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All Textbook Solutions for Introduction to Probability and Statistics

Quantitative or Qualitative? Identify each variable as quantitative or qualitative: a. Ethnic origin of a candidate for public office b. Score (0100) on a placement examination c. Fast-food restaurant preferred by a student(McDonald’s, Burger King, or Carl’s Jr.) d. Mercury concentration in a sample of tunaSymmetric or Skewed? Do you expect the distributions of the following variables to be symmetric or skewed? Explain. a. Size in dollars of nonsecured loans b. Size in dollars of secured loans c. Price of an 8-ounce can of peas d. Height in inches of freshman women at your university e. Number of broken taco shells in a package of 100 shells f. Number of ticks found on each of 50 trapped cottontail rabbitsContinuous or Discrete? Identify each variable as continuous or discrete: a. Number of homicides in Detroit during a 1 -month period b. Length of time between arrivals at an outpatient clinic c. Number of typing errors on a page of manuscript d. Number of defective lightbulbs in a package containing four bulbs e. Time required to finish an examination1.41SEContinuous or Discrete, again Identify each variable as continuous or discrete: a. Number of people in line at a supermarket checkout counter b. Depth of a snowfall c. Length of time for a driver to respond when faced with an impending collision d. Number of aircraft arriving at the Atlanta airport in a given hourAqua Running Aqua running has been suggested as a method of cardiovascular conditioning for injured athletes and others who want a low-impact aerobics program. A study reported in the Journal of Sports Medicine investigated the relationship between exercise cadence and heart rate by measuring the heart rates of 20 healthy volunteers at a cadence of 96 steps per minute.11 The data are listed here: 87 109 79 80 96 95 90 92 96 98 101 91 78 112 94 98 94 107 81 96 Construct a stem and leaf plot to describe the data. Discuss the characteristics of the data distribution.1.44SEAges of Pennies We collected 50 pennies and recorded their ages. by calculating AGE=CURRENTYEARYEARONPENNY. 5 1 9 1 2 20 0 25 0 17 1 4 4 3 0 25 3 3 8 28 52119 9 0 5 0 2 1 0 0 1 19 0 2 0 20 16 22 10 1936230 117 6 0 50 a. Before drawing any graphs. try to visualize what the distribution of penny ages will look like. Will it be mound-shaped, symmetric, skewed right, or skewed left? b. Draw a relative frequency histogram to describe the distribution of penny ages. How would you describe the shape of the distribution?1.46SEPresidential Vetoes Here is a list of the 44 presidents of the United States along with the number of regular vetoes used by each :5 Use an appropriate graph to describe the number of vetoes cast by the 44 presidents. Write a summary paragraph describing this set of data.1.48SE1.49SE1.50SE1.51SE1.52SE1.53SEStudent Heights The self-reported heights of 105 students in a biostatistics class are described in the relative frequency histogram below. Describe the shape of the distribution. Do you see any unusual feature in this histogram? Can you think of an explanation for the two peaks in the histogram? Is there some other factor that is causing the heights to mound up in two separate peaks? What is it?1.55SEPulse Rates A group of 50 biomedical students recorded their pulse rates by counting the numberof beats for 30 seconds and multiplying by 2. a. Why are all of the measurements even numbers? b. Draw a stem and leaf plot to describe the data, splittingeach stem into two lines. c. Construct a relative frequency histogram for the data. d. Write a short paragraph describing the distribution ofthe student pulse rates.1.57SE1.58SE1.59SE1.60SE1.61SEOld Faithful The data below are the wailing times between eruptions of the Old Faithful geyser in Yellowstone National Park.21 Use one of the graphical methods from this chapter to describe the distribution of waiting times. If there are any unusual features in your graph, see if you can think of any practical explanation for them. 56 89 51 79 58 82 52 88 52 78 69 75 77 53 80 54 79 74 65 78 55 87 53 85 61 93 54 76 80 81 59 86 78 71 77 89 45 93 72 71 76 94 75 50 83 82 72 77 75 65 79 72 78 77 79 72 82 74 80 49 75 78 64 80 49 49 88 51 78 85 65 75 77 69 92 91 53 86 49 79 68 87 61 81 55 93 53 84 70 73 93 50 87 77 74 89 87 76 59 801.63SE1.64SE1.65SEThe number of Starhucks coffee shops in cities within 20 miles of the University of California. Riverside is shown in the following table.16 Draw a dotplot to describe the data. Describe the shape of the distribution. Is there another variable that you could measure that might help to explain why some cities have more Starbucks than others? Explain.What’s Normal? The 98.6 degree standard for human body temperature was derived by a German doctor in 1868. In an attempt to verify his claim. Mackowiak. Wasserman. and Levine22 took temperatures from 148 healthy people over a 3-day period. A data set closely matching the one in Mackowiak’s article was derived by Allen Shoemaker, and appears in the Journal of Statistics Education.23 The body temperatures for these 130 individuals are shown in the relative frequency histogram that follows. Describe the shape of the distribution of temperatures. Are there any unusual observations? Can you think of any explanation for these? Locate the 98.6-degee standard on the horizontal axis of the graph. Does it appear to be near the center of the distribution?Experimental Units Identify the experimental units on which the following variables are measured: a. Gender of a student b. Number of errors on a midterm exam c. Age of a cancer patient d. Number of flowers on an azalea plant e. Color of a car entering a parking lotQualitative or Quantitative? Identify each variable as quantitative or qalitative: Amount of time it takes to assemble a simple puzzle Number of students in a first-grade classroom Rating of a newly elected politician (excellent, good, fair, poor) State in which a person livesDiscrete or Continuous? Identify the following quantitative variable as discrete or continuous: Population in a particular area of the United States Weight of newspapers recovered for recycling on a single day Time to complete a sociology exam Number of consumers in a poll of 1000 who consider nutritional labeling on food products to be importantDiscrete or Continuous? Identify each quantitaive variable as discrete or continuous. a. Number of boating accidents along a 50—mile stretch ot the Colorado River b. Time required to complete a questionnaire c. Cost ot a head of lettuce d. Number of brothers and sisters you have e. Yield in kilograms of wheat from a 1 —hectare plot in a wheat fIeldParking on Campus Six vehicles are selected from the vehicles that are issued campus parking permits, and the following data are recorded: What are the experimental units? What are the variable being measured? What types of variables are they? Is this