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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

Recognize anecdotal data and use specific examples to explain why anecdotal data have limited value.
Find the joint distribution, the marginal distributions, and the conditional distributions in a two-way table from software output.
Use the conditional distributions to describe the relationship displayed in a two-way table.
Find and interpret the marginal distributions in a two-way table.
Find and interpret the joint distribution in a two-way table.
Identify the row variable, the column variable, and the cells in a two-way table.
Explain the difference between association and causality when interpreting the relationship between two variables.
Identify lurking variables that can influence the interpretation of relationships between two variables.
Identify outliers and influential observations by examining scatterplots and residual plots.
Use a plot of the residuals versus the explanatory variable to assess the fit of a regression line.
Calculate the residuals for a set of data using the equation of the least-squares regression line and the observed values of the explanatory variable.
Identify situations where using a regression equation for prediction would not be accurate because of extrapolation.
Read the output of statistical software to find the equation of the least-squares regression line and the value of r2
Predict a value of the response variable y for a given value of the explanatory variable x using a regression equation.
Find the equation of the least-squares regression line and draw it on a scatterplot of a set of data.
describing the relationship between two quantitative variables.
Identify the roles of the correlation, a numerical summary, and the scatterplot, a graphical summary, for
Identify situations in which the correlation is not a good measure of association between two quantitative variables.
Use a correlation to describe the direction and strength of a linear relationship between two quantitative variables.
Use different plotting symbols to include information about a categorical variable in a scatterplot.
Use a log transformation to change a curved relationship into a linear relationship.
Use a scatterplot to describe the form, direction, and strength of a relationship and to identify outliers.
Describe the overall pattern in a scatterplot and any striking deviations from that pattern.
Make a scatterplot to examine a relationship between two quantitative variables.
Identify the key characteristics of a data set to be used to explore a relationship.
Classify variables as response variables or explanatory variables.
Compute the effects of a linear transformation on the mean, the median, the standard deviation, and the IQR.
Choose measures of center and spread for a particular set of data.
Identify outliers by using the 1.5×IQR rule.
Describe a distribution or compare data sets measured on the same variable by using boxplots.
Describe a distribution by using the five-number summary.
Describe the spread of a distribution by using the interquartile range (IQR) or the standard deviation.
Describe the center of a distribution by using the mean or the median.
Use a time plot to describe the trend of a quantitative variable that is measured over time.
Identify and describe any outliers in the distribution of a quantitative variable.
Describe the key characteristics of a set of data.
Identify the values of a variable and classify variables as categorical or quantitative.
Identify the variables in a data set and when a variable can be used as a label.
Give examples of cases in a data set.
17.80 Can we improve the capability of the meat-packaging process? Refer to the previous exercise. The average weight of each section can be increased (or decreased) by increasing (or decreasing) the time between slices of the machine.Based on the results of the previous exercise, would a change in
17.79 Capability indexes for the meat-packaging process. Refer to the previous two exercises. The lower and upper specifications for the one-pound sections are 0.95 and 1.09.a. Using these data, estimate Cp and Cpk for this process.b. What may be a reason for the specifications being centered at a
17.78 Determining the natural tolerances for the weight of ground beef.Refer to the previous exercise. Because the distribution of weights appears to be stable, use the data to construct the natural tolerances within which you expect almost all the weights to fall. Also, make sure to check whether
17.77 Control limits for a meat-packaging process. A meat-packaging company produces one-pound packages of ground beef by having a machine slice a long circular cylinder of ground beef as it passes through the machine.The timing between consecutive cuts will alter the weight of each section.17
17.76 Deming speaks. The following comments were made by the quality guru W. Edwards Deming (1900–1993). Choose one of these sayings.Explain carefully what facts about improving quality the saying attempts to summarize.a. “People work in the system. Management creates the system.”b.
17.75 Probability of another out-of-control signal. One out-of-control signal is four out of five successive points being on the same side of the center line and farther thanσ/n from it. Find the probability of this event when a process is in control.
17.74 Probability of an out-of-control signal. There are other out-of-control rules that are sometimes used with x¯charts. One is “15 points in a row within the 1σlevel” (that is, 15 consecutive points falling betweenμ−σ/n andμ+σ/n). This signal suggests either that the value ofσused
17.73 More on the film thickness process. Previously, control of the process was based on categorizing the thickness of each film inspected as satisfactory or not. Steady improvement in process quality has occurred, so that just 15 of the last 5000 films inspected were unsatisfactory.a. What type
17.72 Calculating the percent that meet specifications. Examination of individual measurements shows that they are close to Normal. If the process mean is set to the target value, about what percent of films will meet the specifications?
