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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
16.14 Visit lengths to a statistics help room. Table 5.1 (page 283) gives the length (in minutes) for a sample of 50 visits to a statistics help room. See Example 5.5 (page 283) for more details about these data.a. Make a histogram of the 50 visit lengths. Describe the shape. Is it similar to the
16.13 Standard error versus the bootstrap standard error. We have two ways to estimate the standard deviation of a sample mean x¯ : use the formula s/n for the standard error or use the bootstrap standard error.a. Find the sample standard deviation s for the 60 IQ test scores in Exercise 16.10 and
16.12 Bootstrap distribution of average audio file length. The lengths (in seconds) of audio files found on an iPod (Table 7.3, page 399) are skewed. We previously transformed the data prior to using t procedures.
16.11 Bootstrap distribution of delivery time from COSI. A random sample of eight times (in minutes) it took a delivery robot to bring you lunch from COSI to your dormitory (Example 7.1, page 387) are 13.7 26.3 20.0 45.3 8.5 43.6 10.1 17.3 The distribution has no outliers, but we cannot comfortably
16.10 Bootstrap distribution of average IQ score. The distribution of the 60 IQ test scores in Table 1.1(page 15) is roughly Normal (see Figure 1.7), and the sample size is large enough that we expect a Normal sampling distribution.
16.9 Bootstrap distribution of the time to start a business. We examined the distribution of the time to start a business for 187 countries in Example 1.43 (page 60). The distribution is clearly skewed and has an outlier. We view these data as coming from a process that gives times to start a
16.8 Interpreting bootstrap output. FIGURE 16.9 gives output from R for the sample of six viewing times in Check-in question 16.1 (page 16-6). Summarize the results of the analysis using this output.FIGURE 16.9 R output for the Facebook viewing time bootstrap, Exercise 16.8.Description The output
16.7 Interpreting the output. FIGURE 16.8 gives output from R for the sample of ratios (as a percent)in Exercise 16.3. Summarize the results of the analysis using this output.FIGURE 16.8 R output for the percent change in double stout sales bootstrap, Exercise 16.7.Description The output lists the
16.6 Assessing another bootstrap distribution. Refer to the data in Check-in question 16.1 (page 16-6).FIGURE 16.7 gives a histogram and a Normal quantile plot of 3000 resample means (labeled t*). What do these plots tell you about the sampling distribution of x¯ when n=6 ?FIGURE 16.7 R graphical
16.5 Assessing the bootstrap distribution. Refer to the data in Exercise 16.3. FIGURE 16.6 gives a histogram and a Normal quantile plot of 3000 resample means (labeled t*). What do these plots tell you about the sampling distribution of x¯ when n=6 ?FIGURE 16.6 R graphical output for the percent
16.4 More on the bootstrap standard error. Refer to your work in the previous exercise.a. Do you expect your bootstrap standard error to be larger, smaller, or approximately equal to the standard deviation of the original sample of six regions? Explain your answer.b. Would your answer change if the
16.3 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 Exercise 1.37 (page 44), we examined Gosset’s data on the change in the double stout market before and after World War I
16.2 Describing the method to obtain resamples. Suppose an SRS of size n=10 is obtained from the undergraduates at a large public university to measure the average stress level during dead week.Carefully detail two ways to randomly select resamples in this setting.
16.1 What’s wrong? For each of the following, explain what is wrong and why.a. The standard deviation of the bootstrap distribution will be approximately the same as the standard deviation of the original sample.b. The bootstrap distribution is created by resampling without replacement from the
15.41 Multiple comparisons for cooking pots. Exercise 15.37 outlines how to use the Wilcoxon rank sum test several times for multiple comparisons with overall significance level 0.05 for all comparisons together. Apply this procedure to the data used in Exercises 15.38 and 15.40. Why is is not
15.40 Iron in food cooked in iron pots. The raw data appear to show that food cooked in iron pots has the highest iron content. They also suggest that the three types of food differ in iron content. Is there significant evidence that the three types of food differ in iron content when all are
15.39 Cooking meat and legumes in aluminum and clay pots. Is there a significant difference between the iron content of meat cooked in aluminum and clay? Is the difference between aluminum and clay significant for legumes? Use rank tests.
