Introduction To Operations Research 9th Edition Frederick S. Hillier - Solutions

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1.91 How high is the top 15%? What SAT scores make up the top 15% of all scores?
1.90 Find an ACT percentile. Reports on a student’s ACT or SAT results usually give the percentile as well as the actual score. The percentile is just the cumulative proportion stated as a percent: the percent of all scores that were lower than or equal to this one. Joshua scores 21 on the ACT.
1.89 Find an SAT percentile. Reports on a student’s ACT or SAT results usually give the percentile as well as the actual score. The percentile is just the cumulative proportion stated as a percent: the percent of all scores that were lower than or equal to this one. Renee scores 1360 on the SAT.
1.88 Find the SAT equivalent. Alyssa scores 32 on the ACT. Assuming that both tests measure the same thing, what score on the SAT is equivalent to Alyssa’s ACT score?
1.87 Find the ACT equivalent. Jorge scores 1400 on the SAT. Assuming that both tests measure the same thing, what score on the ACT is equivalent to Jorge’s SAT score?
1.86 Make another comparison. Joshua scores 14 on the ACT. Anthony scores 690 on the SAT.Assuming that both tests measure the same thing, who has the higher score? Report the z -scores for both students.
1.85 Compare an SAT score with an ACT score. Jessica scores 1240 on the SAT. Ashley scores 28 on the ACT. Assuming that both tests measure the same thing, who has the higher score? Report the z -scores for both students.
1.84 High IQ scores. Refer to the previous exercise, The organization MENSA, which calls itself “the high-IQ society,” requires a WAIS score of 130 or higher for membership. What percent of adults would qualify for membership?There are two major tests of readiness for college, the ACT and the
1.83 Find some values of z. The Wechsler Adult Intelligence Scale (WAIS) is the most common IQ test.The scale of scores is set separately for each age group, and the scores are approximately Normal, with mean 100 and standard deviation 15. People with WAIS scores below 70 are considered
1.82 Find more values of z illustrate the result with a sketch. The variable Z has a standard Normal distribution.a. Find the number z that has cumulative proportion 0.68.b. Find the number z such that the event Z > z has proportion 0.122.
1.81 Find some values of z. Find the value z of a standard Normal variable Z that satisfies each of the following conditions. (If you use Table A, report the value of z that comes closest to satisfying the condition.) In each case, sketch a standard Normal curve with your value of z marked on the
1.80 Find more proportions. Using either software or Table A, find the proportion of observations from a standard Normal distribution for each of the following events. In each case, sketch a standard Normal curve and shade the area representing the proportion.a. Z ≤ −1.7b. Z ≥ −1.7c. Z >
1.79 Find some proportions. Using either software or Table A, find the proportion of observations from a standard Normal distribution that satisfies each of the following statements. In each case, sketch a standard Normal curve and shade the area under the curve that is the answer to the
1.78 Use the Normal Curve applet. The 68–95–99.7 rule for Normal distributions is a useful approximation. You can use the Normal Curve applet on the text website to see how accurate the rule is.Drag one flag across the other so that the applet shows the area under the curve between the two
1.77 Use the Normal Curve applet. Use the Normal Curve applet for the standard Normal distribution to say how many standard deviations above and below the mean the quartiles of any Normal distribution lie.
1.76 Three density curves. FIGURE 1.34 displays three density curves, each with three points marked on it. At which of these points on each curve do the mean and the median fall?FIGURE 1.34 Three density curves, Exercise 1.76.Description Three points are marked on each curve, A, B, and C. The first
1.75 Find the mean, the median, and the quartiles. What are the mean and the median of the uniform distribution in Figure 1.33? What are the quartiles?
