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statistics alive
Introduction To Probability And Statistics 15th Edition William Mendenhall Iii , Robert Beaver , Barbara Beaver - Solutions
If we observe a link between smoking status and survival status, is there something that could explain how this happened?
Is smoking actually beneficial to your health, considering that a smaller percentage of smokers died?
How can we describe the relationship between smoking and survival with numbers (descriptive statistics)?
Is climate change occurring, and how serious is it?
How can we investigate which countries emit the highest amounts of carbon dioxide or use the most nonrenewable energy?
How can you predict the selling price of a house?
How can you analyze whether a diet really works?
What ‘hot streaks’ should you expect in basketball?
Is there bias against women in appointing managers?
How likely are you to win the lottery?
What’s the chance your tax return will be audited?
Are cell phones dangerous to your health?
How can you evaluate evidence about global warming?
Then, if s1 and s2 are unknown but we have reason to believe that s1 2s2, we can pool the standard deviations. Using sample sizes n1 and n2, the sample test statistic x1 2x2 has a Student’s t distribution, whereNote: With statistical software, select the pooled variance or equal variance
Expand Your Knowledge: Software Approximation for Degrees of Freedom Given x1 and x2 distributions that are normal or approximately normal with unknown s1 and s2, the value of t corresponding to x1 2x2 has a distribution that is approximated by a Student’s t distribution. We use the convention
Expand Your Knowledge: Confidence Intervals for md Using techniques from Section 7.2, we can find a confidence interval for md. Consider a random sample of n matched data pairs A, B. Let d 5B 2A be a random variable representing the difference between the values in a matched data pair. Compute the
Critical Thinking: Boxplots and Confidence Intervals The distribution of heights of 18-year-old men in the United States is approximately normal, with mean 68 inches and standard deviation 3 inches(U.S. Census Bureau). In Minitab, we can simulate the drawing of random samples of size 20 from this
Each of 300 volunteers collected a random sample of 500 adult voters and recorded the proportion that were in favor of a particular new constitutional amendment.The proportions in the 300 samples are summarized in the following frequency table.Range Frequency 0.33 - 0.35 3 0.35 - 0.37 22 0.37 -
Webster’s Widgets collects random samples of 100 widgets from their widget factory several times per shift and records the mean weight of the sample(in grams). The means of 400 samples are summarized in the following frequency table.Range Frequency 6.8 – 7.2 21 7.2 – 7.6 73 7.6 – 8.0 103
Estimating the Standard Deviation Consumer Reports gave information about the ages at which various household products are replaced. For example, TVs are replaced at an average age of m 5 8 years after purchase, and the (95% of data) range was from 5 to 11 years. Thus, the range was 112 5 5 6
Critical Thinking: Normality Consider the following data. The summary statistics, histogram, and normal quantile plot were generated using Minitab.27 27 27 28 28 28 28 28 28 29 29 29 29 29 29 29 29 29 29 30 30 30 30 30 30 30 30 30 30 30 30 31 31 31 31 31 31 31 31 32 32 32 32 33 33 33 33 33 34
Critical Thinking Look at the two normal curves in Figures 6-12 and 6-13. Which has the larger standard deviation? What is the mean of the curve in Figure 6-12?What is the mean of the curve in Figure 6-13?AppendixLO1 FIGURE 6-12 4 6 8 10 12 14 16 FIGURE 6-13 2 3 4 -co -5
Statistical Literacy Look at the normal curve in Figure 6-11, and find m, m 1s , and s .AppendixLO1 FIGURE 6-11 16 18 20 22
Statistical Literacy Which, if any, of the curves in Figure 6-10 look(s) like a normal curve? If a curve is not a normal curve, tell why.AppendixLO1 FIGURE 6-10 (a) (b) (c) (d)
Interpretation A Harris Poll surveyed 2085 U.S.adults regarding use of cell phones while driving.All the adults were asked their opinion regarding how dangerous it is for a driver to use a cell phone while driving. Graph (a) shows the percentage responding “very dangerous” by age group. Only
Archaeology: Tree-Ring Data The Sand Canyon Archaeological Project, edited by W. D. Lipe and published by Crow Canyon Archaeological Center, contains the stem-and-leaf diagram shown in Figure 2-22. The study uses tree rings to accurately determine the year in which a tree was cut. The figure gives
Criminal Justice: Prisoners The time plot in Figure 2-21 gives the number of state and federal prisoners per 100,000 population (Source: Statistical Abstract of the United States, 120th edition).(a) Estimate the number of prisoners per 100,000 people for 1980 and for 1997.(b) Interpretation During
