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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

17.19 Determining the probability of detection. An x¯ chart plots the means of samples of size 4 against center line CL=715 and control limits LCL=691 and UCL=739 . The process has been in control.a. What are the process mean and standard deviation?b. The process is disrupted in a way that changes
17.18 Causes of variation in the time to respond to an application. The personnel department of a large company records a number of performance measures. Among them is the time required to respond to an application for employment, measured from the time the application arrives. Suggest some
17.17 2σ control charts. Some special situations call for 2σ control charts. That is, the control limits for a statistic Q will beμQ±2σQ . Suppose that you know the process meanμ and standard deviationσ and will plot x¯ and s from samples of size n.a. What are the 2σcontrol limits for an
17.16 Identifying special causes on control charts. The process described in Exercise 17.14 goes out of control. Investigation finds that a new type of yarn was recently introduced. The pH in the kettles is influenced by both the dye liquor and the yarn. Moreover, on a few occasions, a faulty valve
7. Time, 10 30. Sample measurement 1, 64. Sample measurement 2, 44. Sample measurement 3, 34. Sample measurement 4, 34.Sample measurement 5, 54. Average x bar, blank. Range, R, 30. Row 3. Date, 3, 7. Time, 11 45. Sample measurement 1, 34.Sample measurement 2, 44. Sample measurement 3, 54. Sample
17.15 Control charts for a mounting-hole process. FIGURE 17.10 reproduces a data sheet from a factory that makes electrical meters. The sheet shows measurements of the distance between two mounting holes for 18 samples of size 5. The heading informs us that the measurements are in multiples of
17.14 Control limits for a dyeing process. The unique colors of the cashmere sweaters your firm makes result from heating undyed yarn in a kettle with a dye liquor. The pH (acidity) of the liquor is critical for regulating dye uptake and, hence, the final color. There are five kettles, all of which
17.13 Control limits for a milling process. The width of a slot cut by a milling machine is important to the proper functioning of a hydraulic system for large tractors. The manufacturer checks the control of the milling process by measuring a sample of six consecutive items during each hour’s
17.12 More on the tablet compression process. Exercise 17.11 concerns process control data on the hardness of tablets for a pharmaceutical product. TABLE 17.4 gives data for 20 new samples of size 4, with the x¯ and s for each sample. The process has been in control with mean at the target
17.11 Control charts for a tablet compression process. A pharmaceutical manufacturer forms tablets by compressing a granular material that contains the active ingredient and various fillers. The hardness of a sample from each lot of tablets is measured in order to control the compression process.
17.10 Control limits for air conditioner thermostats. A maker of auto air conditioners checks a sample of five thermostatic controls from each hour’s production. The thermostats are set at 72°F and then placed in a chamber where the temperature is raised gradually. The temperature at which the
17.9 More on control limits for label placement. Refer to the previous exercise. What happens to the center line and control limits for the x¯ and s control charts ifa. The distributor samples 10 bottles from each batch?b. The distributor samples two bottles from each batch?c. The distributor uses
17.8 Control limits for label placement. A rum producer monitors the position of its label on the bottle by sampling four bottles from each batch. One quantity measured is the distance from the bottom of the bottle neck to the top of the label. The process mean should beμ=1.75 inches. Past
17.7 Making a Pareto chart. Continue the study of the process of calling in a sandwich order (Exercise 17.1). If you kept good records, you could make a Pareto chart of the reasons (special causes) for unusually long order times. Make a Pareto chart of these reasons. That is, list the reasons,
17.6 Constructing another Pareto chart. A large hospital finds that it is losing money on surgery due to inadequate reimbursement by insurance companies and government programs. An initial study looks at losses broken down by diagnosis. Government standards place cases into diagnosis-related
17.5 Constructing a Pareto chart. Comparisons are easier if you order the bars in a bar graph by height. A bar graph ordered from tallest to shortest bar is sometimes called a Pareto chart, after the Italian economist who recommended this procedure. Pareto charts are often used in quality studies
17.4 Type of control chart. For each of the following measures of quality, state whether variable or attribute control charts should be used in monitoring:a. Daily number of on-the-job accidents.b. Time to ship a product to the customer.c. The thread diameter of a screw.d. The daily proportion of
17.3 Causes of variation. Each week you go to your instructor’s office hours. Her office is on the sixth floor, so you take the elevator. List five possible causes for variation in the time it takes between pushing the elevator button and arriving at her office. Explain whether each is a common
17.2 Determining sources of common cause variation and special cause variation. Refer to the previous exercise. The time it takes from deciding to order a sandwich to receiving the sandwich will vary. List several common causes of variation in this time. Then list several special causes that might
17.1 Constructing a flowchart. Consider the process of ordering a Jimmy John’s sandwich order for delivery to your residence. Make a flowchart of this process, including steps that involve Yes/No decisions.