15.38 Cooking vegetables in different pots. Let’s first concentrate on the 12 observations for the vegetable dish. Does the vegetable dish vary in iron content when cooked in aluminum, clay, and iron pots?a. What do the data appear to show? Check the conditions for one-way ANOVA. Which
15.37 Multiple comparisons for plants and hummingbirds. As in ANOVA, we often want to carry out a multiple comparisons method following a Kruskal-Wallis test to tell us which groups differ significantly.The Bonferroni method (page 374) is a simple method: if we carry out k tests at fixed
15.36 Response times for telephone repair calls. A study examined the time required for the telephone company Verizon to respond to repair calls from its own customers and from customers of CLEC, another phone company that pays Verizon to use its local lines. Here are the data, which are rounded to
15.35 Time spent studying. In Exercise 1.119 (page 69), you compared the time spent studying by men and women. The students in a large first-year college class were asked how many minutes they studied on a typical weeknight. Here are the responses of random samples of 30 women and 30 men from the
15.34 Plants and hummingbirds. Different varieties of the tropical flower Heliconia are fertilized by different species of hummingbirds. Over time, the lengths of the flowers and the forms of the hummingbirds’ beaks have evolved to match each other. Here are data on the lengths in millimeters of
15.33 Do poets die young? In Exercise 12.60 (page 646) you analyzed the age at death for female writers. They were classified as novelists, poets, and nonfiction writers.a. Use the Kruskal-Wallis test to compare the three groups of female writers.b. Compare these results with what you find using
15.32 Jumping and strong bones. In Exercise 12.61 (page 646), you studied the effects of jumping on the bones of rats. Ten rats were assigned to each of three treatments: a 60-centimeter “high jump,” a 30-centimeter “low jump,” and a control group with no jumping. Here are the bone
15.31 Read the output. FIGURE 15.11 gives JMP output for the analysis of the data described in Exercise 15.27. Describe the results given in the output and write a short summary of your conclusions from the analysis.FIGURE 15.11 JMP output for the Kruskal-Wallis test applied to the Facebook data,
15.30 Do we experience emotions differently? In Exercise 12.55 (page 644) you analyzed data related to the way people from different cultures experience emotions. The study subjects were 416 college students from five different cultures. They were asked to record, on a 1 (never) to 7 (always)
15.29 What are the hypotheses? Refer to Exercise 15.27. What are the null hypothesis and the alternative hypothesis? Explain why a nonparametric procedure would be appropriate in this setting.
15.28 Vitamins in bread. Does bread lose its vitamins when stored? Here are data on the vitamin C content (milligrams per 100 grams of flour) in bread baked from the same recipe and stored for one, three, five, or seven days. The 10 observations are from 10 different loaves of bread.Condition
15.27 Number of Facebook friends. An experiment was run to examine the relationship between the number of Facebook friends and the user’s perceived social attractiveness. A total of 134 undergraduate participants were randomly assigned to observe one of five Facebook profiles. Everything about
15.26 Do isoflavones increase bone mineral density? In Exercise 12.59 (page 645) you investigated the effects of isoflavones from kudzu on bone mineral density (BMD). The experiment randomized rats to three diets: control, low isoflavones, and high isoflavones. Here are the data:Treatment
15.25 Vitamin C in wheat-soy blend. The U.S. Agency for International Development provides large quantities of wheat-soy blend (WSB) for development programs and emergency relief in countries throughout the world. One study collected data on the vitamin C content of five bags of WSB at the factory
15.24 Radon detectors. How accurate are radon detectors of a type sold to homeowners? To answer this question, university researchers placed 12 detectors in a chamber that exposed them to 105 picocuries per liter (pCi/l) of radon. The detector readings are as follows:91.9 97.8 111.4 122.3 105.4
15.23 The full moon and behavior. Can the full moon influence behavior? A study observed 15 nursing-home patients with dementia. The number of incidents of aggressive behavior was recorded each day for 12 weeks. Here we call a day a “moon day” if it is the day of a full moon or the day before
15.22 Read the output. The data in Exercise 15.13 are a subset of a larger set of data. FIGURE 15.9 gives Minitab output for the analysis of this larger set of data.a. How many pairs of observations are in the larger data set?b. What is the value of the Wilcoxon signed rank statistic W+ ?c. Report
15.21 Find and interpret the P-value. Refer to the odd-numbered exercises from Exercise 15.13 through Exercise 15.19. Find the P-value for the Wilcoxon signed rank statistic using the Normal approximation with the continuity correction.