1.74 Use a different range for the uniform distribution. Many random number generators allow users to specify the range of the random numbers to be produced. Suppose that you specify that the outcomes are to be distributed uniformly between 0 and 5. Then the density curve of the outcomes has
1.73 A uniform distribution. If you ask a computer to generate “random numbers” between 0 and 1, you will get observations from a uniform distribution. FIGURE 1.33 graphs the density curve for a uniform distribution. Use areas under this density curve to answer the following questions.a. What
1.72 Data from Mexico. Refer to the previous exercise. A similar study in Mexico was conducted with 31 women and 20 men. The women averaged 14,704 words per day, with a standard deviation of 6215.For men the mean was 15,022, and the standard deviation was 7864.a. Answer the questions from the
1.71 Do women talk more? Conventional wisdom suggests that women are more talkative than men.One study designed to examine this stereotype collected data on the speech of 42 women and 37 men in the United States.a. The mean number of words spoken per day by the women was 14,297, with a standard
1.70 Are the NAEP eighth-grade mathematics scores approximately Normal? Refer to the previous exercise. For the NAEP eighth-grade mathematics scores, the mean is 282, and the standard deviation is 40. Here are the reported percentiles:Percentile Score 10% 231 25% 255 50% 282 75% 309 90% 333 Is the
1.69 Are the NAEP eighth-grade geography scores approximately Normal? In Exercise 1.65, we assumed that the NAEP U.S. geography scores for eighth-grade students are approximately Normal with the reported mean and standard deviation, N(261, 31) . Let’s check that assumption. In addition to means
1.68 Compute the percentile scores. Refer to the previous exercise. When scores such as the NAEP assessment scores are reported for individual students, the actual values of the scores are not particularly meaningful. Usually, they are transformed into percentile scores. The percentile score is the
1.67 Standardize some NAEP eighth-grade geography scores. The NAEP geography assessment scores for eighth-grade students are approximately N(261, 31) . Find z -scores by standardizing the following scores: 200, 250, 280, 300, 320.
1.66 NAEP 12th-grade geography scores. Refer to the previous exercise. The scores for 12th-grade students on the geography assessment are approximately N(282, 26) . Answer the questions in the previous exercise for this assessment.
1.65 NAEP eighth-grade geography scores. In Check-in question 1.29 (page 52) we examined the distribution of NAEP scores for the eighth-grade mathematics skills assessment. For eighth-grade students, the average geography score is approximately Normal, with mean 261 and standard deviation 31.a.
1.64 The effect of changing the mean.a. Sketch a Normal curve that has mean 20 and standard deviation 2.b. On the same x axis, sketch a Normal curve that has mean 30 and standard deviation 2.c. How does the Normal curve change when the mean is varied but the standard deviation stays the same?
1.63 The effect of changing the standard deviation.a. Sketch a Normal curve that has mean 20 and standard deviation 2.b. On the same x axis, sketch a Normal curve that has mean 20 and standard deviation 4.c. How does the Normal curve change when the standard deviation is varied but the mean stays
1.62 Means and medians.a. Sketch a symmetric distribution that is not Normal. Mark the location of the mean and the median.b. Sketch a distribution that is skewed to the right. Mark the location of the mean and the median.
1.61 What’s wrong? Explain what is wrong with each of the following:a. Standardized values are always positive.b. Ninety-five percent of the values of a Normal distribution will be within one standard deviation of the mean.c. The standard Normal distribution has mean equal to 1 and standard
1.60 Changing units from centimeters to inches. Refer to Exercise 1.56. Change the measurements from centimeters to inches by multiplying each value by 0.39. Answer the questions from that exercise and explain the effect of the transformation on these data.
1.59 A different type of mean. The trimmed mean is a measure of center that is more resistant than the mean but uses more of the available information than the median. To compute the 10% trimmed mean, discard the highest 10% and the lowest 10% of the observations and compute the mean of the
1.58 Changing units from inches to centimeters. Changing the unit of length from inches to centimeters multiplies each length by 2.54 because there are 2.54 centimeters in an inch. This change of units multiplies our usual measures of spread by 2.54. This is true of IQR and the standard
1.57 Weight gain. A study of diet and weight gain deliberately overfed 15 volunteers for eight weeks.The mean increase in fat was x¯ = 2.31 kilograms, and the standard deviation was s = 1.30 kilograms. What are x¯ and s , in pounds? (A kilogram is 2.2 pounds.)