17.71 Capability of the film thickness process. The specifications call for film thickness 830±25 mm×10−4.a. What is the estimateσ^of the process standard deviation based on the sample standard deviations(after removing Samples 1 and 10)? Estimate the capability ratio Cp and comment on what it
17.70 Recalculating the x¯and s charts. Interviews with the operators reveal that in Samples 1 and 10, mistakes in operating the interferometer resulted in one high-outlier thickness reading that was clearly incorrect. Recalculate x¯and s after removing Samples 1 and 10. Recalculate UCL for the s
17.69 Constructing the s chart. Calculate control limits for s, make an s chart, and comment on control of short-term process variation.
17.68 Selecting the appropriate control chart and limits. At the present time, about 8 out of every 1000 lots of material arriving at a plant site from outside vendors are rejected because they do not meet specifications. The plant receives about 550 lots per week. As part of an effort to reduce
17.67 Detecting special cause variation. Is each of the following examples of a special cause most likely to first result in (i) a sudden change in level on the s or R chart, (ii) a sudden change in level on the x¯chart, or (iii) a gradual drift up or down on the x¯chart? In each case, briefly
17.66 Choice of control chart. What type of control chart or charts would you use as part of efforts to assess quality for each of the following scenarios?Explain your choices.a. Time to get security clearance.b. Percent of job offers accepted.c. Thickness of steel washers.d. Number of dropped
17.65 Constructing a Pareto chart. You manage the customer service operation for a maker of electronic equipment sold to business customers. Traditionally, the most common complaint has been that equipment does not operate properly when installed, but attention to manufacturing and installation
17.64 Describing a process that is in control. A manager who knows no statistics asks you, “What does it mean to say that a process is in control? Is an in-control process a guarantee that the quality of the product is good?” Answer these questions in plain language that the manager can
17.63 More on monitoring a high-quality process. Because the manufacturing quality in the previous exercise is so high, the process of writing up orders is the major source of quality problems: the orderwriting defect rate is 8000 per million opportunities. The manufacturer processes about 500
17.62 Constructing a p chart for a high-quality process. A manufacturer of consumer electronic equipment makes full use not only of statistical process control but also of automated testing equipment that efficiently tests all completed products. Data from the testing equipment show that finished
17.61 Calculating the p chart limits for school absenteeism. Here are data from an urban school district on the number of eighth-grade students with three or more unexcused absences from school during each month of a school year. Because the total number of eighth-graders changes a bit from month
17.60 Constructing the p chart limits for prescription errors. A regional chain of retail pharmacies finds that about 2% of prescriptions it receives from doctors result in prescriptions errors. These errors include filling the order with the wrong dosage, failing to recognize a harmful drug
17.59 Constructing a p chart for missing or deformed rivets. After completion of an aircraft wing assembly, inspectors count the number of missing or deformed rivets. There are hundreds of rivets in each wing, but the total number varies depending on the aircraft type. Recent data for wings with a
17.58 More on constructing a p chart for damaged eggs. Refer to the previous exercise. Suppose that there are two machine operators, each working four-hour shifts. The first operator is very skilled and can inspect 500 eggs per hour. The second operator is less experienced and can inspect only 400
17.57 Constructing a p chart for damaged eggs. An egg farm wants to monitor the effects of some new handling procedures on the percent of eggs arriving at the packaging center with cracked or broken shells. In the past, 1.93% of the eggs were damaged. A machine will allow the farm to inspect 500
17.56 Constructing a p chart for mishandled baggage. The latest report by Société Internationale de Télécommunications Aéronautiques (SITA) states that the mishandled baggage rate has plateaued at 5.7 per thousand passengers. Starting with this information, you plan to sample records for 2500
17.55 Constructing a p chart for unpaid invoices. The controller’s office of a corporation is concerned that invoices that remain unpaid after 30 days are damaging relations with vendors. To assess the magnitude of the problem, a manager searches payment records for invoices that arrived in the
17.54 Constructing a p chart for absenteeism. After inspecting Figure 17.22, you decide to monitor the next four weeks’ absenteeism rates using a center line and control limits calculated from the second two weeks’ data recorded in Table 17.12. Find p¯ for these 10 days and give the new values
17.53 More on interpreting the s chart. Each of the four out-of-control values of s in part (a) of the previous exercise is explained by a single outlier, a very long response time to one call in the sample.You can see these outliers in Figure 17.17(b). What are the values of these outliers, and
17.52 Constructing and interpreting the s chart. Table 17.11 also gives x¯ and s for each of the 50 samples.a. Make an s chart and check for points out of control.b. If the s-type cause responsible is found and removed, what would be the new control limits for the s chart? Verify that no points s
17.51 Choosing the sample. The six calls each shift are chosen at random from all calls received during the shift. Discuss the reasons behind this choice and those behind a choice to time six consecutive calls.