15.20 A summer language institute for teachers. A matched pairs study of the effect of a summer language institute on the ability of teachers to comprehend spoken French had these improvements in scores between the pretest and the posttest for 20 teachers:2 0 6 6 3 3 2 3 −6 6 6 6 3 0 1 1 0 2 3
15.19 Find the mean and the standard deviation. Refer to the odd-numbered exercises from Exercise 15.13 through Exercise 15.17. Use the sample size to find the mean and the standard deviation of the sampling distribution of the Wilcoxon signed rank statistic W+ under the null hypothesis.
15.18 Comparison of two energy drinks with an additional subject. Refer to Exercise 15.14. Let’s suppose that there is an additional subject who expresses a strong preference for energy drink B. Here is the new data set:Drink Subject 1 2 3 4 5 6 7 A 43 83 66 87 78 67 90 B 45 78 64 79 71 62 60
15.17 State the hypotheses. Refer to Exercise 15.13. State the null hypothesis and the alternative hypothesis for this setting.10
15.16 Number of friends on Facebook. A study examined all active Facebook users (more than 10% of the global population) and determined that the average user has 190 friends. This distribution takes only integer values, so it is certainly not Normal. It is also highly skewed to the right, with a
15.15 Find the Wilcoxon signed rank statistic. Using the work that you performed in Exercise 15.13, find the value of the Wilcoxon signed rank statistic W+ .
15.14 Comparison of two energy drinks. Consider the following study comparing two popular energy drinks. For each subject, a coin was flipped to determine which drink to rate first. Each drink was rated on a 0 to 100 scale, with 100 being the highest rating.Drink Subject 1 2 3 4 5 6 A 43 83 66 87
15.13 Fuel efficiency. Computers in some vehicles calculate various quantities related to performance.One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the driver recorded the vehicle’s mpg each time the gas
15.12 Learning math through subliminal messages. A “subliminal” message is below our threshold of awareness but may, nonetheless, influence us. Can subliminal messages help students learn math? A group of students who had failed the mathematics part of the City University of New York Skills
15.11 More data for women and men talking. The data in the previous exercise were a sample of the data collected in a larger study of 42 men and 37 women. Use the larger data set to answer the questions in the previous exercise. Discuss the advisability of using the Wilcoxon rank sum test versus
15.10 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 10 men and 10 women in the United States. The variable recorded is the number of words per day. Here are the data:Men Women
15.9 Find and interpret the P-value. Refer to the odd-numbered exercises from Exercises 15.1 through Exercise 15.7. Find the P-value using the Normal approximation with the continuity correction and interpret the result of the significance test.
15.8 Is civic engagement related to education? A Pew Internet Poll of adults aged 18 and older examined factors related to civic engagement. Participants were asked whether or not they had participated in a civic group or activity in the preceding 12 months. One analysis looked at the relationship
15.7 Find the mean and standard deviation of the distribution of the statistic. The statistic W that you calculated in Exercise 15.3 is a random variable with a sampling distribution. What are the mean and the standard deviation of this sampling distribution under the null hypothesis?