16.92 Comparing two operators. Exercise 7.29 (page 409) gives these data on a delicate measurement of total body bone mineral content made by two operators on the same eight subjects:Operator Subject 1 2 3 4 5 6 7 8 1 1.328 1.342 1.075 1.228 0.939 1.004 1.178 1.286 2 1.323 1.322 1.073 1.233 0.934
16.91 Other ways to look at Jocko’s estimates. Refer to the previous exercise. Let’s consider some other ways to analyze these data.a. For each damaged vehicle, divide Jocko’s estimate by the estimate from the other garage. Perform your analysis on these data. Write a short report that
16.90 Insurance fraud? Jocko’s Garage has been accused of insurance fraud.Data on estimates (in dollars) made by Jocko and another garage were obtained for 10 damaged vehicles. Here is what the investigators found:Car 1 2 3 4 5 Jocko’s 1375 1550 1250 1300 900 Other 1250 1300 1250 1200 950 Car 6
16.89 Comparing the variances for sadness and spending. Refer to the previous exercise. Some treatments in randomized experiments such as this can cause variances to be different. Are the variances of the neutral and sad subjects equal?a. Compute the ratio F*=s12/s22 and compare to the F
16.88 Sadness and spending. Refer to Exercise 7.47 (page 430). A study of sadness and spending randomized subjects to watch videos designed to produce sad or neutral moods. Each subject was given $10, and after watching the video, he or she was asked to trade $0.50 increments of their $10 for an
16.87 Another way to communicate the result. Refer to the previous two exercises. Here is another way to communicate the result:female teenagers are 33% more likely to say they plan to attend a four-year college than male teenagers.a. Explain how the 33% is computed.b. Use the bootstrap to give a
16.86 Use a ratio for females versus males. Refer to the previous exercise.In many settings, researchers prefer to communicate the comparison of two proportions with a ratio. For teenagers planning to attend a four-year college, they would report that females are 1.33 (68/51) times more likely to
16.85 Planning to attend a four-year college. A Pew survey asked U.S.teenagers whether they plan to attend a four-year college. For the boys, 51%of 461 survey participants said they planned to attend a four-year college. For 14 the girls, 68% of 454 survey participants said this. Use the bootstrap
16.84 Are female personal trainers, on average, younger? A fitness center employs 20 personal trainers. Here are the ages, in years, of the female and male personal trainers working at this center:Male 25 26 23 32 35 29 30 28 31 32 29 Female 21 23 22 23 20 29 24 19 22a. Make a back-to-back
16.83 Bootstrap confidence interval for the median. Most software can generate random numbers that have the uniform distribution on 0 to 1. For example, Excel has the RAND() function (page 168) and R has the runif()function. Generate a sample of 50 observations from this distribution.a. Figure 4.9
16.82 Variance for poets. Refer to Exercises 16.79 and 16.81.a. Instead of comparing means, compare variances using the ratio of sample variances as the statistic. Summarize your findings.b. Explain how questions about the equality of standard deviations are related to questions about the equality
16.81 Permutation test for the poets. Refer to Exercise 16.79. Answer part(c) of that exercise using the permutation test. Summarize your findings and compare them with what you found in Exercise 16.79.
16.80 Medians for the poets. Refer to the previous exercise. Use the bootstrap methods of this chapter to compare the medians of the two distributions. Summarize your findings and compare them with part (c) of the previous exercise.