univariate, bivariate, or multivariate data?Past U.S. Presidents A data set consists of the ages at death for each of the 38 past presidents of the United States now deceased. a. Is this set ot meastireiiients a population or a sample? b. What is the variable being measured? c. Is the variable in part h quantitative or qualitative?Voter Attitudes You are a candidate for your state legislature. and you want to survey voter attitudes regarding your chances of winning. Identify the population that is of interest to you and from which you would like to select your sample. How is this population dependent on time?Cancer Survival Times A medical researcher wants to estimate the survival time of a patient after the onset of a particular type of cancer and after a particular regimen of radiotherapy. a. What is the variable of interest to the medical researcher? b. Is the variable in part a qualitative, quantitative discrete, or quantitative continuous? c. Identity the population of interest to the medical researcher. d. Describe how the researcher could select a sample from the population. e. What problems might arise in sampling from this population?New Teaching Methods An educational researcher wants to evaluate the effectiveness of a new method for leaching reading to deaf students. Achievement at the end of a period of teaching is measured by a students score on a reading test. a. What is the variable to be measured? What type of variable is it? b. What is the experimental unit? c. Identify the population of interest to the experimenter.Fifty people are grouped into four categrires A, B, C, and D- and the number of people who fall into each category is shown in the table: a. What is the experimental unit? b. What is the variable being measured? Is it qualitative or quantitative? c. Construct a pie chart to describe the data. d. Construct a bar chart to describe the data. e. Does the shape of the bar chart in part d change depending on the order of presentation of the four categories? Is the order of presentation important? f. What percentage of the people are in category B, C, or D? g. What percentage of the people are not in a category B?Jeans A manufacturer of jeans has plants in California, Arizona, and Texas. A group of 25 pairs of jeans is randomly selected from the computerized database, and the state in which each is produced is recorded: What is the experimental unit? What is the variable being measured? Is it qualitative or quantitative? Construct a pie chart to describe the data. Construct a bar chart to describe the data. What proportion of the jeans are made in Texas? What state produced the most jeans in the group? If you want to find out whether the three plants produced equal numbers of jeans, or whether one produced more jeans than the others, how can you use the charts from parts c and d to help you? What conclusions can you draw from these data?1.12EWant to Be President? Would you want to be the president of the United States? Although many teenagers think that they could grow up to be the president. most don’t want the job. In an opinion poll conducted by ABC News, nearly 80% of the teens were not interested in the job.2 When asked “What’s the main reason you would not want to be president?” they gave these responses: a. Are all of the reasons accounted for in this table? Add another category if necessary. b. Would you use a pie chart or a bar chart to graphically describe the data? Why? c. Draw the chart you chose in part b. d. If you were the person conducting the opinion poll, what other types of questions might you want to investigate?Facebook Fanatics The social networking site called Facebook has grown quickly since its inception in 2004. In fact, Facebook’s United States user base grew from 42 million users to 103 million users between 2009 and 2010. The table below shows the age distribution of Facebook users (in thousands) as it changed from January 2009 to January 2010.3 a. Define the variable that has been measured in this table. b. Is the variable quantitative or qualitative? c. What do the numbers represent? d. Construct a pie chart to describe the age distribution of Facebook users as of January 4, 2009. e. Construct a pie chart to describe the age distribution of Facebook users as of January 4, 2010. f. Refer to parts d and e. How would you describe the changes in the age distributions of Facebook users during this 1-year period?Back to Work How long does it take you to adjust to your normal work routine after coming back from vacation? A bar graph with data from the Snapshots section of USA Today is shown below:4 a. Are all of the opinions accounted for in the table? Add another category if necessary. b. Is the bar chart drawn accurately? That is, are the three bars in the correct proportion to each other? c. Use a pie chart to describe the opinions. Which graph is more interesting to look at?Construct a stem and leaf plot for these 50 measurements: 3.1 4.9 2.8 3.6 2.5 4.5 3.5 3.7 4.1 4.9 2.9 2.1 3.5 4.0 3.7 2.7 4.0 4.4 3.7 4.2 3.8 6.2 2.5 2.9 2.8 5.1 1.8 5.6 2.2 3.4 2.5 3.6 5.1 4.8 1.6 3.6 6.1 4.7 3.9 3.9 4.3 5.7 3.7 4.6 4.0 5.6 4.9 4.2 3.1 3.9 a. Describe the shape of the data distribution. Do you see any outliers? b. Use the stem and leaf plot to find the smallest observation. c. Find the eighth and ninth largest observations.Refer to Exercise 1.16. Construct a relative frequency histogram for the data. a. Approximately how many class intervals should you use? b. Suppose you decide to use classes starting at 1.6 with a class width of.5 (i.e., 1.6 to <2.1,2.1 to <2.6). Construct the relative frequency histogram for the data. c. What fraction of the measurements are less than 5.1? d. What fraction of the measurements are larger than 3.6? e. Compare the relative frequency histogram with the stem and leaf plot in Exercise 1.1 6. Are the shapes similar?1.18EA discrete variable can take on oniy the values 0. 1. or 2. A set of 20 measurements on this variable is shown here: 1 2 1 0 2 2 1 1 0 0 2 2 1 1 0 01 2 1 1 a. Construct a relative frequency histogram for the data. b. What proportion of the measurements are greater than 1? c. What proportion of the measurements are less than 2? d. If a measurement is selected at random from the 20 measurements shown, what is the probability that it isa2? e. Describe the shape of the distribution. Do you see any outliers?1.20E1.21E1.22ECheeseburgers Create a dotplot for the number of cheeseburgers consumed in a given week by 10 college students. 5421 3427 a. How would you describe the shape of the distribution? b. What proportion of the students ate more than 4 cheeseburgers that week?1.24E1.25E1.26EEducation Pays Off! Education pays off, according to a snapshot provided by the Bureau of Labor Statistics.8 The median weekly earnings for six different levels of education are shown in the table: What graphical methods could you use to describe the data? Select the method from part a that you think best describes the data and create the appropriate graph. How would you summarize the information that you see in the graph regarding educational levels and salary?Preschool The ages (in months) at which 50 children were first enrolled in a preschool are listed below. 