17.50 More on Six Sigma quality. The originators of the Six Sigma quality idea reasoned as follows. Short-term process variation is described byσ . In the long term, the process mean μ will also vary. Studies show that, in most manufacturing processes,±1.5σ is adequate to allow for changes in
17.49 Cp and Six Sigma.A process with Cp≥2 is sometimes said to have “Six Sigma quality.” Sketch the specification limits and a Normal distribution of individual measurements for such a process when it is properly centered. Explain from your sketch why this is called Six Sigma quality.
17.48 Will these actions help the capability? Based on the results of the previous exercise, you conclude that the capability of the bearing-making process is inadequate. Here are some suggestions for improving the capability of this process. Comment on the usefulness of each action suggested.a.
17.47 Assessing the capability of the skateboard bearings process. Recall the skateboard bearings process described in Exercise 17.38 (page 17-34). The bore diameter has specifications (7.9920, 8.000)mm. The process is monitored by x¯ and s charts based on samples of five consecutive bearings each
17.46 Calculating capability indexes for the DRG 209 hospital losses. Table 17.8 (page 17-33) gives data on a hospital’s losses for 120 DRG 209 patients, collected as 15 monthly samples of 8 patients each.The process has been in control, and losses have a roughly Normal distribution. The hospital
17.45 Estimating capability indexes for the distance between holes. Figure 17.10 (page 17-18)displays a record sheet on which operators have recorded 18 samples of measurements on the distance between two mounting holes on an electrical meter. Sample 5 was out of control on an s chart. We remove it
17.44 An alternative estimate for Cpk of the waterproofing process. In Exercise 17.42(b), you found C^pk for specifications LSL=1500 and USL=3500 using the standard deviation s=383.8 for all 80 individual jackets in Table 17.1. Repeat the calculation using the control chart estimateσ^=s¯/c4 . You
17.43 Capability of a characteristic with a uniform distribution. Suppose that a quality characteristic has the uniform distribution on 0 to 1. Figure 4.9 (page 229) shows the density curve. You can see that the process mean (the balance point of the density curve) isμ=1/2 . The standard deviation
17.42 Capability indexes for the waterproofing process (continued). We could improve the performance of the waterproofing process discussed in the previous exercise by making an adjustment that moves the center of the process toμ=2500 mm , the center of the specifications. We should do this even
17.41 Capability indexes for the waterproofing process. Table 17.1 (page 17-9) gives 20 process control samples of the water resistance of a particular outdoor jacket. In Example 17.11, we estimated from these samples thatμ^=x== 2750.7 mm and σ^=s=383.8 mm.a. The original specifications for water
17.40 Monitoring weight. Joe has recorded his weight, measured at the gym after a workout, for several years. The mean is 181 pounds, and the standard deviation is 1.7 pounds, with no signs of lack of control. An injury keeps Joe away from the gym for several months. The data below give his weight,
17.39 Transforming the quality measure. In Chapter 7 (page 399), we discussed the use of a transformation to turn a variable with a skewed distribution into a variable whose distribution is often close to Normal. Describe how a transformation might be used to approximate the natural tolerances of a
17.38 Control charts for the bore diameter of a bearing. A sample of five skateboard bearings is taken near the end of each hour of production. TABLE 17.10 gives x¯ and s for the first 21 samples, coded in units of 0.001 mm from the target value. The specifications allow a range of±0.004 mm about
17.37 Assessing the Normality of the distance between holes. Make a Normal quantile plot of the 85 distances in the data file MOUNT that remain after removing Sample 5. How does the plot reflect the limited precision of the measurements (all of which end in 4)? Is there any departure from Normality
17.36 Assessing the Normality of the densitometer measurements. Are the 60 individual measurements in Table 17.9 at least approximately Normal so that the natural tolerances you calculated in Exercise 17.34 can be trusted? Make a Normal quantile plot (or another graph, if your software is limited)
17.35 Determining the percent of meters that meet specifications. The record sheet in Figure 17.10 gives the specifications as 0.6054±0.0010 inch.That’s 54±10 as the data are coded on the record. Assuming that the distance varies Normally from meter to meter, about what percent of meters meet