15.6 Do the calculations by hand. Use the data in Exercise 15.2 for children telling Story 2 to carry out by hand the steps in the Wilcoxon rank sum test.a. Arrange the 10 observations in order and assign ranks. There are no ties.b. Find the rank sum W for the five high-progress readers. What are
15.5 State the hypotheses for study times. Refer to Exercise 15.1. State appropriate null and alternative hypotheses for this setting.
15.4 Repeat the analysis for Story 2. Repeat the analysis of Exercise 15.2 for the scores when children retell a story they have heard and seen illustrated with pictures (Story 2).
15.3 Find the rank sum statistic for study times. Refer to Exercise 15.1. Compute the value of the Wilcoxon statistic. Take the first sample to be the students in the early section.
15.2 Storytelling and the use of language. A study of early childhood education asked kindergarten students to retell two fairy tales that had been read to them earlier in the week. The 10 children in the study included five high-progress readers and five low-progress readers. Each child told two
15.1 Time spent studying. A first-year college class had two large sections. One met at 8:30 a.m.(early), and the other met at 4:00 p.m. (late). A sample of 6 students from each section were interviewed about their class experience, including how much time they spent studying during a typical week
14.38 Tipping behavior in Canada. The Consumer Report on Eating Share Trends (CREST)contains data from all provinces of Canada detailing away-from-home food purchases by roughly 4000 households per quarter. Researchers recently restricted their attention to restaurants at which tips would normally
14.37 Finding the best model. In Example 14.14 (page 14-18), we looked at a multiple logistic regression for movie profitability based on three explanatory variables. Complete the analysis by looking at the three models that include two explanatory variables and the three models that include only
14.36 Is there an effect of sex? In this exercise, we investigate the effect of sex (coded as 0 for males and 1 for females) on the odds of getting a high GPA.a. Use sex to predict HIGPA using a logistic regression. Summarize the results.b. Perform a logistic regression using sex and the two SAT
14.35 Use high school grades and SAT scores to predict high grade point averages. Run a logistic regression to predict HIGPA using the three high school grade summaries and the two SAT scores as explanatory variables. We want to produce an analysis that is similar to that done for the case study in
14.34 Use SAT scores to predict high grade point averages. Use a logistic regression to predict HIGPA using the SATM and SATCR scores as explanatory variables.6a. Summarize the results of the hypothesis test that the coefficients for both explanatory variables are zero.b. Give the coefficient for
14.33 Use high school grades to predict high grade point averages. Use a logistic regression to predict HIGPA using the three high school grade summaries as explanatory variables.a. Summarize the results of the hypothesis test that the coefficients for all three explanatory variables are zero.b.
14.32 Predicting physical activity. Participation in physical activities typically declines between high school and young adulthood. This suggests that postsecondary institutions may be an ideal setting to address physical activity. A study looked at the association between physical activity and
14.31 Another example of Simpson’s paradox. Refer to Exercises 2.105 and 2.106 (page 139).Using Exercise 14.29 as a guide, analyze these data using logistic regression.