1.56 Longleaf pine trees. The Wade Tract in Thomas County, Georgia, is an old-growth forest of longleaf pine trees (Pinus palustris) that has survived in a relatively undisturbed state since before the settlement of the area by Europeans. A study collected data on 584 of these trees. One of the
1.55 Imputation. Various problems with data collection can cause some observations to be missing.Suppose a data set has 20 cases. Here are the values of the variable x for 10 of these cases:18 9 12 15 20 23 9 12 16 21 The values for the other 10 cases are missing. One way to deal with missing data
1.54 Mean and median for six observations. In the Mean and Median applet, place six observations on the line by clicking below it, five close together near the center of the line, and one somewhat to the right of these five.a. Pull the single rightmost observation out to the right. (Place the
1.53 Mean and median for two observations. The Mean and Median applet allows you to place observations on a line and see their mean and median visually. Place two observations on the line by clicking below it. Why does only one arrow appear?
1.52 Metabolic rates. Calculate the mean and standard deviation of the metabolic rates in Example 1.28(page 36), showing each step in detail. First find the mean x by summing the seven observations and dividing by 7. Then find each of the deviations xi − x¯ and their squares.Check that the
1.51 How does the median change? The firm in Exercise 1.49 gives no raises to the clerks and junior accountants, while the owner’s take increases to $950,000. How does this change affect the mean? How does it affect the median?
1.50 Be careful about how you treat the zeros. In computing the median income of any group, some federal agencies omit all members of the group who had no income. Give an example to show that the reported median income of a group can go down even though the group becomes economically better off. Is
1.49 Mean versus median. A small accounting firm pays each of its six clerks $40,000, four junior accountants $46,000 each, and the firm’s owner $700,000. What is the mean salary paid at this firm?How many of the employees earn less than the mean? What is the median salary?
1.48 Create a data set. Create a data set with five observations for which the median would change by a large amount if the largest observation were deleted.
1.47 Median versus mean for net worth. A report on the assets of American households says that the median net worth of U.S. families is $97,300. The mean net worth of these families is $692,100. What explains the difference between these two measures of center?25 26
1.46 Calories in beer. Refer to the previous two exercises. The data set also lists calories per 12 ounces of beverage.a. Analyze the data and summarize the distribution of calories for these 160 brands of beer.b. Are there any outliers? If yes, list them by name. How do these outliers compare with
1.45 Outliers for alcohol content of beer. Refer to the previous exercise.a. Calculate the mean with and without the outliers. Do the same for the median. Explain how these values change when the outliers are excluded.b. Calculate the standard deviation with and without the outliers. Do the same
1.44 The alcohol content of beer. Brewing beer involves a variety of steps that can affect the alcohol content. A website gives the percent alcohol for 160 domestic brands of beer. Use graphical and numerical summaries of your choice to describe the data. Give reasons for your choice.
