6.29. Let’s run through the steps needed to obtain the power of 0.92 whenμ=22.015 .a. Given that we reject H0 if z≤−1.96 or z≥1.96 and z=x¯-220.01/5, for what values of x¯do we reject H0 ?b. Now assuming
6.83 What is the power of the test? A study is run to test H0: μ=50 using the twosided alternative and theα=0.01 significance level.For each of the following settings, give the power of the test whenμ=55 .a. The probability of a Type II error whenμ=55 is 0.43.b. The probability of a Type II
6.82 Choose the appropriate distribution, continued. Refer to Exercise 6.80. Another possible decision procedure is to reject H0 if X≤3 .a. Find the probabilities of a Type I and Type II error under this decision procedure.b. Which decision procedure, X≤2 or X≤3 , do you prefer?Explain your
6.81 Percent energy from added sugars: Type I and Type II errors. A test of significance was performed in Example 6.15 (page 357).a. Describe the Type I and Type II errors in this example.b. Based on the conclusion, which type of error might have occurred? Explain your answer.
6.80 Choose the appropriate distribution. You must decide which of two discrete distributions a random variable X has. We will call the distributions p0 and p1 . Here are the probabilities assigned to the values x of X:x 0 1 2 3 4 5 6 p0 0.0 0.1 0.1 0.1 0.4 0.2 0.1 p1 0.2 0.4 0.2 0.1 0.1 0.0 0.0
6.79 What are the Type I and Type II errors? A smartphone manufacturer gets its phone batteries from one supplier. For each shipment, a random sample of n=6 batteries are tested to ensure that the shipment complies with specifications, such as capacity and discharge curve levels.a. Specify the
6.78 A role as a statistical consultant. You are the statistical expert for a graduate student planning her PhD research. After you carefully present the mechanics of significance testing, she suggests usingα=0.20 for the study because she would be more likely to obtain statistically significant
6.77 Significance using the Bonferroni procedure. Refer to the previous exercise. A researcher has performed 12 tests of significance and wants to apply the Bonferroni procedure withα=0.05 . The calculated Pvalues are 0.141, 0.519, 0.186, 0.753, 0.001, 0.008, 0.646, 0.038, 0.898, 0.013,
6.76 An adjustment for multiple tests. One way to deal with the problem of misleading Pvalues when performing more than one significance test is to adjust the criterion you use for statistical significance. The Bonferroni method does this in a simple way. If you perform two tests and want to use
6.75 Interpreting a very small P-value. Assume that you are performing a large number of significance tests. Let n be the number of these tests. How large would n need to be for you to expect about one Pvalue to be 0.00001 or smaller? Use this information to write an explanation of how to interpret
6.74 More on searching for significance. You perform 1000 significance tests usingα=0.05 . Assuming that all null hypotheses are true, about how many of the test results would you expect to be statistically significant? Explain how you obtained your answer.
6.73 Searching for significance. A research team is looking for risk factors associated with Alzheimer’s disease. The team has decided to investigate roughly 500 different factors, testing each at theα=0.05 level. Explain why these test results would lead to misleading conclusions.
6.72 Predicting success of trainees. What distinguishes managerial trainees who eventually become executives from those who, after expensive training, don’t succeed and leave the company? We have abundant data on past trainees—data on their personalities and goals, their college preparation and
6.71 Drug treatment to stop smoking. A company matches 200 smokers who signed up for the company’s drug treatment with 200 smokers from the general population. Matching was done on length of smoking, number of packs per day, age, and sex. The company then followed the smokers for six months and
6.70 Find journal articles. Find two journal articles that report results with statistical analyses.For each article, summarize how the results are reported and write a critique of the presentation. Be sure to include details regarding use of significance testing at a particular level of
6.69 More on statistical versus practical significance. A study with 14 subjects reported a result that failed to achieve statistical significance at the 5% level. The P-value was 0.051. Write a short summary of how you would interpret these findings.
6.68 Statistical versus practical significance. A study with 7500 subjects reported a result that was statistically significant at the 5% level. Explain why this result might not be particularly important.
