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Elementary Statistics 10th Edition Mario F. Triola - Solutions
19. Eye Contact In a study of facial behavior, people in a control group are timed for eye contact in a 5-minute period. Their times are normally distributed with a mean of 184.0 s and a standard deviation of 55.0 s (based on data from “Ethological Study of Facial Behavior in Nonparanoid and
18. Birth Weights Birth weights in Norway are normally distributed with a mean of 3570 g and a standard deviation of 500 g. Repeat Exercise 17 for babies born in Norway. Is the result very different from the result found in Exercise 17?
17. Birth Weights Birth weights in the United States are normally distributed with a mean of 3420 g and a standard deviation of 495 g. If a hospital plans to set up special observation conditions for the lightest 2% of babies, what weight is used for the cutoff separating the lightest 2% from the
16. Designing Caskets The standard casket has an inside length of 78 in.a. What percentage of men are too tall to fit in a standard casket, and what percentage of women are too tall to fit in a standard casket? Based on those results, does it appear that the standard casket size is adequate?b. A
15. Designing Doorways The standard doorway height is 80 in.a. What percentage of men are too tall to fit through a standard doorway without bending, and what percentage of women are too tall to fit through a standard doorway without bending? Based on those results, does it appear that the current
13. Beanstalk Club Height Requirement The Beanstalk Club, a social organization for tall people, has a requirement that women must be at least 70 in. (or 5 ft 10 in.) tall.What percentage of women meet that requirement?14. Height Requirement for Women Soldiers The U.S. Army requires women’s
Women’s heights are normally distributed with mean 63.6 in. and standard deviation 2.5 in.
12. Find the IQ score separating the top 85% from the others.In Exercises 13–16, use this information (based on data from the National Health Survey):● Men’s heights are normally distributed with mean 69.0 in. and standard deviation 2.8 in.
11. Find the IQ score separating the top 35% from the others.
10. Find P60, which is the IQ score separating the bottom 60% from the top 40%.
9. Find P10, which is the IQ score separating the bottom 10% from the top 90%.
8. Find the probability that a randomly selected adult has an IQ between 110 and 120(referred to as bright normal).
7. Find the probability that a randomly selected adult has an IQ between 90 and 110 (referred to as the normal range).
6. Find the probability that a randomly selected adult has an IQ greater than 131.5 (the requirement for membership in the Mensa organization).
5. Find the probability that a randomly selected adult has an IQ that is less than 130.
4. z Scores and Areas What is the difference between a z score and an area under the graph of a normal probability distribution? Can a z score be negative? Can an area be negative?IQ Scores. In Exercises 5–12, assume that adults have IQ scores that are normally distributed with a mean of 100 and
3. Random Digits Computers are often used to randomly generate digits of telephone numbers to be called when conducting a survey. Can the methods of this section be used to find the probability that when one digit is randomly generated, it is less than 5? Why or why not? What is the probability of
2. Normal Distributions The distribution of IQ scores is a nonstandard normal distribution with a mean of 100 and a standard deviation of 15. What are the values of the mean and standard deviation after all IQ scores have been standardized using z (x ) ?
1. Normal Distributions What is the difference between a standard normal distribution and a nonstandard normal distribution?
46. Refer to the graph of the triangular probability distribution of the continuous random variable x. (See the margin graph.)a. Find the value of the constant c.b. Find the probability that x is between 0 and 3.c. Find the probability that x is between 2 and 9.
45. Sketch a graph representing a cumulative distribution for (a) a uniform distribution and (b) a normal distribution.
44. In a continuous uniform distribution, Find the mean and standard deviation for the uniform distribution represented in Figure 6-2.
43. Assume that z scores are normally distributed with a mean of 0 and a standard deviation of 1.a. If P(0 z a) 0.3907, find a.b. If P(b z b) 0.8664, find b.c. If P(z c) 0.0643, find c.d. If P(z d) 0.9922, find d.e. If P(z e) 0.4500, find e.
42. If a continuous uniform distribution has parameters of 0 and 1, then the minimum is and the maximum isa. For this distribution, find P(1 x 1).b. Find P(1 x 1) if you incorrectly assume that the distribution is normal instead of uniform.c. Compare the results from parts (a) and
41. For a standard normal distribution, find the percentage of data that area. within 1 standard deviation of the mean.b. within 1.96 standard deviations of the mean.c. between 3 and 3 .d. between 1 standard deviation below the mean and 2 standard deviations above the mean.e. more than 2
4. Areas If you determine that for the graph of a standard normal distribution the cumulative area to the left of a z score is 0.4, what is the cumulative area to the right of that z score?
3. Standard Normal Distribution What requirements are necessary for a normal probability distribution to be a standard normal probability distribution?
2. Normal Distribution A normal distribution is informally and loosely described as a probability distribution that is “bell-shaped” when graphed. What is the “bell shape”?