Expand Your Knowledge: Back-to-Back Stem Plot In archaeology, the depth (below surface grade) at which artifacts are found is very important. Greater depths sometimes indicate older artifacts, perhaps from a different archaeological period. Figure 2-20 is a back-toback stem plot showing the depths
Split Stem: Golf The U.S. Open Golf Tournament was played at Congressional Country Club, Bethesda, Maryland, with prizes ranging from $465,000 for first place to $5000. Par for the course was 70. The tournament consisted of four rounds played on different days. The scores for each round of the 32
Health Care: Hospitals The American Medical Association Center for Health Policy Research, in its publication State Health Care Data: Utilization, Spending, and Characteristics, included data, by state, on the number of community hospitals and the average patient stay (in days). The data are shown
feel it is important to stay connected when in vehicle.The graph shows the percentage of respondents in each age category who agree strongly or somewhat agree to each of the six statements.Make a time-series graph displaying the data. For more information, visit the web site for Pyramid Lake
Interpretation Consider the two graphs depicting the influence of advertisements on making large purchases for two different age groups, those 18–34 years old and those 45–54 years old (based on a Harris Poll of about 2500 adults aged 18 or older). Note: Other responses such as “not sure”
Interpretation The graph below is information from a survey of 1000 pet owners who were asked to identify the types of pets they had at home.Is it possible to make a circle graph using the information provided in the graph? Explain why or why not.AppendixLO1 Percentage 60% 53 50% 39 40% 30% 20% 17
Interpretation Consider graphs (a) and (b) below showing the interest rates of a mortgage loan for new homes from 2015 to 2019.(a) What are the differences in graph (a) and graph (b)?(b) Suppose someone claimed that interest rates have increased dramatically. Explain whether or not this is a true
Interpretation Consider graph (a) of Reasons People Like Texting on Cell Phones based on a GfK Roper survey of 1000 adults.(a) Do you think respondents could select more than one response? Explain.(b) Could the same information be displayed in a circle graph? Explain.(c) Is graph (a) a Pareto
Expand Your Knowledge: Dotplot Another display technique that is somewhat similar to a histogram is a dotplot. In a dotplot, the data values are displayed along the horizontal axis. A dot is then plotted over each data value in the data set.How to Make a Dotplot Display the data along a horizontal
Critical Thinking The following ogives come from different distributions of 50 whole numbers between 1 and 60.Labels on each point give the cumulative frequency and the cumulative percentage of data.(a) In which distribution does the most data fall below 20.5?(b) In which distribution does the most
Interpretation The ogives shown on the next page are based on U.S. Census data and show the average annual personal income per capita for each of the 50 states. The data are rounded to the nearest thousand dollars.(a) How were the percentages shown in graph (ii)computed?(b) How many states have
Interpretation How much time is spent socially interacting with friends in-person? The following histograms are based on different random samples of size 100 drawn from the same population on the weekly number of hours an individual spent interacting with friends in-person.(a) Identify the midpoint
Interpretation Histograms of random sample data are often used as an indication of the shape of the underlying population distribution. Histograms (i), (ii), and (iii) on this page are based on random samples of size 30, 50, and 100 from the same population.(i) Sample of size 30(ii) Sample of size
Critical Thinking Look at the histogram in Figure 2-11(a), which shows mileage, in miles per gallon(mpg), for a random selection of older passenger cars(Reference: Consumer Reports).(a) Is the shape of the histogram essentially bimodal?(b) Jose looked at the raw data and discovered that the 54 data
Critical Thinking An agricultural study is comparing the harvest volume of two types of barley. The site for the experiment is bordered by a river. The field is divided into eight plots of approximately the same size.The experiment calls for the plots to be blocked into four plots per block. Then,
In his exciting novel Congo, Michael Crichton describes a search by Earth Resources Technology Service (ERTS), a geological survey company, for deposits of boron-coated blue diamonds, diamonds that ERTS believes to be the key to a new generation of optical computers.10 In the novel, ERTS is racing
20. Tossing a Coin How many times should a coin be tossed to obtain a probability equal to or greater than .9 of observing at least one head?