1.43 Potatoes. A quality product is one that is consistent and has very little variability in its characteristics. Controlling variability can be more difficult with agricultural products than with products that are manufactured. The following table gives the weights, in ounces, of the 25 potatoes
1.42 Are there outliers in the smolt data? Refer to the previous two exercises.a. Find the IQR for the smolt data.b. Use the 1.5 × IQR rule to identify any outliers.c. Make a boxplot for the smolt data and describe the distribution using only the information in the boxplot.d. Make a modified
1.41 Measures of spread for smolts. Refer to the previous exercise.a. Find the standard deviation of the reflectance for these smolts.b. Find the quartiles of the reflectance for these smolts.c. Do you prefer the standard deviation or the quartiles as a measure of spread for these data? Give
1.40 Smolts. Smolts are young salmon at a stage when their skin becomes covered with silvery scales, and they start to migrate from freshwater to the sea. The reflectance of a light shined on a smolt’s skin is a measure of the smolt’s readiness for the migration. Here are the reflectances, in
1.39 Are there outliers in the double stout data? Refer to the previous two exercises.a. Find the IQR for these data.b. Use the 1.5 × IQR rule to identify and name any outliers.c. Make a boxplot for these data and describe the distribution using only the information in the boxplot.d. Make a
1.38 Measures of spread for the double stout data. Refer to the previous exercise.a. Compute the standard deviation for these data.22 23b. Compute the quartiles for these data.c. Which measure do you prefer for describing the spread of this distribution? Explain your answer.(You may include a
1.37 Gosset’s data on double stout sales. William Sealy Gosset worked at the Guinness Brewery in Dublin and made substantial contributions to the practice of statistics. In his work at the brewery, he collected and analyzed a great deal of data. Archives with Gosset’s handwritten tables,
1.36 Potassium sources. The data for potassium absorption in the previous exercise were expressed in milligrams (mg). Convert the data to grams (g) and answer the questions given in the previous exercise.There are 1000 mg in 1 g, so 3000 mg is the same as 3 g. In what ways are your answers here
1.35 Compare the potatoes with the supplement. Refer to Exercises 1.15 and 1.16 (page 22).a. Use a back-to-back stemplot to display the data for the two sources of potassium. Compare the two distributions and write a short summary of your findings.b. Use side-by-side boxplots to display the data
1.34 Potassium from a supplement. Refer to Exercise 1.16 (page 22), where you examined the potassium absorption of a group of 29 adults who ate a controlled diet that included 40 mEq of potassium from a supplement for five days. In Exercise 1.16, you used a stemplot to examine the distribution of
1.33 Potassium from potatoes. Refer to Exercise 1.15 (page 22), where you examined the potassium absorption of a group of 27 adults who ate a controlled diet that included 40 mEq of potassium from potatoes for five days. In Exercise 1.15, you used a stemplot to examine the distribution of the
1.32 Potassium from a supplement. Refer to Exercise 1.16 (page 22), where you examined the potassium absorption of a group of 29 adults who ate a controlled diet that included 40 mEq of potassium from a supplement for five days.a. Compute the standard deviation for these data.b. Compute the
1.31 Potassium from potatoes. Refer to Exercise 1.15 (page 22), where you examined the potassium absorption of a group of 27 adults who ate a controlled diet that included 40 mEq of potassium from potatoes for five days.a. Compute the standard deviation for these data.b. Compute the quartiles for
1.30 Potassium from a supplement. Refer to Exercise 1.16 (page 22), where you examined the potassium absorption of a group of 29 adults who ate a controlled diet that included 40 mEq of potassium from a supplement for five days.a. Compute the mean for these data.b. Compute the median for these
1.29 Potassium from potatoes. Refer to Exercise 1.15 (page 22), where you examined the potassium absorption of a group of 27 adults who ate a controlled diet that included 40 mEq of potassium from potatoes for five days.a. Compute the mean for these data.b. Compute the median for these data.c.
1.28 What’s wrong? Explain what is wrong with each of the following:a. The mean is a resistant measure of the center of a distribution.b. If you multiply a variable by 10, you do not change the value of the mean.c. The five number summary includes the mean and the standard deviation.
