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Introduction To Probability And Statistics 15th Edition William Mendenhall Iii , Robert Beaver , Barbara Beaver - Solutions

6.84 More best of five Refer to the previous exercise which asked for the distribution of the number of games played in a best of 5 series when team A wins with probability 50%.a. Find the expected number of games played in a best of five series.b. Find the expected number of games played when team
6.85 Family size in Gaza The Palestinian Central Bureau of Statistics (www.pcbs.gov.ps) asked mothers of age 20–24 about the ideal number of children. For those living on the Gaza Strip, the probability distribution is approximately P112 = 0.01, P122 = 0.10, P132 = 0.09, P142 = 0.31, P152 = 0.19,
6.86 Longest streak made In basketball, when the probability of making a free throw is 0.50 and successive shots are independent, the probability distribution of the longest streak of shots made has m = 4 for 25 shots, m = 5 for 50 shots, m = 6 for 100 shots, and m = 7 for 200 shots.a. How does the
5.75 Health insurance According to a 2006 census bureau report, 59% of Americans have private health insurance, 25%have government health insurance (Medicare or Medicaid or military health care), and 16% have no health insurance.a. Estimate the probability that a patient has health insurance.b.
5.83 A dice game Consider a game in which you roll two dice, and you win if the total is 7 or 11 and you lose if the total is 2, 3, or 12. You keep rolling until one of these totals occurs. Using conditional probability, find the probability that you win.
5.101 FIFA World Cup In the FIFA World Cup, 32 teams play 64 matches. Of these, 50% qualify for the round of 16 series of matches. Half of the teams in the round of 16 continue to the quarterfinals, and of these, 50% continue to the semifinals. The winners of the semifinals compete in the final
5.102 How good is a probability estimate? In Example 8 about Down syndrome, we estimated the probability of a positive test result (predicting that Down syndrome is present) to be P1POS2 = 0.257, based on observing 1355 positive results in 5282 observations. How good is such an estimate?From
5.103 Protective bomb Before the days of high security at airports, there was a legendary person who was afraid of traveling by plane because someone on the plane might have a bomb. He always brought a bomb himself on any plane flight he took, believing that the chance would be astronomically small
5.104 Streak shooter Sportscaster Maria Coselli claims that players on the New York Knicks professional basketball team are streak shooters. To make her case, she looks at the statistics for all the team’s players over the past three games and points out that one of them (Joe Smith)made six shots
5.105 Multiple choice Choose ALL correct responses. For two events A and B, P1A2 = 0.5 and P1B2 = 0.2. Then P(A or B) equalsa. 0.10, if A and B are independentb. 0.70, if A and B are independentc. 0.60, if A and B are independentd. 0.70, if A and B are disjoint
5.106 Multiple choice Which of the following is always true?a. If A and B are independent, then they are also disjoint.b. P1A•B2 + P1A•Bc2 = 1c. If P1A•B2 = P1B•A2, then A and B are independent.d. If A and B are disjoint, then A and B cannot occur at the same time.
5.107 Multiple choice: Coin flip A balanced coin is flipped 100 times. By the law of large numbers:a. There will almost certainly be exactly 50 heads and 50 tails.b. If we got 100 heads in a row, almost certainly the next flip will be a tail.c. For the 100 flips, the probability of getting 100
5.108 Multiple choice: Dream come true You have a dream in which you see your favorite movie star in person. The very next day, you are visiting Manhattan and you see her walking down Fifth Avenue.a. This is such an incredibly unlikely coincidence that you should report it to your local
5.109 Multiple choice: Comparable risks Mammography is estimated to save about 1 life in every 1000 women.“Participating in annual mammography screening … has roughly the same effect on life expectancy as reducing the distance one drives each year by 300 miles” (Gigerenzer 2002, pp. 60, 73).
5.110 True or false Answer true of false for each part.a. When you flip a coin ten times, you are more likely to get the sequence HHHHHTTTTT than the sequence HHHHHHHHHH.b. When you flip a coin ten times, you are more likely to get a sequence that contains five heads than a sequence that contains
5.111 Obesity in Australia A report published in 2016 (www.huffingtonpost.com.au) showed that 63.4% of Australian adults were overweight or obese in 2014–2015. When you randomly select two adults from the entire population for estimating the percentage of obese Australians, what is wrong in the
5.112 Driving versus flying In the United States in 2002, about 43,000 people died in auto crashes and 0 people died in commercial airline accidents. G. Gigerenzer (2002, p. 31)states, “The terrorist attack on September 11, 2001, cost the lives of some 3,000 people. The subsequent decision of
5.113 Prosecutor’s fallacy An eyewitness to the crime says that the person who committed it was male, between 15 and 20 years old, Hispanic, and drove a blue Honda. The prosecutor points out that a proportion of only 0.001 people living in that city match all those characteristics, and one of
5.114 Generalizing the addition rule For events A, B, and C such that each pair of events is disjoint, use a Venn diagram to explain why P1A or B or C2 = P1A2 + P1B2 + P1C2.