38 40 30 35 39 40 48 36 31 36 47 35 34 43 41 36 41 43 48 40 32 34 41 30 46 35 40 30 46 37 55 39 33 32 32 45 42 41 36 50 42 50 37 39 33 45 38 46 36 31 a. Construct a stem and leaf display for the data. b. Construct a relative frequency histogram for these data. Start the lower boundary of the first class at 30 and use a class width of 5 months. c. Compare the graphs in parts a and b. Are there any significant differences that would cause you to choose one as the better method for displaying the data? d. What proportion of the children were 35 months (2 years. 11 months) or older, but less than 45 months (3 years. 9 months) of age when first enrolled in preschool? e. If one child were selected at random from this group of children, what is the probability that the child was less than 50 months old (4 years. 2 months) when first enrolled in preschool?1.29E1.30E1.31E1.32E1.33ERBC Counts The red blood cell count of a healthy person was measured on each of 15 days. The number recorded is measured in 106 cells per microliter (L). 5.4 5.2 5.0 5.2 5.5 5.3 5.4 5.2 5.1 5.3 5.3 4.9 5.4 5.2 5.2 a. Use an appropriate graph to describe the data. b. Describe the shape and location of the red blood cell counts. c. If the person’s red blood cell count is measured today as 5.7106L, would you consider this unusual? What conclusions might you draw?1.35E1.36EHazardous Waste How safe is your neighborhood? Are there any hazardous waste sites nearby? The table shows the number of hazardous waste sites in each of the 50 states and the District of Columbia in the year 2009:5 What variable is being measured? Is the variable discrete or continuous? Describe the shape of the data distribution using the stem and leaf plot shown here. Identify the unusually large measurements marked “HI” by state. Can you think of any reason these five states would have a large number of hazardous waste sites? What other variable might you measure to help explain why the data behave as they do?Raisins The number of raisins in each of 14miniboxes (1/2 -ounce size) was counted for ageneric brand and for Sunmaid brand raisins. Thetwo data sets are shown here: a. What are the mean and the standard deviation for thegeneric brand? b. What are the mean and the standard deviation for theSunmaid brand? c. Compare the centers and variabilities of the twobrands using the results of parts a and b.2.55SE2.56SEA Recurring IIIness Refer to Exercise 1.26 and data set EX0126. The lengths of time (in months) between the onset of a particular illness and its recurrence were recorded: a. Find the range. b. Use the range approximation to find an approximate value for s. c. Compute s for the data and compare it with your approximation from part b.2.58SE2.59SETuna Fish, again Refer to Exercise 2.8. Theprices of a 6-ounce can or a 7.06-ounce pouch for 14different brands of water-packed light tuna, based onprices paid nationally in supermarkets, are reproduced here.4 Calculate the five-number summary. Construct a box plot for the data. Are there anyoutliers? The value x=1.92 looks large in comparison to the other prices. Use a z-score to decide whether this isan unusually expensive brand of tuna.2.61SEChloroform According to the EPA, Chloroform, which in its gaseous form is suspected of being a cancer-causing agent. ¡s present in small quantities in all of thecountry’s 240,000 public water sources. If the mean andstandard deviation of the amounts of chloroform presentin the water sources are 34 and 53 micrograms per liter,respectively, describe the distribution for the populationof all public water sources.2.63SESleep and the College Student How muchsleep do you get on a typical school night? A group of10 college students were asked to report the number ofhours that they slept on the previous night with the following results: Find the mean and the standard deviation of thenumber of hours of sleep for these 10 students. Calculate the z-score for the largest value (x=8.5.) Is this an unusually sleepy college student? What is the most frequently reported measurement?What is the name for this measure of center? Construct a box plot for the data. Does the boxplot confirm your results in part b? [HINT: Since thez-score and the box plot are two unrelated methodsfor detecting outliers, and use different types ofstatistics, they do not necessarily have to (butusually do) produce the same results.]2.65SE2.66SEPolluted Seawater Petroleum pollution in seasand oceans stimulates the growth of some types ofbacteria. A count of petroleumlytic micro-organisms (bacteria per 100 milliliters) in 10 portions of seawatergave these readings: 49, 70, 54, 67, 59, 40, 61, 69, 71, 52 Guess the value for s using the rangeapproximation. Calculate x and s arid compare with the rangeapproximation of part a. Construct a box plot for the data and use it to describe the data distribution.2.68SE2.69SE2.70SE2.71SE2.72SE2.73SE2.74SETV Commercials The mean duration oftelevision commercials on a given network is 75 seconds, with a standard deviation of 20 seconds.Assume that durations are approximately normallydistributed. a. What is the approximate probability that acommercial will last less than 35 seconds? b. What is the approximate probability that acommercial will last longer than 55 seconds?2.76SE2.77SE2.78SE2.79SE2.80SE2.81SE2.82SE2.83SE2.84SE2.85SEYou are given n=5 measurements: 0, 5, 1, 1,3. Draw a dotplot for the data. (HInt: If twomeasurements are the same, place one dot above theother.) Guess the approximate “center.” Find the mean, median, and mode. Locate the three measures of center on the dotplotin part a. Based on the relative positions of the meanand median, are the measurements symmetric orskewed?2.2E2.3EAuto Insurance The cost of automobile insurance has become a sore subject in California because insurance rates are dependent on so many different variables, such as the city in which you live, the number of cars you insure, and the company with which you are insured. The website www.insurance.ca.gov reports the annual 2010 premium for a male, licensed for 6-8 years, who drives a Honda Accord 12,600-15,000 miles per year and has no violations or accidents.1 a. What is the average premium for GEICO Insurance? b. What is the average premium for 21st Century Insurance? c. If you were a consumer, would you be interested in the average premium cost? If not, what would you be interested in?2.5E2.6E2.7E2.8E2.9E2.10E2.11E2.12EYou are given n=5 measurements: 2, 1, 1,3,5. a. Calculate the sample mean, x. b. Calculate the sample variance, s2, using the formulagiven by the definition. c. Find the sample standard deviation, s. d. Find s2 arid s using the computing formula. Comparethe results with those found in parts b and c.2.14E2.15E2.16E2.17EUtility Bills