6.67 Practical significance and sample size. Every user of statistics should understand the distinction between statistical significance and practical importance. A sufficiently large sample will declare very small effects statistically significant. Consider the study of elite female Canadian
6.66 Do you agree? State whether or not you agree with each of the following statements and provide a short summary of the reasons for your answers.a. If the P-value is larger than 0.05, the null hypothesis is true.b. Practical significance is not the same as statistical significance.c. We can
6.65 How far do rich parents take us? How much education children get is strongly associated with the wealth and social status of their parents, termed “socioeconomic status,” or SES. The SES of parents, however, has little influence on whether children who have graduated from college continue
6.64 Vitamin C and colds. In a study to investigate whether vitamin C prevents colds, 400 subjects are assigned at random to one of two groups. The experimental group takes a vitamin C tablet daily, while the control group takes a placebo. At the end of the experiment, the researchers calculate the
6.63 What a test of significance can answer. Explain whether a test of significance can answer each of the following questions.a. Is the sample or experiment properly designed?b. Is the observed effect compatible with the null hypothesis?c. Is the observed effect important?
6.62 Selective publication based on results. In addition to statistical significance, selective publication can also be due to the observed outcome. A review of 74 studies of antidepressant agents found 38 studies with positive results and 36 studies with negative or questionable results. All but 1
6.61 What do you know? A research report described two results that both achieved statistical significance at the 5% level. The P-value for the first is 0.048; for the second it is 0.0002. Do the Pvalues add any useful information beyond that conveyed by the statement that both results are
6.60 What other information is needed? An observational study that involved n=27,391 subjects concluded that those who were frequent thumb-suckers as children lived longer than those who were not frequent thumb-suckers(P=0.00035) . What other information from this study would you like to know
6.59 More on using Table A and Table D to find a P-value. Refer to the previous exercise. Find the Pvalue for z=−1.88 .
6.58 Using Table A and Table D to find a P-value. Consider a significance test for a null hypothesis versus a two-sided alternative. Between what values from Table D does the P-value for an outcome z=1.88 lie? Calculate the P-value using Table A and verify that it lies between the values you found
6.57 More on using Table D to find a P-value. You have performed a one-sided test of significance and obtained a value of z=1.03 .Use Table D to find the approximate P-value for this test when the alternative is greater than.
6.56 Using Table D to find a P-value. You have performed a two-sided test of significance and obtained a value of z=2.08 . Use Table D to find the approximate P-value for this test.
6.55 Test statistic and levels of significance. Consider a significance test for a null hypothesis versus a two-sided alternative. Give a value of z that will give a result significant at the 1% level but not at the 0.5% level.
6.54 More on understanding levels of significance. You are told that a significance test is significant at the 5% level. From this information, can you determine whether or not it is significant at the 1% level?Explain your answer.
6.53 Understanding levels of significance. Explain in plain language why a significance test that is significant at the 1% level must always be significant at the 5% level.
6.52 Other changes and the P-value. Refer to the previous exercise.a. What happens to the P-values when you change the significance levelα to 0.01? Explain the result.b. What happens to the P-values when you change the sample size n from 10 to 100? Explain the result.
6.51 Changing to a two-sided alternative, continued. Repeat the previous exercise but with the two-sided alternative hypothesis. How does this change affect the P-values associated with each x¯ ? Explain why the Pvalues change in this way.
6.50 Impact of x¯ on the P-value. We can study the P-value by using the Statistical Significance applet. Set the null hypothesis toμ=0 , the alternative hypothesis to μ>0 , the sample size to n=10 , the standard deviation to σ=1 , the significance level toσ=0.05 , and the observed x¯ to x¯=1
6.49 Changing the sample size. Refer to Exercise 6.46. Suppose that you increase the sample size n from 10 to 100. Again, make a table giving x¯ and the results of the significance tests at the 0.05 significance level. What do you conclude?
6.48 Changing to a two-sided alternative. Repeat the previous exercise but with the twosided alternative hypothesis. How does this change affect which values of x¯ are far enough away from μ0 to be statistically significant at the 0.01 level?
6.47 Effect of changingα on significance. Repeat the previous exercise with significance levelα=0.01 . How does the choice of α affect which values of x¯ are far enough away from μ0 to be statistically significant?