1. Normal Distribution When we refer to a “normal” distribution, does the word “normal”have the same meaning as in ordinary language, or does it have a special meaning in statistics? What exactly is a normal distribution?
3. Credit Card Usage Astudent of the author conducted a survey of credit card usage by 25 of her friends. Each subject was asked how many times he or she used a credit card within the past seven days, and the results are listed as relative frequencies in the accompanying table.a. Does the table
2. Determining the Effectiveness of an HIV Training Program The New York State Health Department reports a 10% rate of the HIV virus for the “at-risk” population. In one region, an intensive education program is used in an attempt to lower that 10% rate.After running the program, a follow-up
1. Weights: Analysis of Last Digits The accompanying table lists the last digits of weights of the subjects listed in Data Set 1 in Appendix B. The last digits of a data set can sometimes be used to determine whether the data have been measured or simply reported. The presence of disproportionately
3. Reasons for Being Fired “Inability to get along with others” is the reason cited in 17%of worker firings (based on data from Robert Half International, Inc.). Concerned about her company’s working conditions, the personnel manager at the Boston Finance Company plans to investigate the five
2. TV Ratings The television show Cold Case has a 15 share, meaning that while it is being broadcast, 15% of the TV sets in use are tuned to Cold Case (based on data from Nielsen Media Research). A special focus group consists of 12 randomly selected households (each with one TV set in use during
1. Multiple-Choice Test Because they are so easy to correct, multiple-choice questions are commonly used for class tests, SAT tests, MCAT tests for medical schools, and many other circumstances. The table in the margin describes the probability distribution for the number of correct responses when
Unusual outcomes: This chapter stressed the importance of interpreting results by distinguishing between outcomes that are usual and those that are unusual. We used two different criteria: the range rule of thumb and the use of probabilities.Using the range rule of thumb to identify unusual
APoisson probability distribution applies to occurrences of some event over a specific interval, and its probabilities can be computed with Formula 5-9.
In a binomial distribution, the mean and standard deviation can be easily found by calculating the values of npand .
In a binomial distribution, there are two categories of outcomes and a fixed number of independent trials with a constant probability. The probability of x successes among n trials can be found by using the binomial probability formula, or Table A-1, or software (such as STATDISK, Minitab, or
Important characteristics of a probability distribution can be explored by constructing a probability histogram and by computing its mean and standard deviation using these formulas:
A probability distribution consists of all values of a random variable, along with their corresponding probabilities. A probability distribution must satisfy two requirements: and, for each value of x, 0 P(x) 1.
A random variable has values that are determined by chance.
16. Poisson Approximation to Binomial For a binomial distribution with n 10 and p 0.5, we should not use the Poisson approximation because the conditions n 100 and np 10 are not both satisfied. Suppose we go way out on a limb and use the Poisson approximation anyway. Are the resulting
15. Poisson Approximation to Binomial The Poisson distribution can be used to approximate a binomial distribution if n 100 and np 10. Assume that we have a binomial distribution with n 100 and p 0.1. It is impossible to get 101 successes in such a binomial distribution, but we can compute the
14. Earthquakes For a recent period of 100 years, there were 93 major earthquakes (at least 6.0 on the Richter scale) in the world (based on data from the World Almanac and Book of Facts). Assuming that the Poisson distribution is a suitable model, find the mean number of major earthquakes per
13. Homicide Deaths In one year, there were 116 homicide deaths in Richmond, Virginia(based on “A Classroom Note on the Poisson Distribution: A Model for Homicidal Deaths in Richmond, Va for 1991,” by Winston A. Richards in Mathematics and Computer Education). For a randomly selected day, find
12. Deaths from Horse Kicks A classical example of the Poisson distribution involves the number of deaths caused by horse kicks to men in the Prussian Army between 1875 and 1894. Data for 14 corps were combined for the 20-year period, and the 280 corps-years included a total of 196 deaths. After
11. Radioactive Decay Radioactive atoms are unstable because they have too much energy.When they release their extra energy, they are said to decay. When studying cesium-137, it is found that during the course of decay over 365 days, 1,000,000 radioactive atoms are reduced to 977,287 radioactive
10. Phone Calls The author found that in one month (30 days), he made 47 cell phone calls, which were distributed as follows: No calls were made on 17 days, 1 call was made on each of 7 days, 3 calls were made on each of two days, 4 calls were made on each of two days, 12 calls were made on one
9. Dandelions Dandelions are studied for their effects on crop production and lawn growth. In one region, the mean number of dandelions per square meter was found to be 7.0 (based on data from Manitoba Agriculture and Food).a. Find the probability of no dandelions in an area of 1 m2.b. Find the