17.34 Determining the natural tolerances for the densitometer. Remove any samples in Table 17.9 that your work in Exercise 17.32 showed to be out of control on either chart. Estimate the mean and standard deviation of individual measurements on the phantom. What are the natural tolerances for these
17.33 Determining the natural tolerances for the distance between holes. Figure 17.10 (page 17-18)displays a record sheet for 18 samples of distances between mounting holes in an electrical meter. In Exercise 17.15 (page 17-18), you found that Sample 5 was out of control on the process-monitoring s
17.32 Monitoring the calibration of a densitometer. Loss of bone density is a serious health problem for many people, especially older women. Conventional X-rays often fail to detect loss of bone density until the loss reaches 25% or more. New equipment, such as the Lunar bone densitometer, is much
17.31 Improving the capability of the process. Refer to the previous exercise. The center of the specifications for waterproofing is 2500 mm, but the center of our process is 2750 mm. We can improve capability by adjusting the process to have center 2500 mm. This is an easy adjustment that does not
17.30 The percent of products that meet specifications. If the water resistance readings for individual jackets follow a Normal distribution, we can describe capability by giving the percent of jackets that meet specifications. The old specifications for water resistance are 1000 to 4000 mm. The
17.29 Checking the Normality of losses. Do the losses on the 120 individual patients in Table 17.8 appear to come from a single Normal distribution? Make a Normal quantile plot and discuss what it shows. Are the natural tolerances you found in the previous exercise trustworthy? Explain your answer.
17.28 Determining the tolerances for losses from DRG 209 patients. Table 17.8 gives data on hospital losses for samples of DRG 209 patients. The distribution of losses has been stable over time. What are the natural tolerances within which you expect losses on nearly all such patients to fall?
17.27 Efficient process control. A company that makes smartphones requires that its microchip supplier practice statistical process control and submit control charts for verification. This allows the company to eliminate inspection of the microchips as they arrive, which is a considerable cost
17.26 Estimating the control chart limits from past data. TABLE 17.8 gives data on the losses (in dollars) incurred by a hospital in treating DRG 209 (major joint replacement) patients. The hospital has taken from its records a random sample of eight such patients each month for 15 months.a. Make
17.25 Reevaluating the process parameters. The x¯ and s control charts for the waterproofing example were based onμ=2750 mm andσ=430 mm . Table 17.1(page 17-9) gives the 20 most recent samples from this process.a. Estimate the processμ andσ based on these 20 samples.b. Your calculations
17.24 Altering the sampling plan. Refer to the previous exercise. Suppose now that each sample contains an equal number of invoices from each clerk.a. Sketch the x¯ and s chart patterns that will result.b. The process in this case will appear to be in control. When might this be an acceptable
17.23 Control chart for an unusual sampling situation. Invoices are processed and paid by two clerks, one very experienced and the other newly hired. The experienced clerk processes invoices quickly. The new hire often refers to the procedures handbook and is much slower. Both are quite consistent
17.22 Setting up another control chart. Refer to the previous exercise. TABLE 17.7 contains another set of 24 samples. Repeat parts (a), (b), and (c) of the previous exercise using this data set.TABLE 17.7 x¯ and s for 24 samples of label placement (in inches)Sample x¯ s Sample x¯ s 1 2.0309
17.21 Setting up a control chart. In Exercise 17.8 (page 17-21), the x¯ and s control charts for the placement of the rum label were based on historical results. Suppose that a new labeling machine has been purchased, and new control limits need to be determined. TABLE 17.6 contains the means and
17.20 Alternative control limits. American and Japanese practice uses 3σ control charts. That is, the control limits are 3 standard deviations on either side of the mean. When the statistic being plotted has a Normal distribution, the probability of a point outside the limits is about 0.003 (or

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