16.79 Poetry: An occupational hazard. According to William Butler Yeats,“She is the Gaelic muse, for she gives inspiration to those she persecutes. The Gaelic poets die young, for she is restless, and will not let them remain long on earth.” One study designed to investigate this issue examined
16.78 Bootstrap confidence interval for the ratio. Here is one conclusion from the data in Table 16.3, described in Exercise 16.68: “The mean serum retinol level in uninfected children was 1.255 times the mean level in the infected children. A 95% confidence interval for the ratio of means in the
16.77 Bootstrap confidence interval for the difference in proportions.Refer to Exercise 16.70 (page 16-49). We want a 95% confidence interval for the change from 2015 to 2020 in the proportions of U.S. residents who report that they have listened to at least one podcast. Bootstrap the sample data.
16.76 Use the regression slope. Refer to the previous exercise, where we used correlations to address the question of whether or not the relationship between GPA and high school math grades is the same for men and women. In Exercise 16.50 (page 16-37), we used the bootstrap to examine the slope of
16.75 Compare the correlations. In Exercise 16.45 (page 16-37), we compared the mean GPA for males and females using the bootstrap. In Exercise 16.46, we used the bootstrap to examine the correlation between GPA and high school math grades. Find the correlations for men and women separately and
16.74 Change the trim. Refer to the previous exercise. Change the statistic of interest to the 10% trimmed mean. Answer the questions in the previous exercise and also compare your new interval with the one you found there.
16.73 Sex and GPA. In Example 16.7 (page 16-15), you used the bootstrap to find a 95% confidence interval for the 25% trimmed mean of GPA. Let’s change the statistic of interest to the 5% trimmed mean. Using Examples 16.5 through 16.7 as a guide, find the corresponding 95% confidence
16.18. Discuss any differences you find and how you would report the results to the nursing home staff.12
16.72 The effect of outliers (continued). In Exercise 16.52 (page 16-38), we studied the effect of outliers on the bootstrap distribution and confidence intervals. For this exercise, perform the permutation test without the three patients with very small differences and compare the results with
16.71 Sex and GPA. In Exercise 16.45 (page 16-37), we used the bootstrap to compare the mean GPA scores for men and women.a. Use permutation methods to compare the means for men and women.b. Use permutation methods to compare the standard deviations for men and women.c. Write a short paragraph
16.70 Listening to podcasts. A 2015 Edison Research study asked U.S. individuals age 12 and older whether or not they had ever listened to a podcast. The survey was repeated with different users in 2020. For the 2015 survey, 660 of the 2002 people surveyed reported that they had listened to at
16.69 Methods of resampling. In Exercise 16.68, we did a permutation test for the hypothesis “no difference between infected and uninfected children,” using the ratio of mean serum retinol levels to measure “difference.” We might also want a bootstrap confidence interval for the ratio of
16.3 gives the serum retinol levels for both groups, in micromoles per liter.a. The researchers are interested in the proportional reduction in serum retinol. Verify that the mean for infected children is 0.620 and that the mean for uninfected children is 0.778.11b. There is no standard test for
16.68 Comparing serum retinol levels. The formal medical term for vitamin A in the blood is serum retinol. Serum retinol has various beneficial effects, such as protecting against fractures. Medical researchers working with children in Papua New Guinea asked whether recent infections reduce the
16.67 Comparing standard deviations. In Example 12.17 (page 620), the modified Levene’s test was used to compare standard deviations. Let’s now consider performing a permutation test using the F statistic (the ratio of the largest and smallest sample variances) as your statistic. What do you
16.66 Comparing mpg calculations. Exercise 7.25 (page 408) gives data on a comparison of driver and computer mpg calculations. This is a matched pairs study, with mpg values for 20 fill-ups.a. Carry out the matched pairs t test. That is, state hypotheses, calculate the test statistic, and give its
16.65 Testing the correlation between price and rating. Example 16.14 (page 16-34) uses the bootstrap to find a confidence interval for the correlation between price and rating for 24 laundry detergents. Let’s use a permutation test to examine this correlation.a. State the null and alternative
16.64 Compare the medians. Refer to the previous exercise. Use a permutation test to compare the medians. Write a short summary of your results and conclusions. Include a comparison of what you found here with what you found in the previous exercise.