1.27 The Boston Marathon. Women were allowed to enter the Boston Marathon in 1972. TABLE 1.3 gives the times (in minutes, rounded to the nearest minute) for the winning women from 1972 to 2019.TABLE 1.3 Boston Marathon winning times for women Year Time Year Time Year Time Year Time 1972 190 1984
1.26 Describe the self-concept scores. Based on a suitable graph, briefly describe the distribution of self-concept scores for the students in TABLE 1.2. Be sure to identify any suspected outliers.TABLE 1.2 Educational data for 78 seventh-grade students Obs GPA IQ Gender Self Obs GPA IQ Gender Self
1.25 Sketch a skewed distribution. Sketch a histogram for a distribution that is skewed to the left.Suppose that you and your friends emptied your pockets of coins and recorded the year marked on each coin. The distribution of dates would be skewed to the left. Explain why.17
1.24 Describe the IQ scores. Make a graph of the distribution of IQ scores for the seventh-grade students in Table 1.2. Describe the shape, center, and spread of the distribution, as well as any outliers.IQ scores are usually said to be centered at 100. Is the midpoint for these students close to
1.23 Grades and self-concept. TABLE 1.2 presents data on 78 seventh-grade students in a rural midwestern school. The researcher was interested in the relationship between the students’ “selfconcept”and their academic performance. The data we give here include each student’s grade point
1.22 Vehicle colors. Vehicle colors differ among regions of the world. Here are data on the most popular colors for vehicles in North America:Color Percent White 24 Black 19 Silver 16 Gray 15 Red 10 Blue 7 15 16 Brown 5 Other 4a. Describe these data with a bar graph.b. Describe these data with a
1.21 Garbage. The formal name for garbage is “municipal solid waste.” In the United States, approximately 250 million tons of garbage are generated in a year. Here is a breakdown of the materials 14 that made up American municipal solid waste in a recent year:Material Weight(million
1.20 Cheap colors. Refer to the previous exercise. The same study also asked people about what colors they associate with the words cheap/inexpensive. Here are the results: brown, 13%; green, 6%; gray, 8%;orange, 26%; other, 3%; purple, 4%; red, 9%; white; 9%; yellow, 22%. Answer the questions from
1.19 Least favorite colors. What is your least favorite color? One survey produced the following summary of responses to that question: brown, 23%; green, 4%; gray, 12%; orange, 30%; other, 1%;purple, 13%; white; 4%; yellow, 13%.a. Make a bar graph of the percents with the colors ordered
1.18 Energy consumption in a different year. Refer to the previous exercise. Here are the data for the previous year:Month Energy(quadrillion Btu)Month Energy(quadrillion Btu)January 8.99 July 8.27 February 8.02 August 8.17 March 8.38 September 7.64 April 7.52 October 7.72 May 7.62 November 8.14
1.17 Energy consumption. The U.S. Energy Information Administration reports data summaries of various energy statistics. Let’s look at the total amount of energy consumed, in quadrillions of British thermal units (Btu), for each month in a recent year. Here are the data:Month Energy(quadrillion
1.16 Potassium from a supplement. Refer to the previous exercise. Data were also recorded for 29 subjects who received a potassium salt supplement with 40 mEq of potassium. Answer the questions in the previous exercise for the supplemented subjects.
1.15 Potassium from potatoes. The 2015 Dietary Guidelines for Americans notes that the average potassium (K) intake for U.S. adults is about half of the recommended amount. A major source of potassium is potatoes. Nutrients in the diet can have different absorption depending on the source. One
1.14 Facebook users by country. The following table gives the numbers of active Facebook users by country for the top 11 countries based on the number of users in July 2019.Country Facebook users(in millions)India 270 United States 190 Indonesia 130 Brazil 120 Mexico 82 Philippines 68 Vietnam 58
1.13 Pie chart for frequent users of social media. Refer to the previous exercise.a. Use a pie chart to describe the percents of frequent users of these sites and write a short summary of the data based on your chart.b. Compare this pie chart with the bar graph that you produced for the previous
1.12 Frequent users of social media. A recent survey by the Pew Research Center asked social media users about how often they visited various sites. Pew defined a frequent user to be someone who visited a site several times a day. Here are the percents of users who are frequent users for several
1.11 Which graphical display should you use? For each of the following scenarios, decide which graphical display (pie chart, bar graph, stemplot, or histogram) you would use to describe the distribution of the variable. Give a reason for your choice and, if there is an alternative choice that would
1.10 What’s wrong? Explain what is wrong with each of the following:a. A stemplot can be used to display the distribution of a categorical variable.b. A symmetric distribution can be skewed to the right.c. Always discard outliers before doing an analysis of a set of data.