19. Blood Tests To reduce testing costs, blood tests are conducted on a pooled sample of blood collected from a group of n people. If the test is negative in the pooled blood sample, none have the disease. If the test is positive, the blood of each individual must be tested. The individual tests
18. Blood Pressure Meds A physician compared the effectiveness of two blood pressure drugs A and B using four pairs of identical twins—drug A was given to one twin; drug B to the other. If, in fact, there is no difference in the effects of the drugs, what is the probability that the drop in blood
17. Mass Transit Only 40% of all people in a community favor the development of a mass transit system. If four citizens are selected at random from the community, what is the probability that all four favor the mass transit system? That none favors the mass transit system?
16. Quality Control A quality-control plan calls for accepting a large lot of crankshaft bearings if a sample of seven is drawn and none are defective. What is the probability of accepting the lot if none in the lot are defective? If 1/10 are defective? If 1/2 are defective?
15. Waiting Times The probability of waiting 5 minutes or longer for checkout at a particular supermarket counter is .2. On a given day, a man and his wife decide to shop individually at the market, each checking out at different counters. They both reach the counters at the same time.a. What is
14. Work Tenure Suppose the probability of remaining with a particular company 10 years or longer is 1/6.A man and a woman start work at the company on the same day.a. What is the probability that the man will work there less than 10 years?b. What is the probability that both the man and the woman
13. Contract Negotiations Experience has shown that, 50% of the time, a particular union–management contract negotiation led to a settlement within a 2-week period, 60% of the time the union strike fund was adequate to support a strike, and 30% of the time both conditions were satisfied. What is
12. The Match Game Two men each toss a coin. They obtain a “match” if either both coins are heads or both are tails. Suppose the tossing is repeated three times.a. What is the probability of three matches?b. What is the probability that all six tosses (three for each man) result in tails?c.
10. ACL/MCL Tears The American Journal of Sports Medicine published a study of 810 women collegiate rugby players with two common knee injuries: medial cruciate ligament (MCL) sprains and anterior cruciate ligament (ACL) tears.7 For backfield players, it was found that 39% had MCL sprains and 61%
9. The Birthday Problem, continued If n people enter a room, find these probabilities:A: None of the people have the same birthday B: At least two of the people have the same birthday Solve fora. n3b. n4[note: Surprisingly, P(B) increases rapidly as n increases. For example, for n20,
8. The 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
7. Lottery, continued Refer to Exercise 6. Hours after the rigging of the Pennsylvania state lottery was announced on September 19, 1980, Connecticut state lottery officials were stunned to learn that their winning number for the day was 666.a. All evidence indicates that the Connecticut selection
6. Pennsylvania Lottery Probability played a role in the rigging of the April 24, 1980, Pennsylvania state lottery. To determine each digit of the three-digit winning number, each of the numbers 0, 1, 2, . . . , 9 was written on a Ping-Pong ball, the 10 balls were blown into a compartment, and the
5. Rental Trucks A rental truck agency services its vehicles on a regular basis to check for mechanical problems. The agency has six moving vans, two of which have brake problems. During a routine check, the vans are tested one at a time.a. What is the probability that the last van with brake
4. A Reticent Salesman The probability that a salesperson makes a sale during the first contact with a client is.4 but improves to .55 on the second contact, given that the client did not buy during the first contact. This salesperson makes one and only one callback to a client.a. What is the
3. Interstate Commerce A shipping container contains seven complex electronic systems. Unknown to the purchaser, three are defective. Two of the seven are selected for thorough testing and are then classified as defective or nondefective. What is the probability that no defectives are found?