16.63 Assessing a summer language institute. Exercise 7.105 (page 447) gives data on a study of the effect of a summer language institute on the ability of high school language teachers to understand spoken French. This is a matched pairs study, with scores for 20 teachers at the beginning
16.62 Testing the correlation between BMI and physical activity. In Exercise 16.47 (page 16-37), we assessed the significance of the correlation between BMI and physical activity by creating bootstrap confidence intervals. If a 95% confidence interval does not cover 0, the observed correlation is
16.61 When is a permutation test valid? You want to test the equality of the means of two populations. Sketch density curves for two populations for whicha. a permutation test is valid but a t test is not.b. both permutation and t tests are valid.c. a t test is valid but a permutation test is not.
16.60 Standard deviation of the estimated P-value. The estimated P-value for the DRP study(Example 16.16, page 16-40) based on 1000 resamples is P=0.014 . Suppose that we obtained the same P-value based on 4000 resamples. What is the approximate standard deviation of each of these P-values?
16.59 Low-calorie sweeteners. Examples 7.18 and 7.19 (page 407) examine data on an experiment to compare weight change in subjects who were asked to consume a sweetened beverage in addition to their normal diet. The sweetened beverage was sweetened with either saccharin or sucralose. In Example
16.58 Product labels with animals? Participants in a study were asked to indicate their attitude toward a product on a seven-point scale (from 1=dislike very much to 7=like very much) . A bottle of MagicCoat pet shampoo, with a picture of a collie on the label, was the product. Prior to indicating
16.57 A small-sample permutation test. For this exercise, perform a permutation test by hand for a small random subset of the DRP data (Example 16.15, page 16-39). Here are the data:Treatment group 57 67 Control group 53 42 42 37a. Calculate the difference in means x¯treatment−x¯control between
16.56 Compare the correlations. Refer to the previous exercise. Suppose that you calculate the correlation between the satisfaction of these two features for each phone. Outline the steps needed to compare these two correlations using a permutation test.
16.55 Features of smartphones. Refer to Exercise 16.53. Before asking the students to provide an overall satisfaction rating, you asked them to provide ratings, using the same scale of 1 to 100, for several features of the smartphone. Two of these were satisfaction with the touchscreen and
16.54 Marketing smartphones (continued). Refer to the previous exercise. Suppose that you had each of the 40 students use both smartphones. Each is used for one week, and the order is randomly determined. Outline the steps needed to compare the means for the two smartphones using a permutation test.
16.53 Marketing smartphones. You received two prototypes of a new smartphone and designed an experiment to help decide which one to market. Forty students were each randomly assigned to use one of the two phones for two weeks. Their overall satisfaction with the phone was recorded on a subjective
16.52 The effect of outliers. We know that outliers can strongly influence statistics such as the mean and the least-squares line. A study of dementia patients in nursing homes recorded various types of disruptive behaviors every day for 12 weeks. Days were classified as moon days if they were in a
16.51 Predicting BMI. Continue your study of the relationship between BMI and physical activity, begun in Exercise 16.47. Bootstrap the least-squares regression line using physical activity as the explanatory variable.a. Examine the shape and bias of the bootstrap distribution of the slope b1 of
16.50 Predicting GPA. Continue your study of GPA and high school math grades, begun in Exercise 16.46, by performing a regression to predict GPA using high school math grades as the explanatory variable.a. Plot the residuals against the math grades and make a Normal quantile plot of the residuals.
16.49 Predicting ratings of laundry detergents. Refer to Example 16.13 (page 16-33).a. Find the least-squares regression line for predicting rating from price.b. Bootstrap the regression line and give a 95% confidence interval for the slope of the population regression line.c. Compare the bootstrap
16.48 Bootstrap distribution for the slopeβ1 . Describe carefully how to resample from data on an explanatory variable x and a response variable y to create a bootstrap distribution for the slope b1 of the least-squares regression line.