14.30 Reducing the number of workers. To be competitive in global markets, many corporations are undertaking major reorganizations. Often, these involve “downsizing” or a “reduction in force”(RIF), where substantial numbers of employees are terminated. Federal and various state laws require
14.29 An example of Simpson’s paradox. Here is an example of Simpson’s paradox: the reversal of the direction of a comparison or an association when data from several groups are combined to form a single group. The data concern the comparison of success rates for 2-pointers and 3-pointers for
14.28 z and the X2 statistic. Use the three outputs in Figure 14.8 (page 14-16) to explore the relationship between the z statistic and the X2 statistic that we have discussed in this chapter (page 14-12).a. Use the information in each output to calculate the z statistic. Verify that they are
14.27 Interpret the fitted model. If we apply the exponential function to the fitted model in Example 14.9 (page 14-9), we get odds=e−256 + 1.125x=e−2.56×e1.125x 5Show that for any value of the quantitative explanatory variable x, the odds ratio for increasing x by 1, oddsx+1oddsx is
14.26 More exergaming in Canada. Refer to the previous exercise. Another explanatory variable reported in this study was the amount of television watched per day. Of the 54 students who reported that they watched no TV, 11.1% were exergamers; for the 776 students who watched some TV but less than
14.25 Exergaming in Canada. Exergames are active video games such as rhythmic dancing games, virtual bicycles, balance board simulators, and virtual sports simulators that require a screen and a console. A study of exergaming by students in grades 10 and 11 in Montreal, Canada, examined many
14.24 Compare the multiple logistic regression analysis with the two-way table. The data analyzed in Figure 14.11 were studied in Exercises 9.1, 9.3, 9.5, and 9.7 (pages 503 to 504) using a 2×6 table of counts.Compare these two approaches to the analysis of these data. Describe some strengths and
14.23 Another logistic model for cell phones and age. Refer to Exercise 14.5 (page 14-10).Suppose that you use the actual value of age in years as the explanatory variable in a logistic regression model.a. Describe the statistical model for logistic regression in this setting.b. Interpret β1 in
14.22 Give a 95% confidence interval for the odds ratio. Refer to Exercise 14.15 and the outputs for teeth and military service in Figure 14.4 (page 14-10). Using the estimate b1 and its standard error, find the 95% confidence interval for the odds ratio and verify that this agrees with the
14.12 (page 14-15). Suppose that you wanted to report a 99% confidence interval forβ1 . Show how you would use the information provided in the outputs shown in Figure 14.8 (page 14-16) to compute this interval.
14.21 Give a 99% confidence interval forβ1 . Refer to Example
14.20 Odds ratios for the multiple logistic regression model. Refer to the two previous exercises.a. Give the odds ratio for each explanatory variable.b. Give the 95% confidence interval for each odds ratio.c. Give a brief description of the meaning of each odds ratio in this analysis.
14.19 Inference for the multiple logistic regression model. Refer to the previous exercise.a. Describe and interpret the significance test that tests the null hypothesis that all regression coefficients are zero.b. Using the information provided in the output in Figure 14.11, calculate and
14.18 Teeth and military service with six age categories. In Exercises 14.4, 14.13, and 14.15, we used logistic regression to study the relationship between being rejected for military service because a recruit did not have enough teeth and age, categorized into two groups, under 20 and 40 or over.
14.17 What’s wrong? For each of the following, explain what is wrong and why.a. For a multiple logistic regression with four explanatory variables, the null hypothesis that the regression coefficients of all the explanatory variables are zero is tested with an F test.b. For a logistic regression,
14.16 Odds ratio for high blood pressure and cardiovascular disease. The results describing the relationship between blood pressure and cardiovascular disease are given in terms of the change in log odds in Exercise 14.14.a. Transform b1 to the odds ratio and the 95% confidence interval forβ1 to a
14.15 Odds ratio for teeth and military service. Refer to Exercise 14.13.a. Give the odds ratio for this analysis.b. Give the 95% confidence interval for the odds ratio.c. Give a brief description of the meaning of the odds ratio in this analysis.
14.14 High blood pressure and cardiovascular disease. Refer to the study of cardiovascular disease and blood pressure in Exercise 14.6 (page 14-10). Computer output for a logistic regression analysis of these data gives the b1=0.7505 with standard error SEb1=0.2578 .a. Give a 95% confidence
14.13 Inference for teeth and military service. Refer to Exercise 14.4 (page 14-10), where you described a logistic regression model for a study of whether U.S. recruits had enough teeth for adequate nutrition during the Spanish–American War of 1898.a. Give the estimates of the regression
14.12 Internet use in Canada. A recent study used data from the Canadian Internet Use Survey(CIUS) to explore the relationship between certain A variables and Internet use by individuals in Canada. The response variable refers to the use of the Internet from any location within the last 12 months.