2. DVRs A retailer sells two styles of digital video recorders (DVR) that are in equal demand. (Fifty percent of all potential customers prefer style 1, and 50% favor style 2.) If the retailer stocks four of each, what is the probability that the first four customers seeking a DVR all purchase the
1. Whistle Blowers Although there is legal protection for “whistle blowers”—employees who report illegal or unethical activities in the workplace—it has been reported that approximately 23% of those who reported fraud suffered reprisals such as demotion or poor performance ratings. Suppose
14. Cheating on Your Taxes? Suppose 5% of all people filing the long income tax form seek deductions that they know are illegal, and an additional 2% incorrectly list deductions because they are unfamiliar with income tax regulations. Of the 5% who are guilty of cheating, 80% will deny knowledge of
13. Medical Diagnostics Different illnesses can produce identical symptoms. Suppose a particular set of symptoms, which we will denote as event H, occurs only when any one of three illnesses—A, B, or C—occurs. (For the sake of simplicity, we will assume that illnesses A, B, and C are mutually
12. No Pass, No Play Under the “no pass, no play”rule for athletes, an athlete who fails a course cannot participate in sports activities during the next grading period. Suppose the probability that an athlete who has not previously been disqualified will be disqualified is .15 and the
11. Football Strategies A football team is known to run 30% of its plays to the left and 70% to the right.A linebacker on an opposing team notices that, when plays go to the right, the right guard shifts his stance most of the time (80%) and that he uses a balanced stance the remainder of the time.
10. Airport Security Suppose that, in a particular city, airport A handles 50% of all airline traffic, and airports B and C handle 30% and 20%, respectively. The detection rates for weapons at the three airports are .9, .8, and.85, respectively. If a passenger at one of the airports is found to be
9. Worker Error A worker-operated machine produces a defective item with probability .01 if the worker follows the machine’s operating instructions exactly, and with probability .03 if he does not. If the worker follows the instructions 90% of the time, what proportion of all items produced by
8. Violent Crime City crime records show that 20% of all crimes are violent and 80% are nonviolent, involving theft, forgery, and so on. Ninety percent of violent crimes are reported versus 70% of nonviolent crimes.a. What is the overall reporting rate for crimes in the city?b. If a crime in
7. Law of Total Probability A population can be divided into two subgroups that occur with probabilities 60% and 40%, respectively. An event A occurs 30% of the time in the first subgroup and 50% of the time in the second subgroup. What is the unconditional probability of the event A, regardless of
Refer to Example 4.24. Find the probability that the person selected was 65 years of age or older, given that the person owned at least five pairs of wearable sneakers.
37. Jordan and Durant Two stars of the LA Clippers and the Golden State Warriors are very different when it comes to making free throws.ESPN.com reports that DeAndre Jordan makes 81% of his free throw shots while Kevin Durant makes 62% of his free throws.5 Assume that the free throws are
32. Smoke Detectors A smoke-detector system uses two devices, A and B. If smoke is present, the probability that it will be detected by device A is .95; by device B, .98; and by both devices, .94.a. If smoke is present, find the probability that the smoke will be detected by device A or device B or
31. Smoking and Cancer A survey of people in a given region showed that 20% were smokers. The probability of death due to lung cancer, given that a person smoked, was roughly 10 times the probability of death due to lung cancer, given that a person did not smoke. If the probability of death due to
30. Inspection Lines A certain manufactured item is visually inspected by two different inspectors. When a defective item comes through the line, the probability that it gets by the first inspector is .1. Of those that get past the first inspector, the second inspector will “miss”5 out of 10.