16.47 The correlation between BMI and physical activity. Figure 10.3 (page 518) shows a relatively weak negative relationship between BMI and physical activity. Use the bootstrap to perform statistical inference for the correlation.a. Describe the shape and bias of the bootstrap distribution. Do
16.46 The correlation between GPA and high school math grades. The study described in Example 16.5 (page 16-12) used high school grades to predict GPA. For this exercise, we will look at the correlation between GPA and high school math grades.a. Describe the distribution of GPAs. Do the same for
16.45 Bootstrap confidence intervals for the difference in GPAs. Example 16.9 (page 16-17)considers the difference in mean GPAs of men and women. The bootstrap distribution appeared reasonably Normal. Give the 95% BCa confidence interval for the difference in mean GPAs. Is this interval comparable
16.44 Bootstrap confidence interval for the GPA data. The GPA data for females from Example 16.8(page 16-16) are strongly skewed to the left and have a cluster of observations at 4.a. Bootstrap the mean of the data. Based on the bootstrap distribution, which bootstrap confidence intervals would you
16.43 The effect of decreasing the sample size. Exercise 16.15 (page 16-11) gives an SRS of 10 of the visit lengths from Table 5.1 (page 283). Describe the bootstrap distribution of x¯ from this sample. Give a 95% confidence interval for the population meanμ based on these data and a method of
16.42 Bootstrap confidence intervals for the standard deviation. We would like a 95% confidence interval for the standard deviationσ of 150 GPAs. In Exercise 16.23 (page 16-22), we considered the bootstrap t interval. Now we have a more accurate method. Bootstrap s and report all three 95%
16.41 Bootstrap confidence intervals for visits to a help room. The distribution of the visit lengths to a statistics help room that you used in Exercise 16.21 (page 16-21) is skewed. In that exercise, you found a bootstrap t confidence interval for the population meanμ , even though the bootstrap
16.40 Bootstrap confidence intervals for the average audio file length. In Check-in question 16.4(page 16-16), you found a bootstrap t confidence interval for the population meanμ . Careful examination of the bootstrap distribution reveals a slight skewness in the right tail. Is this something to
16.39 BCa interval for the correlation coefficient. Find the 95% BCa confidence interval for the correlation between price and rating, from the data in Example 16.13 (page 16-33). Is this more accurate interval in general agreement with the 95% bootstrap t and percentile intervals? Do you still
16.38 More on using bootstrapping to check traditional methods. Continue to work with the data given in Exercise 16.36.a. Find the 95% BCa confidence interval.b. Does your opinion of the robustness of the one-sample t confidence interval change when you compare it with the BCa interval?c. To check
16.37 Comparing bootstrap confidence intervals. Although the graphs in Figure 16.13 (page 16-18)do not appear to show any important skewness in the bootstrap distribution of the difference in means for Example 16.9, there is evidence that the right tail is slightly longer than expected. Using the
16.36 Using bootstrapping to check traditional methods. Bootstrapping is a good way to check whether traditional inference methods are accurate for a given sample. Consider the following data:98 107 113 104 94 100 107 98 112 97 99 95 97 90 109 102 89 101 93 95 95 87 91 101 119 116 91 95 95 104a.
16.35 Confidence interval for a Normal data set. In Exercise 16.25 (page 16-22), you bootstrapped the mean of a simulated SRS from the standard Normal distribution N(0, 1) and found the 95% standard t and bootstrap t confidence intervals for the mean.a. Find the 95% bootstrap percentile confidence
16.34 Confidence interval for the average IQ score. The distribution of the 60 IQ test scores in Table 1.1 (page 15) is roughly Normal, and the sample size is large enough that we expect a Normal sampling distribution. We will compare confidence intervals for the population mean IQμ based on this
16.33 Summarize the output. FIGURES 16.23 and 16.24 show R output regarding a comparison of two variances with samples n1=25 and n2=40 . Much as in Example 16.12 (page 16-31), the ratio is used as the statistic. Summarize the information in the output, making sure to explain your choice of
16.32 Find the 95% bootstrap percentile confidence interval. A farmer is interested in the average weight of his pigs at six months of age. The mean of a sample of n=20 pigs is x¯=225.7 pounds, and the standard deviation is s=15.3 pounds. The mean of the bootstrap distribution is x¯=224.6 , and
16.31 The effect of non-Normality. The populations in the two previous exercises have the same mean and standard deviation, but one is Normal, and the other is strongly non-Normal. Based on your work in these exercises, how does non-Normality of the population affect the bootstrap distribution of
16.30 The effect of increasing the sample size. Refer to the data in Exercise 16.9 (page 16-11). Let’s think of the times from the 187 countries as the population for this exercise. This means the population distribution of times is very non-Normal with one outlier.a. Find the meanμand the
16.29 Bootstrap versus sampling distribution. Most statistical software includes a function to generate samples from Normal distributions. Set the mean to 26 and the standard deviation to 27. You can think of all the numbers that would be produced by this function if it ran forever as a population
16.28 Variation in the bootstrap distributions. Consider the variation in the bootstrap for each of the following situations with two scenarios, S1 and S2. In comparing the variation, do you expect, in general, that S1 will have less variation than S2, that S2 will have less variation than S1, or
16.27 Bootstrap distribution of the mpg standard deviation. The Environmental Protection Agency(EPA) establishes the tests to determine the fuel economy of new cars, but it often does not perform them. Instead, the test protocols are given to the car companies, and the companies perform the tests
16.26 Bootstrap distribution of the median. We will see in Section 16.3 that bootstrap methods often work poorly for the median. To illustrate this, bootstrap the sample median of the 21 viewing times we studied in Example 16.1 (page 16-3). Why is the bootstrap t confidence interval not justified?