7. If 0.5, find P(3). 8. If 3.25, find P(5).In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.
5. If 5, find P(4). 6. If 3 4, find P(2).
4. Poisson Binomial An experiment involves rolling a die 6 times and counting the number of 2s that occur. If we calculate the probability of x 0 occurrences of 2 using the Poisson distribution, we get 0.368, but we get 0.335 if we use the binomial distribution. Which is the correct probability
2. Poisson Distribution The random variable x represents the number of phone calls received in an hour, and it has a Poisson distribution with a mean of 9. What is its standard deviation? What is its variance?3. Parameters When attempting to apply the Poisson distribution, which of the following
1. Poisson Distribution What are the conditions for using the Poisson distribution?
22. Acceptable Defective Products Mario’s Pizza Parlor has just opened. Due to a lack of employee training, there is only a 0.8 probability that a pizza will be edible. An order for five pizzas has just been placed. What is the minimum number of pizzas that must be made in order to be at least
21. Using the Empirical Rule An experiment is designed to test the effectiveness of the MicroSort method of gender selection, and 100 couples try to have baby girls using the MicroSort method. Assume that boys and girls are equally likely and also assume that the method of gender selection has no
20. Test of Touch Therapy Nine-year-old Emily Rosa conducted this test: A professional touch therapist put both hands through a cardboard partition and Emily would use a coin toss to randomly select one of the hands. Emily would place her hand just above the hand of the therapist, who was then
19. Cholesterol Drug In a clinical trial of Lipitor (atorvastatin), a common drug used to lower cholesterol, 863 patients were given a treatment of 10-mg atorvastatin tablets.That group consists of 19 patients who experienced flu symptoms (based on data from Pfizer, Inc.). The probability of flu
18. Cell Phones and Brain Cancer In a study of 420,095 cell phone users in Denmark, it was found that 135 developed cancer of the brain or nervous system. If we assume that such cancer is not affected by cell phones, the probability of a person having such a cancer is 0.000340.a. Assuming that cell
17. Voting In a past presidential election, the actual voter turnout was 61%. In a survey, 1002 subjects were asked if they voted in the presidential election.a. Find the mean and standard deviation for the numbers of actual voters in groups of 1002.b. In the survey of 1002 people, 701 said that
16. Mendelian Genetics When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 580 peas, and Mendel theorized that 25%of them would be yellow peas.a. If Mendel’s theory is correct, find the mean and standard deviation for the numbers of yellow peas in
15. Deciphering Messages The Central Intelligence Agency has specialists who analyze the frequencies of letters of the alphabet in an attempt to decipher intercepted messages.In standard English text, the letter r is used at a rate of 7.7%.a. Find the mean and standard deviation for the number of
14. Gender Selection In a test of the MicroSort method of gender selection, 51 babies are born to couples trying to have baby boys, and 39 of those babies are boys (based on data from the Genetics & IVF Institute).a. If the gender-selection method has no effect and boys and girls are equally
13. Gender Selection In a test of the MicroSort method of gender selection, 325 babies are born to couples trying to have baby girls, and 295 of those babies are girls (based on data from the Genetics & IVF Institute).a. If the gender-selection method has no effect and boys and girls are equally
12. Are 14% of M&M Candies Yellow? Mars, Inc., claims that 14% of its M&M plain candies are yellow, and a sample of 100 such candies is randomly selected.a. Find the mean and standard deviation for the number of yellow candies in such groups of 100.b. Data Set 13 in Appendix B consists of a random
11. Are 20% of M&M Candies Orange? Mars, Inc., claims that 20% of its M&M plain candies are orange, and a sample of 100 such candies is randomly selected.a. Find the mean and standard deviation for the number of orange candies in such groups of 100.b. Data Set 13 in Appendix B consists of a random
10. Guessing Answers Several economics students are unprepared for a multiple-choice quiz with 25 questions, and all of their answers are guesses. Each question has five possible answers, and only one of them is correct.>>>m 1 2s m 2 2s m s sm s continued 228 Chapter 5 Discrete Probability
9. Guessing Answers Several psychology students are unprepared for a surprise true false test with 16 questions, and all of their answers are guesses.a. Find the mean and standard deviation for the number of correct answers for such students.b. Would it be unusual for a student to pass by guessing
4. Mean and Standard Deviation A researcher conducts an observational study, then uses the methods of this section to find that the mean is 5.0 while the standard deviation is -2.0. What is wrong with these results?Finding u, , and Unusual Values. In Exercises 5–8, assume that a procedure yields
3. Variance A researcher plans an experimental design in such a way that when randomly selecting treatment groups of people, the mean number of females is 3.0 and the standard deviation is 1.2 females. What is the variance? (Express the answer including the appropriate units.)