in Southern CaliforniaThe monthly utility bills for a household in Riverside, California, were recorded for 12 consecutive months starting in January 2010: a. Calculate the range of the utility bills for the year2010. b. Calculate the average monthly utility bill for theyear 2010. c. Calculate the standard deviation for the 2010 utilitybills.2.19E2.20EA distribution of measurements is relatively mound-shaped with mean 50 and standard deviation lo. What proportion of the measurements will fall between 40 and 60? What proportion of the measurements will fall between 30 and 70? What proportion of the measurements will fall between 30 and 60? If a measurement is chosen at random from this distribution, what is the probability that it will be greater than 60?2.22E2.23EPackaging Hamburger Meat The data listed here are the weights (in pounds) of27 packages of ground beef in a supermarket meatdisplay: Construct a stem and leaf plot or a relativefrequency histogram to display the distribution ofweights. Is the distribution relatively mound- shaped? Find the mean and standard deviation of the dataset. Find the percentage of measurements in the intervals xs,x2s, and x3s. How do the percentages obtained in part c comparewith those given by the Empirical Rule? Explain. How many of the packages weigh exactly I pound’?Can you think of any explanation for this?Breathing Rates Is your breathing rate normal? Actually, there is no standard breathing rate forhumans. It can vary from as low as 4 breaths perminute to as high as 70 or 75 for a person engaged instrenuous exercise. Suppose that the resting breathingrates for college-age students have a relative frequency distribution that is mound-shaped, with a mean equal to12 and a standard deviation of 2.3 breaths per minute.What fraction of all students would have breathingrates in the following intervals? a. 9.7 to 14.3 breaths per minute b. 7.4 to 16.6 breaths per minute c. More than 18.9 or less than 5. 1 breaths per minute2.26ESocial Security Numbers A group of70 students were asked to record the last digit oftheir social security number. Draw a relative frequency histogram using the values 0 through 9 as the class midpoints. What is the shape of the distribution? Based on the shape,what would be your best estimate for the mean of the data set? Use the range approximation to guess the value of sfor this set. Use your calculator to find the actual values of xand s. Compare with your estimates in parts a and b.2.28E2.29E2.30ETimber Tracts To estimate the amount of lumber in a tract of timber, an owner decidedto count the number of trees with diameters exceeding12 inches in randomly selected 50-by-50-foot squares.Seventy 50-by-50-foot squares were chosen, and theselected trees were counted in each tract. The data arelisted here: Construct a relative frequency histogram to describethe data. Calculate the sample mean x as an estimate of , themean number of timber trees for all 50-by-50-footsquares in the tract. Calculate s for the data. Construct the intervals xs,x2s, and x3s.Calculate the percentageof squares falling into each of the three intervals,and compare with the corresponding percentagesgiven by the Empirical Rule and Tchebysheff’sTheorem.2.32E2.33E2.34E2.35E2.36E2.37E2.38E2.39E2.40EFind the five-number summary and the IQR forthese data: 19, 12, 16, 0, 14, 9, 6, 1, 12, 13, 10, 19, 7, 5, 8Given the following data set: 2.3, 1.0, 2.1, 6.5, 2.8, 8.8, 1.7, 2.9, 4.4, 5.1, 2.0 a. Find the positions of the lower and upper quartiles. b. Sort the data from smallest to largest and find thelower and upper quartiles. c. Calculate the IQR.Given the following data set: .23, .30, .35, .41, .56, .58, .76, .80 a. Find the lower and tipper quartiles. b. Calculate the IQR. c. Calculate the lower and tipper fences. Are there any outliers?Construct a box plot for these data and identifyany outliers: 25, 22, 26, 23, 27, 26, 28, 18, 25, 24, 12Construct a box plot for these data and identifyany outliers: 3, 9, 10,2, 6, 7,5,8,6,6,4,9,22If you scored at the 69th percentile on a placement test, how does your score compare with others?Mercury Concentration in DolphinsEnvironmental scientists are increasinglyconcerned with the accumulation of toxic elementsin marine mammals and the transfer of such elementsto the animals’ offspring. The striped dolphin (Stenellacoeruleoalba), considered to be a toppredator in the marine food chain, was the subjectof one such study. The mercury concentrations(micrograms/gram) in the livers of 28 maIe stripeddolphins were as follows: Calculate the five-number summary for the data. Construct a box plot for the data. Are there any outliers? If you knew that the first four dolphins were allless than 3 years old, while all the others were more than 8 years old, would this information help explainthe difference in the magnitude of those four observations? Explain.Hamburger Meat The weights (in pounds) of the 27 packages of ground beef from Exercise 2.24 (see data set EX0224) are listed here in order from smallest to largest: Confirm the values of the mean and standard deviation, calculated in Exercise 2.24 as and The two largest packages of meat weigh 1.38 and 1.41 pounds. Are these two packages unusually heavy? Explain. Construct a box plot for the package weights. What does the position of the median line and the length of the whiskers tell you about the shape of the distribution?Comparing NFL Quarterbacks How does Aaron Rodgers, quarterback for the 2011 Super Bowl winners, the Green Bay Packers, compare to Drew Brees, quarterback for the 2010 Super Bowl winners, the New Orleans Saints? The table below shows the number of completed passes for each athlete during the 2010 NFL football season:11 Calculate five-number summaries for the number of passes completed by both Aaron Rodgers and Drew Brees. Construct box plots for the two sets of data. Are there any outliers? What do the box plots tell you about the shapes of the two distributions? Write a short paragraph comparing the number of pass completions for the two quarterbacks.Presidential Vetoes The set of presidential vetoes in Exercise 1.47 arid data set EX0147 is listed here, along with a box plot generated by MINITAB. Use the box plot to describe the shape of the distribution and identify any outliers.Survival Times Altman and Bland report the survival times for patients with active hepatitis, half treated with prednisone and half receiving no treatment.12 The survival times (in months) (Exercise 1.25 and EXO125) are adapted from their data for those treated with prednisone. Can you tell by looking at the data whether it is roughly symmetric? Or is it skewed? Calculate the mean and the median. Use these measures to decide whether or not the data are symmetric or skewed. Draw a box plot to describe the data. Explain why the box plot confirms your conclusions in part b.Utility Bills in Southern California, again The monthly utility bills for a household in Riverside, California, were recorded for 