6.46 Impact of x¯ on significance. The Statistical Significance applet illustrates statistical tests with a fixed level of significance for Normally distributed data with known standard deviation. Open the applet and set the null hypothesis toμ=0 , the alternative hypothesis to μ>0 , the sample
6.45 Fuel economy. In 2017, the Environmental Protection Agency (EPA) changed how it calculates window-sticker gas mileage. For many models, this meant a 1−2 miles per gallon(mpg) reduction in combined highway/city mileage. Here are some combined mpg test values for the 2018 Toyota RAV4 LE:25.9
6.44 Nutritional intake among Canadian high-performance athletes. Since previous studies have reported that elite athletes are often deficient in their nutritional intake (for example, total calories, carbohydrates, protein), a group of researchers decided to evaluate Canadian high-performance
6.43 Attitudes toward school. The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures the motivation, attitude toward school, and study habits of students. Each scores ranges from 0 to 200. The mean attitude score for U.S. college students is about 110, and the
6.42 Who is the author? Statistics can help decide the authorship of literary works. Sonnets by a certain Elizabethan poet are known to contain an average of 18 19μ=8.9 new words (words not used in the poet’s other works). The standard deviation of the number of new words isσ=2.5 . Now a
6.41 Are the pine trees randomly distributed from east to west? Answer the questions in the previous exercise for the east–west direction, for which the sample mean is 113.8.
6.40 Are the pine trees randomly distributed from north to south? In Example 6.1 (page 329), we looked at the distribution of longleaf pine trees in the Wade Tract. One way to formulate hypotheses about whether or not the trees are randomly distributed in the tract is to examine the average
6.39 Alcohol awareness among college students. A study of alcohol awareness among college students reported a higher awareness for students enrolled in a health and safety class than for those enrolled in a statistics class. The difference is described as being statistically significant. Explain
6.38 Symbol of wealth in ancient China? Every society has its own symbols of wealth and prestige. In ancient China, it appears that owning pigs was such a symbol. Evidence comes from examining burial sites. If the skulls of sacrificed pigs tend to appear along with expensive ornaments, that
6.37 Peer pressure and choice of major. A study followed a cohort of students entering a business/economics program. All students followed a common track during the first three semesters and then chose to specialize in either business or economics. Through a series of surveys, the researchers were
6.36 Change in consumption of sweet snacks? Refer to Exercise 6.13 (page 345). A similar study performed four years earlier reported the average consumption of sweet snacks among healthy-weight children aged 12 to 19 years to be 369.4 kilocalaries per day (kcal/d). Does this current study suggest a
6.35 Average starting salary, continued. Refer to Exercise 6.12 (page 345). Use the information presented in the exercise to test whether the average income of graduates from your institution is different from the national average(α=0.05) . Write a short paragraph summarizing your conclusions.
6.34 Consumption of neurotoxic insecticide delays migration. Some Canadian researchers recently investigated the impact of songbirds consuming a widely used insecticide. They reported that whitecrowned sparrows fed 1.2 mg/kg body mass of imidacloprid had an average mass loss of 3%16(P=0.005) just
6.33 More on computing the P-value. A test of the null hypothesis H0: μ=μ0 gives test statistic z=−2.59 .a. What is the P-value if the alternative is Ha: μ>μ0 ?b. What is the P-value if the alternative is Ha: μ
6.32 Computing the P-value. A test of the null hypothesis H0: μ=μ0 gives test statistic z=1.37 .a. What is the P-value if the alternative is Ha: μ>μ0 ?b. What is the P-value if the alternative is Ha: μ
6.31 Translating research questions into hypotheses. Translate each of the following research questions into appropriate H0 and Ha .a. U.S. Census Bureau data show that the mean household income in the area served by a shopping mall is $78,800 per year. A market research firm questions shoppers at
6.30 Even more on determining hypotheses. In each of the following situations, state an appropriate null hypothesis H0 and alternative hypothesis Ha .Be sure to identify the parameters that you use to state the hypotheses. (We have not yet learned how to test these hypotheses.)a. A sociologist asks

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