1.9 Alcohol-impaired driving fatalities. A report on drunk-driving fatalities in the United States gives the number of alcohol-impaired driving fatalities for each year from 1982 to 2017. Discuss at least three different ways that these numbers could be converted to rates. Give the advantages and
1.8 Attending college in your state or in another state. The U.S. Census Bureau collects a large amount of information concerning higher education.For example, the bureau provides a table that includes the following variables:state, number of students from the state who attend college, and number
1.7 How would you rate colleges? Popular magazines rank colleges and universities on their “academic quality” in serving undergraduate students.Describe five variables that you would like to see measured for each college if you were choosing where to study. Give reasons for each of your choices.
1.6 How would you rank cities? Various organizations rank cities and produce lists of the 10 or the 100 best cities, based on various measures. Create a list of criteria that you would use to rank cities. Include at least eight variables and give reasons for your choices. Say whether each variable
1.5 Employee application data. The human resources (HR) department keeps records on all employees in a company. Here is the information HR keeps in one of its data files: employee identification number, last name, first name, middle initial, department, number of years with the company, salary,
1.4 An experiment on haptic feedback. A group of technology students is interested in whether haptic feedback (forces and vibrations applied through a joystick) is helpful in navigating a simulated game environment they created.To investigate this, they randomly assign 20 students to each of three
1.3 A survey of graduates. A college surveys its graduates who have earned a bachelor’s degree each year. For a recent year, the data include the responses of 1255 employed students who responded to the request for information about their job. The variables collected include an ID numbered 1 to
1.2 Coffee ratings. A website ranks 50 different varieties of coffee. The data include the following variables: name of the coffee, price for a 12-ounce serving, overall rating (0 to 100), roast (light, medium, or dark), flavor, aroma, and body ratings (0 to 10).a. What are the cases?b. Identify
1.1 Student organizations. A university website gives information about 30 student organizations. You have created a data set that summarizes this information. The variables in your data set are the name of the organization, whether the majority of the members are undergraduates or graduate
Consider the general m n, two-person, zero-sum game.Let pij denote the payoff to player 1 if he plays his strategy i (i 1, . . . , m) and player 2 plays her strategy j (j 1, . . . , n).Strategy 1 (say) for player 1 is said to be weakly dominated by strategy 2 (say) if p1j p2j for j 1, . . . , n
Consider the linear programming model for player 1 given near the end of Sec. 14.5 for variation 3 of the political campaign problem (see Table 14.6). Verify the optimal mixed strategies for both players given in Sec. 14.5 by applying an automatic routine for the simplex method to this model to
Consider variation 3 of the political campaign problem(see Table 14.6). Refer to the resulting linear programming model for player 1 given near the end of Sec. 14.5. Ignoring the objective function variable x3, plot the feasible region for x1 and x2 graphically (as described in Sec. 3.1). (Hint:
Consider the linear programming models for players 1 and 2 given near the end of Sec. 14.5 for variation 3 of the political campaign problem (see Table 14.6). Follow the instructions of Prob. 14.5-6 for these two models.
Section 14.5 presents a general linear programming formulation for finding an optimal mixed strategy for player 1 and for player 2. Using Table 6.14, show that the linear programming problem given for player 2 is the dual of the problem given for player 1.(Hint: Remember that a dual variable with a
Follow the instructions of Prob. 14.5-3 for the game having the following payoff table.
Refer to the last paragraph of Sec. 14.5. Suppose that 3 were added to all the entries of Table 14.6 to ensure that the corresponding linear programming models for both players have feasible solutions with x3 0 and y4 0. Write out these two models. Based on the information given in Sec. 14.5,
Consider the odds and evens game introduced in Sec. 14.1 and whose payoff table is shown in Table 14.1.(a) Use the approach described in Sec. 14.5 to formulate the problem of finding optimal mixed strategies according to the minimax criterion as two linear programming problems, one for player 1
The A. J. Swim Team soon will have an important swim meet with the G. N. Swim Team. Each team has a star swimmer(John and Mark, respectively) who can swim very well in the 100-yard butterfly, backstroke, and breaststroke events. However, the rules prevent them from being used in more than two of

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Introduction To Operations Research 9th Edition Frederick S. Hillier - Solutions

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