12 consecutive months starting in January 2010: a. Construct a box plot for the monthly utility costs. b. What does the box plot tell you about thedistribution of utility costs for this household?What’s Normal? again Refer to Exercise1.67 and data set EX0167. In addition to the normalbody temperature in degrees Fahrenheit for the 130individuals, the data record the gender of the individuals. Box plots for the two groups, male and female, are shown below:13 Box plots for Exercise 2.53 How would you describe the similarities and differences between male and female temperatures in thisdata set?3.21SECheese, Please! Health-conscious Americans often consult the nutritional information on food packages in an attempt to avoid foods with large amounts of fat, sodium, or cholesterol. The following information was taken from eight different brands of American cheese slices: a. Which pairs of variables do you expect to he strongly related? b. Draw a scatterplot for fat and saturated fat. Describe the relationship. c. Draw a scatterplot for fat and calories. Compare the pattern to that found in part b. d. Draw a scatterplot for fat versus sodium and another for cholesterol versus sodium. Compare the patterns. Are there any clusters or outliers? e. For the pairs of variables that appear to be linearly related, calculate the correlation coefficients. f. Write a paragraph to summarize the relationships you can see in these data. Use the correlations and the patterns in the four scatterplots to verify your conclusions.3.23SE3.24SE3.25SE3.26SE3.27SE3.28SE3.29SE3.30SE3.31SEPottery, continued Here is the percentage of aluminum oxide. the percentage of iron oxide. and the percentage of magnesium oxide in five samples collected at Ashley Rails in the United Kingdom. a. Find the correlation coefficients describing the relationships between aluminum and iron oxide content, between iron oxide and magnesium oxide, and between aluminum oxide and magnesium oxide. b. Write a sentence describing the relationships between these three chemicals in the pottery samples.Gestation Times and Longevity The table below shows the gestation time in days and the average longevity in years for a variety of mammals in captivity; the potential life span of animals is rarely attained for animals and in the wild. 11 a. Draw a scatterplot for the data. b. Describe the form, direction, and strength for the pattern in the scatterplot. c. Are there any outliers or other unusual data points in the set? If so, to which animals do these data points correspond? d. Remove the outliers or unusual data points from the set, and reconstruct the scatterplot. Does it appear that a straight line is appropriate for describing the data?3.34SEArmspan and Height Leonardo da Vinci (1452-1519) drew a sketch of a man, indication that a person’s armspan (meausring across the back with arms outstretched to make a “T”) is roughly equal to the person’s height. To test this claim, we measured eight people with the following results: Draw sactterplot for armspan and height. Use the same scale on both the horizontal and vertical axes. Describe the relationship between the two variables. Calculate the correlation coefficient relating armspan and height. If you were to calculate the regression line for predicting height based on a person’s armspan, how would you estimate the slope of this line? Find the regression line relating armspan to a person’s height. If a person has an armspan of 62 inches, what would you predict the person’s height to be?Armspan and Height Leonardo da Vinci(14521519) drew a sketch of a man, indicatingthat a person’s armspan (measuring across theback with arms outstretched to make a “T”) is roughlyequal to the person’s height. To test this claim, we measuredeight people with the following results: a. Draw a scatterplot for armspan and height. Use thesame scale on both the horizontal and vertical axes.Describe the relationship between the two variables. b. Calculate the correlation coefficient relating armspanand height. c. If you were to calculate the regression line for predictingheight based on a person’s armspan, howwould you estimate the slope of this line? d. Find the regression line relating armspan to a person’sheight. e. If a person has an armspan of 62 inches, what wouldyou predict the person’s height to be?3.37SE3.38SE3.39SERain and Snow Is there a correlation between the amount of rain and the amount of snow that falls in a particular location? The table below shows the average annual rainfall (inches) and the average annual snowfall (inches) for 10 cities in the United States.15 a. Construct a scatterplot for the data. b. Calculate the correlation coefficient r between rainfall and snowfall. Describe the form, direction, and strength of the relationship between rainfall and snowfall. c. Are there any outliers in the scatterplol? If so, which city does this outlier represent? d. Remove the outlier that you found in part c from the data set and recalculate the correlation coefficient r for the remaining nine cities. Does the correlation between rainfall and snowfall change, and, if so, in what way?3.41SE1CS2CS3CSGender Differences Male and female respond ents to a questionnaire about gender differences are categorized into three groups according to their answers to the first question: a. Create side-by-side pie charts to describe these data. b. Create a side—by—side bar chart to describe these data. c. Draw a stacked bar chart to describe these data. d. Which of the three charts best depicts the difference or similarity of the responses of men and women?3.2EConsumer Spending The table below shows the average amounts spent per week by men and women in each of four spending categories: a. What possible graphical methods could you use to compare the spending patterns of women and men? b. Choose two different methods of graphing and display the data in graphical form. c. What can you say about the similarities or differences in the spending patterns for men and women? d. Which of the two methods used in part h provides a better descriptive graph?3.4E3.5EConsumer Price Index The price of living in the United States has increased dramatically in the past decade, as demonstrated by the consumer price indexes (CPIs) for housing and transportation. These CPI are listed in the table for the years 1996 through the first half of 2010.3 a. Create side-by-side comparative bar charts to describe the CPIs over time. b. Draw two line charts on the same set of axes to describe the CPIs over time. c. What conclusions can you draw using the two graphs in parts a and b? Which is the most effective?3.7E3.8ESuppose that the relationship between two variables x and y can be described by the regression line y=2.0+0.5x. What is the change in y for a one-unit change in x? Do the values of y increase or decrease as x increases? At what point does the line cross the y-axis? What is the name given to this value? If x=2.5, use the least squares equation to