29. Starbucks 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
28. Profitable Stocks An investor has the option of investing in three of five recommended stocks.Unknown to her, only two will show a substantial profit within the next 5 years. If she selects the three stocks at random (giving every combination of three stocks an equal chance of selection), what
27. Guided Missiles The failure rate for a guided missile control system is 1 in 1000. Suppose that a duplicate, but completely independent, control system is installed in each missile so that, if the first fails, the second can take over. The reliability of a missile is the probability that it
26. Bus or Subway A man takes either a bus or the subway to work with probabilities .3 and .7, respectively.When he takes the bus, he is late 30% of the days.When he takes the subway, he is late 20% of the days. If the man is late for work on a particular day, what is the probability that he took
23. Grant Funding 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
22. Drug Testing In testing prospective employees for drug use, companies need to remember that the tests are not 100% reliable. Suppose a company uses a test that is 98% accurate—that is, it correctly identifies a person as a drug user or nonuser with probability .98—and to reduce the chance
21. Two Dice Two 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
20. Are events A and C independent? Mutually exclusive?
19. Are events A and B independent? Mutually exclusive?
17. Suppose that P(A).4 and P(A B).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?Dice An experiment consists of tossing a single die and observing the number of dots that show on the upper face.Use events A, B, and C, defined as
16. Suppose that P(A).3 and P(B).5. If events A and B are mutually exclusive, find these probabilities:a. P(A B)b. P(A B)
15. Suppose that P(A).4 and P(B) .2. If events A and B are independent, find these probabilities:a. P(A B)b. P(A B)
14. Independence and Mutually Exclusive Suppose that P(A).3 and P(B).4.a. If P(A B).12, are A and B independent? Justify your answer.b. If P(A B).7, what is P(A B)? 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)
Two cards are drawn from a deck of 52 cards. Calculate the probability that the draw includes an ace and a ten.
Toss two coins and observe the outcome. Define these events:A: Head on the first coin B: Tail on the second coin Are events A and B independent?
In a color preference experiment, eight toys are placed in a container. The toys are identical except for color—two are red, and six are green. A child is asked to choose two toys at random.What is the probability that the child chooses the two red toys?
Two fair coins are tossed, and the outcome is recorded. These are the events of interest:A: Observe at least one head B: Observe at least one tail Define the events A, B, AB, AU B, and A as collections of simple events, and find their probabilities.
30. 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?
29. 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 in order—say, 3 triangles, then 3 squares, and so on—would you suspect that the monkey associates identically shaped figures? Calculate
28. Cramming A student prepares for an exam by studying a list of 10 problems. She can solve 6 of them.For the exam, the instructor selects 5 questions at random from the list of 10. What is the probability that the student can solve all 5 problems on the exam?
27. Unbiased Choices A woman brought a complaint of gender discrimination to an eight-member HR committee.The committee, composed of five females and three males, voted 5–3 in favor of the woman, the five females voting for the woman and the three males against. Has the board been affected by
26. 100-Meter Run, again Refer to Exercise 36(Section 4.2), in which a 100-meter sprint is run by John, Bill, Ed, and Dave. Assume that all of the runners are equally qualified, so that any order of finish is equally likely. Use the mn Rule or permutations to answer these questions:a. How many
24. Traffic Problems Two city council members are to be selected from a total of five to form a subcommittee to study the city’s traffic problems.a. How many different subcommittees are possible?b. If all possible council members have an equal chance of being selected, what is the probability
23. A Hospital Survey A study is to be conducted in a hospital to determine the attitudes of nurses toward various administrative procedures. If a sample of 10 nurses is to be selected from a total of 90, how many different samples can be selected? (hint: Is order important in determining the
22. Poker II Refer to Exercise 21. You have a poker hand containing “four of a kind”—a card with the same face value in each of the four suits.a. How many possible poker hands can be dealt?b. In how many ways can you receive four cards of the same face value and one card from the other 48
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