14.11 Salt in the diet and CVD. Refer to the previous exercise. Use x=1 for the high-salt diet and x=0 for the low-salt diet.a. Find the estimates b0 and b1 .b. Give the fitted logistic regression model.c. What is the odds ratio for a high-salt versus low-salt diet?d. When the probability of an
14.10 Salt intake and cardiovascular disease. In Example 9.13 (page 501), the relative risk of developing cardiovascular disease (CVD) for people with low- and high-salt diets was estimated. Let’s reanalyze these data using the methods in this chapter. Here are the data:Developed CVD Salt in diet
14.9 Convert the odds to probabilities. Refer to the previous exercise. For each opening-weekend revenue, compute the estimated probability that the movie is profitable.
14.8 Will a movie be profitable? In Example 14.9 (page 14-9), we described a model to predict whether a movie is profitable based on log opening-weekend revenue(x=LOpening) . What are the predicted odds of a movie being profitable if the opening-weekend revenue isa. $20 million dollars (x=3.10)?b.
14.7 What’s wrong? For each of the following, explain what is wrong and why.a. The intercept β0 is equal to the odds of an event when x=0 .b. The log odds of an event are 1 minus the probability of the event.c. If b1=3 in a logistic regression analysis with one explanatory variable, we estimate
14.6 High blood pressure and cardiovascular disease. There is much evidence that high blood pressure is associated with increased risk of death from cardiovascular disease. A major study of this association examined 3351 men with high blood pressure and 2654 men with low blood pressure. During the
14.5 A logistic model for cell phones. Refer to Exercise 14.2. Suppose that you want to investigate differences in cell phone use among customers of different ages. You create an indicator explanatory variable x that has the value 1 if the customer is 25 years of age or less and 0 if the customer
14.4 A logistic regression for teeth and military service. Exercise 8.40 (page 481) describes data on the numbers of U.S. recruits who were rejected for service in a war against Spain because they did not have enough teeth. The exercise compared the rejection rate for recruits who were under the
14.3 Find some odds. For each of the following probabilities, find the odds: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9. Make a plot of the odds versus the probabilities and describe the relationship.
14.2 How did you use your cell phone? One question in a Pew Internet Poll on cell phone use asked whether, during the past 30 days, the person had used their phone while in a store to call a friend or family member for advice about a purchase they were considering. The poll surveyed 1003 adults
14.1 What purchases will be made? A poll of 1200 adults aged 18 or older asked about purchases they intended to make for the upcoming holiday season. A total of 543 adults listed gift card as a planned purchase.a. What proportion of adults plan to purchase a gift card as a present?b. What are the
13.71 Does color saturation and purchase goal affect willingness to pay? Refer to the previous two exercises. Assuming that the model conditions are approximately satisfied, use a two-way ANOVA to assess the relationship between willingness to pay and the two factors. Write a short paragraph
13.70 Checking conditions for inference. Refer to the previous exercise. Use the data to check whether the model conditions are approximately satisfied. For each condition, provide a numerical or graphical summary to support your conclusion.
13.69 Does color saturation increase perceived size? Some researchers hypothesize that an object with high color saturation is perceived of as larger than the same object with less color saturation. In one study, the researchers randomly assigned 156 participants a goal: to buy a carryon suitcase
13.68 Power for a two-way ANOVA. In Section 12.2 (pages 635–637), we discussed power calculations for the one-way ANOVA. Would power calculations for a two-way ANOVA require inputting any additional study-specific factors? Explain your answer.PUTTING IT ALL TOGETHER
13.67 Are insects more attracted to male plants? Some scientists wanted to determine if there are sex-related differences in the level of herbivory for the jack-in-the-pulpit, a spring-blooming perennial plant common in deciduous forests. A study was conducted in southern Maryland in forests
13.66 More on the analysis using multiple one-way ANOVAs. Perform the tasks described in Exercise 13.62 for the two response variables in the PLANTS2 data set.
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