predict the value of y. What value would you predict if x=4.0?3.10EA set of bivariate data consists of these measurements on two variables. x and y: (3.6) (5,8) (2.6) (1.4) (4.7) (4.6) a. Draw a scatterplot to describe the data. b. Does there appear to be a relationship between x and y? If so. how do you describe it? c. Calculate the correlation coefficient,r,using the computing formula given in this section. d. Find the best—fitting line using the computing formulas. Graph the line on the scatterplot from part a. Does the line pass through the middlle of the points?3.12EConsider this set of bivariate data: a. Draw a scatterplot to describe the data. b. Does there appear to be a relationship between x and y? If so, how do you describe it? c. Calculate the correlation coefficient. r. Does the value of r confirm your conclusions in part b? Explain.3.14EGrocery Costs These data relating the amount spent on groceries per week and the number of household members are from Example 3.3: Find the best-fitting line for these data. Plot the points and the best-fitting line on the same graph. Does the line summarize the information in data points? What would you estimate a household of six to spend on groceries per week? Should you use the fitted line to estimate this amount? Why or why not?3.16E3.17E3.18E3.19E3.20EPlaying the Slots A slot machine has three slots; each will show a cherry, a lemon, a star, or a bar when spun. The player wins if all three slots show the same three items. If each of the four items is equally likely to appear on a given spin, what is your probability of winning?Whistle Blowers Although there is legal protectionfor “whistle blowers”—employees who report illegalor unethical activities in the workplace—it has beenreported that approximately 23% of those who reportedfraud suffered reprisals such as demotion or poor performanceratings. Suppose the probability that an employeewill fail to report a case of fraud is .69. Find the probabilitythat an employee who observes a case of fraudwill report it and will subsequently suffer some form ofreprisal.4.102SERefer to Exercise 4.102. By summing the probabilities of simple events, find P(A),P(B),P(AB),P(AB),P(C),P(AC), and P(AC) .DVRs A retailer sells two styles of digital videorecorders (DVR) that are in equal demand. (Fiftypercent of all potential customers prefer style 1, and50% favor style 2.) If the retailer stocks four of each,what is the probability that the first four customersseeking a DVR all purchase the same style?4.105SE4.106SE4.107SEFire Alarms A fire-detection device uses three temperature-sensitive cells acting independently of one another in such a manner that any one or more can activate the alarm. Each cell has a probability p=.8 of activating the alarm when the temperature reaches 135°F or higher. Let x equal the number of cells activating the alarm when the temperature reaches 135°F. a. Find the probability distribution of x. b. Find the probability that the alarm will function when the temperature reaches 135°F. c. Find the expected value and the variance for the random variable x.4.109SE4.110SE4.111SE4.112SE4.113SE4.114SE4.115SE4.116SE4.117SE4.118SE4.119SE4.120SE4.121SEContract Negotiations Experience has shown that,50% of the time, a particular unionmanagement contractnegotiation led to a settlement within a 2-weekperiod, 60% of the time the union strike fund wasadequate to support a strike, and 30% of the time bothconditions were satisfied. What is the probability of asettlement given that the union strike fund is adequateto support a strike? Is settlement of a contract within a2-week period dependent on whether the union strikefund is adequate to support a strike?4.123SE4.124SE4.125SE4.126SEMass Transit Only 40% of all people in a communityfavor the development of a mass transit system. Iffour citizens are selected at random from the community,what is the probability that all four favor the mass transitsystem? That none favors the mass transit system?4.128SE4.129SE4.130SEFlextime A survey to determine the availability of flextime schedules in the California workplace provided the following information for 220 firms located in two California cities. A company is selected at random from this pool of 220 companies. a. What is the probability that the company is located in city A? b. What is the probability that the company is located in city B and offers flextime work schedules? c. What is the probability that the company does not have flextime schedules? d. What is the probability that the company is located in city B, given that the company has flextime schedules available?4.132SEPepsi™ or Coke™? A taste-testing experiment is conducted at a local supermarket, where passing shoppers are asked to taste two soft-drink samples—one Pepsi and one Coke—and state their preference. Suppose that four shoppers are chosen at random and asked to participate in the experiment, and that there is actually no difference in the taste of the two brands. a. What is the probability that all four shoppers choose Pepsi? b. What is the probability that exactly one of the four shoppers chooses Pepsi?Viruses A certain virus afflicted the families in three adjacent houses in a row of 12 houses. If houses were randomly chosen from a row of 12 houses, what is the probability that the three houses would be adjacent? Is there reason to believe that this virus is contagious?4.135SEIndependence and Mutually Exclusive Suppose that P(A)=.3 and P(B)=.4 . a. If P(AB)=.12 are A and B independent? Justify your answer. b. If P(AB)=.7 what is P(AB) ? Justify your answer. c. If A and B are independent, what is P(A|B) ? d. If A and B are mutually exclusive, what is P(A|B) ?4.137SETossing a Die An experiment involves tossing a single die. These are some events: A: Observe a 2 B: Observe an even number C: Observe a number greater than 2 D: Observe both A and B E: Observe A or B or both F: Observe both A and C a. List the simple events in the sample space. b. List the simple events in each of the events A through F. c. What probabilities should you assign to the simple events? d. Calculate the probabilities of the six events A through F by adding the appropriate simple-event probabilities.4.2EA sample space contains 10 simple events: E1,E2,...,E10 . If P(E1)=3P(E2)=.45 and the remaining simple events are equiprobable, find the probabilities of these remaining simple events.Free Throws A particular basketball player hits 70% of her free throws. When she tosses a pair of free throws, the four possible simple events and three of their associated probabilities are as given in the table: a. Find the probability that the player will hit on the first throw and miss on the second. b. Find the probability that the player will hit on at least one of the two free throws.Four Coins A jar contains four coins: a nickel, a dime, a quarter, and a half-dollar. Three coins are randomly selected from the jar. a. List the simple events in S. b. What is the probability that the