16.25 Bootstrapping a Normal data set. The following data are “really Normal.” They are an SRS from the standard Normal distribution N(0, 1), produced by a software Normal random number generator.0.01 −0.04 −1.02 −0.13 −0.36 −0.03 −1.88 0.34 −0.00 1.21−0.02 −1.01 0.58 0.92
16.24 Bootstrap comparison of tree diameters. In Exercise 7.57 (page 432), you were asked to compare the mean diameter at breast height (DBH) for trees from the northern and southern halves of a land tract using a random sample of 30 trees from each region.a. Use a back-to-back stemplot or
16.23 Bootstrap distribution of the standard deviation s. For Example 16.5 (page 16-12), we bootstrapped the 25% trimmed mean of 150 GPAs. Another statistic whose sampling distribution is unfamiliar to us is the standard deviation s. Bootstrap s for these data. Discuss the shape and bias of the
16.22 Another bootstrap distribution of the trimmed mean. Bootstrap distributions and quantities based on them differ randomly when we repeat the resampling process. A key fact is that they do not differ very much if we use a large number of resamples. Figure 16.11 (page 16-14) shows one bootstrap
16.21 Bootstrap t confidence interval for help room visit lengths. Return to or re-create the bootstrap distribution of the sample mean for the 50 visit lengths in Exercise 16.14 (page 16-11).a. What is the bootstrap estimate of the bias? Verify from the graphs of the bootstrap distribution that
16.20 Bootstrap t confidence interval for average delivery time by a robot. Return to or re-create the bootstrap distribution of the sample mean for the eight lunch delivery times in Exercise 16.11 (page 16-11).a. Although the sample is small, verify using graphs and numerical summaries of the
16.19 Bootstrap t confidence interval for the time to start a business. In Exercise 16.9 (page 16-17), we examined the bootstrap distribution for the times to start a business. Return to or re-create the bootstrap distribution of the sample mean for these 187 observations.a. Find the bootstrap t
16.18 Using the mean instead of the trimmed mean. In Example 16.7 (page 16-15), bootstrap results for the 25% trimmed mean of GPA were presented. Here are some results using the mean as the statistic:ORDINARY NONPARAMETRIC BOOTSTRAP Call:boot(data = GPA, statistic = theta, R = 3000)Bootstrap
16.17 Should you use the bootstrap t confidence interval? For each of the following situations, explain whether or not you would use the bootstrap standard error and the t distribution for the confidence interval. Give reasons for your answers.a. The bootstrap distribution of the mean is
16.16 Use the bootstrap standard error and the t distribution for the confidence interval. Suppose you collect GPA data similar to that of Example 16.5 (page 16-12) from your university. You only collect n=100 GPAs, so the 25%trimmed mean is based on the middle n=50 observations. The mean of the
16.15 More on help room visit lengths. Here is an SRS of 10 of the visit lengths from Exercise 16.14:20 35 142 41 60 150 30 5 165 55 We expect the sampling distribution of x¯ to be less close to Normal for samples of size 10 than for samples of size 50 from a skewed distribution.a. Create and

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