2. Identifying Unusual Values A manufacturing process has a defect rate of 10%, meaning that 10% of the items produced are defective. If batches of 80 items are produced, the mean number of defects per batch is 8.0 and the standard deviation is 2.7. Would it be unusual to get only five defects in a
1. Identifying Unusual Values If we consider an experiment of generating 100 births and recording the genders of the babies, the mean number of girls is 50 and the standard deviation is 5 girls. Would it be unusual to get 70 girls in 100 births? Why or why not?
36. Improving Quality The Write Right Company manufactures ballpoint pens and has been experiencing a 5% rate of defective pens. Modifications are made to the manufacturing process in an attempt to improve quality, and the manager claims that the modified procedure is better, because a test of 50
35. Identifying Gender Discrimination After being rejected for employment, Kim Kelly learns that the Bellevue Credit Company has hired only two women among the last 20 new employees. She also learns that the pool of applicants is very large, with an approximately equal number of qualified men and
34. Author’s Slot Machine The author purchased a slot machine that is configured so that there is a 1 2000 probability of winning the jackpot on any individual trial. Although no one would seriously consider tricking the author, suppose that a guest claims that she played the slot machine 5 times
33. Overbooking Flights Air America has a policy of booking as many as 15 persons on an airplane that can seat only 14. (Past studies have revealed that only 85% of the booked passengers actually arrive for the flight.) Find the probability that if Air America books 15 persons, not enough seats
32. Affirmative Action Programs A study was conducted to determine whether there were significant differences between medical students admitted through special programs(such as affirmative action) and medical students admitted through the regular admissions criteria. It was found that the
31. Acceptance Sampling The Medassist Pharmaceutical Company receives large shipments of aspirin tablets and uses this acceptance sampling plan: Randomly select and test 24 tablets, then accept the whole batch if there is only one or none that doesn’t meet the required specifications. If a
30. IRS Audits The Hemingway Financial Company prepares tax returns for individuals.(Motto: “We also write great fiction.”) According to the Internal Revenue Service, individuals making $25,000$50,000 are audited at a rate of 1%. The Hemingway Company prepares five tax returns for individuals
29. TV Viewer Surveys The CBS television show 60 Minutes has been successful for many years. That show recently had a share of 20, meaning that among the TV sets in use, 20% were tuned to 60 Minutes (based on data from Nielsen Media Research). Assume that an advertiser wants to verify that 20%
28. Find the probability that at least one subject experiences headaches. Is it unusual to have at least one of six subjects experience headaches?
27. Find the probability that more than one subject experiences headaches. Is it unusual to have more than one of six subjects experience headaches?
26. Find the probability that at most two subjects experience headaches. Is it unusual to have at most two of six subjects experience headaches?
25. Find the probability that at least five of the subjects experience headaches. Is it unusual to have at least five of six subjects experience headaches?
14. Finding Probabilities When Guessing Answers A test consists of multiple-choice questions, each having four possible answers (a,b, c, d), one of which is correct. Assume that you guess the answers to six such questions.a. Use the multiplication rule to find the probability that the first two
13. Finding Probabilities When Guessing Answers Multiple-choice questions each have five possible answers (a,b, c,d, e), one of which is correct. Assume that you guess the answers to three such questions.a. Use the multiplication rule to find the probability that the first two guesses are wrong and
12. Determining whether each of 500 defibrillators is acceptable or defective
11. Surveying 250 married couples and recording whether there is a “yes” response when they are asked if they have any children
10. Recording the number of children in 250 families
9. Recording the genders of 250 newborn babies
8. Treating 50 smokers with Nicorette and recording whether there is a “yes” response when they are asked if they experience any mouth or throat soreness
7. Treating 50 smokers with Nicorette and asking them how their mouth and throat feel
6. Surveying 12 jurors and recording whether there is a “no” response when they are asked if they have ever been convicted of a felony
5. Randomly selecting 12 jurors and recording their nationalities
4. Binomial Probabilities When trying to find the probability of getting exactly two 6s when a die is rolled five times, why can’t the answer be found as follows: Use the multiplication rule to find the probability of getting two 6s followed by three outcomes that are not 6, which is (1 6)(1 6)(5
3. Table A-1 Because the binomial probabilities in Table A-1 are so easy to find, why don’t we use that table every time that we need to find a binomial probability?and the counting rule for the number of arrangements of n items when x of them are identical to each other and the other n x are
2. Independence Assume that we want to use the binomial probability distribution for analyzing the genders when 12 jurors are randomly selected from a large population of potential jurors. If selection is made without replacement, are the selections independent?Can the selections be treated as
1. Notation When using the binomial probability distribution for analyzing guesses on a multiple-choice quiz, what is wrong with letting p denote the probability of getting a correct answer while x counts the number of wrong answers?
28. Labeling Dice to Get a Uniform Distribution Assume that you have two blank dice, so that you can label the 12 faces with any numbers. Describe how the dice can be labeled so that, when the two dice are rolled, the totals of the two dice are uniformly distributed so that the outcomes of 1, 2, 3,
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