selection will contain the half-dollar? c. What is the probability that the total amount drawn will equal 160c or more?Preschool or Not? On the first day of kindergarten, the teacher randomly selects 1 of his 25 students and records the student’s gender, as well as whether or not that student had gone to preschool. a. How would you describe the experiment? b. Construct a tree diagram for this experiment. How many simple events are there? c. The table below shows the distribution of the 25 students according to gender and preschool experience. Use the table to assign probabilities to the simple events in part b. d. What is the probability that the randomly selected student is male? What is the probability that the student is a female and did not go to preschool?4.7EThe Urn Problem, continued Refer to Exercise 4.7. A ball is randomly selected from the bowl containing three red and two yellow balls. Its color is noted, and the ball is returned to the bowl before a second ball is selected. List the additional five simple events that must be added to the sample space in Exercise 4.7.Need Eyeglasses? A survey classified a large number of adults according to whether they were judged to need eyeglasses to correct their reading vision and whether they used eyeglasses when reading. The proportions falling into the four categories are shown in the table. (Note that a small proportion, .02, of adults used eyeglasses when in fact they were judged not to need them.) If a single adult is selected from this large group, find the probability of each event: a. The adult is judged to need eyeglasses. b. The adult needs eyeglasses for reading but does not use them. c. The adult uses eyeglasses for reading whether he or she needs them or not.4.10EJury Duty Three people are randomly selected to report for jury duty. The gender of each person is noted by the county clerk. a. Define the experiment. b. List the simple events in S. c. If each person is just as likely to be a man as a woman, what probability do you assign to each simple event? d. What is the probability that only one of the three is a man? e. What is the probability that all three are women?4.12ETea Tasters A food company plans to conduct an experiment to compare its brand of tea with that of two competitors. A single person is hired to taste and rank each of three brands of tea, which are unmarked except for identifying symbols A, B, and C. a. Define the experiment. b. List the simple events in S. c. If the taster has no ability to distinguish a difference in taste among teas, what is the probability that the taster will rank tea type A as the most desirable? As the least desirable?-Meter Run Four equally qualified runners, John, Bill, Ed, and Dave, run a 100-meter sprint, and the order of finish is recorded. a. How many simple events are in the sample space? b. If the runners are equally qualified, what probability should you assign to each simple event? c. What is the probability that Dave wins the race? d. What is the probability that Dave wins and John places second? e. What is the probability that Ed finishes last?Fruit Flies In a genetics experiment, the researchermated two Drosophila fruit flies and observed the traitsof 300 offspring. The results are shown in the table. One of these offspring is randomly selected and observedfor the two genetic traits. a. What is the probability that the fly has normal eyecolor and normal wing size? b. What is the probability that the fly has vermillion eyes? c. What is the probability that the fly has either vermillioneyes or miniature wings, or both?4.16EYou have two groups of distinctly different items, 10 in the first group and 8 in the second. If you select one item from each group, how many different pairs can you form?4.18E4.19ECombinations Evaluate these combinations: a. C35 b. C910 c. C66 d. C120Choosing People In how many ways can youselect five people from a group of eight if the order ofselection is important?Choosing People, again In how many ways canyou select two people from a group of 20 if the order ofselection is not important?4.23ECoins Four coins are tossed. How many simple events are in the sample space?The Urn Problem, again Three balls are selectedfrom a box containing 10 balls. The order of selectionis not important. How many simple events are in thesample space?What to Wear? You own 4 pairs of jeans, 12 cleanT-shirts, and 4 wearable pairs of sneakers. How manyoutfits (jeans, T-shirt, and sneakers) can you create?Itineraries A businessman in New York is preparingan itinerary for a visit to six major cities. Thedistance traveled, and hence the cost of the trip, willdepend on the order in which he plans his route.How many different itineraries (and trip costs) arepossible?Vacation Plans Your family vacation involves across-country air flight, a rental car, and a hotel stay inBoston. If you can choose from four major air carriers,five car rental agencies, and three major hotel chains,how many options are available for your vacationaccommodations?A Card Game Three students are playing a cardgame. They decide to choose the first person to play byeach selecting a card from the 52-card deck and lookingfor the highest card in value and suit. They rank thesuits from lowest to highest: clubs, diamonds, hearts,and spades. a. If the card is replaced in the deck after each studentchooses, how many possible configurations of thethree choices are possible? b. How many configurations are there in which eachstudent picks a different card? c. What is the probability that all three students pickexactly the same card? d. What is the probability that all three students pickdifferent cards?Dinner at Gerard’s A French restaurant offers aspecial summer menu in which, for a fixed dinner cost,you can choose from one of two salads, one of twoentrees, and one of two desserts. How many differentdinners are available?Playing Poker Five cards are selected from a52-card deck for a poker hand. a. How many simple events are in the sample space? b. A royal flush is a hand that contains the A, K, Q,J, and 10, all in the same suit. How many ways are there to get a royal flush? c. What is the probability of being dealt a royal flush?4.32EA Hospital Survey A study is to be conducted in ahospital to determine the attitudes of nurses toward variousadministrative procedures. If a sample of 10 nurses is to beselected from a total of 90, how many different samplescan be selected? (HINT: Is order important in determiningthe makeup of the sample to be selected for the survey?)Traffic Problems Two city council members are tobe selected from a total of five to form a subcommitteeto study the city’s traffic problems. a. How many different subcommittees are possible? b. If all possible council members have an equal chanceof being selected, what is the probability that membersSmith and Jones are both selected?4.35E4.36E4.37ECramming A student prepares for an exam bystudying a list of 10 problems. She can solve 6 of them.For the exam, the instructor selects 5 questions at randomfrom the list of 10. What is the probability that thestudent can solve all 5 problems on the exam?Monkey Business A monkey is given 12 blocks:3 shaped like squares, 3 like rectangles, 3 like triangles,and 3 like circles. If it draws three of each kind inorder—say, 3 triangles, then 3 squares, and so on—would you suspect that the monkey associates identicallyshaped figures? Calculate the probability of thisevent.An experiment can result in one of five equally likely simple events, E1,E2,...,E5 . Events A, B, and C are defined as follows: A:E1,E3P(A)=.4B:E1,E2,E4,E5P(B)=.8C:E3,E4P(C)=.4 Find the probabilities associated with the following events by listing the simple events in each. a. Ac b. AB <c. BC d. AB e. B|C f. A|B g. ABC h. (AB)cRefer to Exercise 4.40. Use the definition of a complementary event to find these probabilities: a. P(Ac) b. P(( AB)c) Do the results agree with those obtained in Exercise 4.40?Refer to Exercise 4.40. Use the definition of conditional probability to find these probabilities: a. P(A|B) b. P(B|C) Do the results agree with those obtained in Exercise 4.40?Refer to Exercise 4.40. Use the Addition and Multiplication Rules to find these probabilities: a. P(AB) b. P(AB) c. P(BC) Do the results agree with those obtained in Exercise 4.40?4.44ESuppose P(A)=.1 and P(B)=.5 . a. If P(A|B)=.1 , what is P(AB) ? b. If P(A|B)=.1 , are A and B independent? c. If P(AB)=0 , are A and B independent? d. If P(AB)=.65 , are A and B mutually exclusive?Dice An experiment consists of tossing a single die and observing the number of dots that show on the upper face. Events A, B, and C are defined as follows: A: Observe a number less than 4 B: Observe a number less than or equal to 2 C: Observe a number greater than 3 Find the probabilities associated with the events below using either the simple event approach or the rules and definitions from this section. a. S b. A|B c. B d. ABC e. AB f. AC g. BC h AC i. BC4.47ETwo fair dice are tossed. a. What is the probability that the sum of the number of dots shown on the upper faces is equal to 7? To 11? b. What is the probability that you roll “doubles”—that is, both dice have the same number on the upper face? c. What is the probability that both dice show an odd number?Suppose that P(A)=.4 and P(B)=.2 . If events A and B are independent, find these probabilities: a. P(AB) b. P(AB)4.50ESuppose that P(A)=.4 and P(AB)=.12 . a. Find P(B|A) . b. Are events A and B mutually exclusive? c. If P(B)=.3 , are events A and B independent?4.52E4.53E4.54EGrant Funding Suppose a group of research proposals was evaluated by a panel of experts to decide whether or not they were worthy of funding. When these same proposals were submitted to a second independent panel of experts, the decision to fund was reversed in 30% of the cases. If the probability that a proposal is judged worthy of funding by the first panel is .2, what are the probabilities of these events? a. A worthy proposal is approved by both panels. b. A worthy proposal is disapproved by both panels. c. A worthy proposal is approved by one panel.Drug Offenders A study of drug offenders who have been treated for drug abuse suggests that the likelihood of conviction within a 2-year period after treatment may depend on the offender’s education. The proportions of the total number of cases that fall into four education/conviction categories are shown in the table below. Suppose a single offender is selected from the treatment program. Here are the events of interest: A: The offender has 10 or more years of education B: The offender is convicted within 2 years after completion of treatment Find the appropriate probabilities for these events: a. A b. B c. AB d. AB e. Ac f. (AB)c g. (AB)c h. A given that B has occurredM i. B given that A has occurred4.57EThe Birthday Problem Two people enter a room and their birthdays (ignoring years) are recorded. a. Identify the nature of the simple events in S. b. What is the probability that the two people have a specific pair of birthdates? c. Identify the simple events in event A: Both people have the same birthday. d. Find P(A) . e. Find P(Ac) .4.59EStarbucks or Peet’s®? A college student frequents one of two coffee houses on campus, choosing Starbucks 70% of the time and Peet’s 30% of the time. Regardless of where she goes, she buys a cafe mocha on 60% of her visits. a. The next time she goes into a coffee house on campus, what is the probability that she goes to Starbucks and orders a cafe mocha? b. Are the two events in part a independent? Explain. c. If she goes into a coffee house and orders a café mocha, what is the probability that she is at Peet’s? d. What is the probability that she goes to Starbucks or orders a cafe mocha or both?4.61ESmoking and Cancer A survey of people in a givenregion showed that 20% were smokers. The probabilityof death due to lung cancer, given that a person smoked,was roughly 10 times the probability of death due tolung cancer, given that a person did not smoke. If theprobability of death due to lung cancer in the region is.006, what is the probability of death due to lung cancergiven that a person is a smoker?Smoke Detectors A smoke-detector system usestwo devices, A and B. If smoke is present, the probabilitythat it will be detected by device A is .95; by deviceB, .98; and by both devices, .94. a. If smoke is present, find the probability that thesmoke will be detected by device A or device B orboth devices. b. Find the probability that the smoke will not bedetected.Plant Genetics In 1865, Gregor Mendel suggesteda theory of inheritance based on the science of genetics.He identified heterozygous individuals for flower color that had two alleles ( r= recessive white color allele and R= dominant red color allele). When these individualswere mated, 3/4 of the offspring were observed to have red flowers and 1/4 had white flowers. The table summarizesthis mating; each parent gives one of its allelesto form the gene of the offspring. We assume that each parent is equally likely to give eitherof the two alleles and that, if either one or two of thealleles in a pair is dominant (R), the offspring will havered flowers. a. What is the probability that an offspring in this matinghas at least one dominant allele? b. What is the probability that an offspring has at leastone recessive allele? c. What is the probability that an offspring has onerecessive allele, given that the offspring has redflowers?4.65EChoosing a Mate Men and women often disagreeon how they think about selecting a mate. Suppose thata poll of 1000 individuals in their twenties gave the followingresponses to the question of whether it is moreimportant for their future mate to be able to communicatetheir feelings (F) than it is for that person to makea good living (G). If an individual is selected at random from this group of1000 individuals, calculate the following probabilities: a. P(F) b. P(G) c. P(FM) d. P(FW) e. P(MF) f. P(WG)