5.115 Generalizing the multiplication rule For events A, B, and C, explain why P1A and B and C2 =P1A2 * P1B•A2 * P1C•A and B2.
5.116 Bayes’ rule Suppose we know P1A2, P1B•A2, and P1Bc •Ac2, but we want to find P1A•B2. a. Using the definition of conditional probability for P(A|B) and for P(B|A), explain why P(A|B) - P(A and B)/P(B) [P(A)P(B|A)]/P(B). b. Splitting the event that B occurs into two parts, according to
5.100 Marijuana leads to heroin? Nearly all heroin addicts have used marijuana sometime in their lives. So, some argue that marijuana should be illegal because marijuana users are likely to become heroin addicts. Use a Venn diagram to illustrate the fallacy of this argument by sketching sets for M
5.99 Mrs. Test Mrs. Test (see www.mrstest.com) sells diagnostic tests for various conditions. Its website gives only imprecise information about the accuracy of the tests. The test for pregnancy is said to be “over 99% accurate.” Describe at least four probabilities to which this could refer.
5.82 Waste dump sites A federal agency is deciding which of two waste dump projects to investigate. A top administrator estimates that the probability of federal law violations is 0.30 at the first project and 0.25 at the second project. Also, he believes the occurrences of violations in these two
5.84 No coincidences Over time, you have many conversations with a friend about your favorite actress, favorite musician, favorite book, favorite TV show, and so forth for 100 topics. On any given topic, there’s only a 0.02 probability that you agree. If you did agree on a topic, you would
5.85 Amazing roulette run? A roulette wheel in Monte Carlo has 18 even-numbered slots, 18 odd-numbered slots, a slot numbered zero, and a double zero slot. On August 18, 1913, it came up even 26 times in a row.8 As more and more evens occurred, the proportion of people betting on an odd outcome
5.86 Death penalty and false positives For the decision about whether to convict someone charged with murder and give the death penalty, consider the variables reality (defendant innocent, defendant guilty) and decision (convict, acquit).a. Explain what the two types of errors are in this
5.87 Screening smokers for lung cancer An article about using a diagnostic test (helical computed tomography) to screen adult smokers for lung cancer warned that a negative test may cause harm by providing smokers with false reassurance, and a false-positive test results in an unnecessary operation
5.88 Screening for heart attacks Biochemical markers are used by emergency room physicians to aid in diagnosing patients who have suffered acute myocardial infarction(AMI), or what’s commonly referred to as a heart attack.One type of biochemical marker used is creatine kinase (CK). Based on a
5.89 Screening for colorectal cancer Gigerenzer (2002, p. 105) reported that on the average, “Thirty out of every 10,000 people have colorectal cancer. Of these 30 people with colorectal cancer, 15 will have a positive hemoccult test. Of the remaining 9,970 people without colorectal cancer, 300
5.90 Color blindness For genetic reasons, color blindness is more common in men than women: 5 in 100 men and 25 in 10,000 women suffer from color blindness.a. Define events and identify in words these proportions as conditional probabilities.b. If the population is half male and half female, what
5.91 HIV testing For a combined ELISA-Western blot blood test for HIV positive status, the sensitivity is about 0.999 and the specificity is about 0.9999 (Gigerenzer 2002, pp. 124, 126).a. Consider a high-risk group in which 10% are truly HIV positive. Construct a tree diagram to summarize this
5.92 Prostate cancer A study of the PSA blood test for diagnosing prostate cancer in men (by R. M. Hoffman et al., BMC Family Practice, vol. 3, p. 19, 2002) used a sample of 2620 men who were 40 years and older. When a positive diagnostic test result was defined as a PSA reading of at least 4, the
5.94 Win again Exercise 5.65 discussed how to use simulation and the table of random digits to estimate an expected value. Referring to the previous exercise, conduct a simulation consisting of at least 20 repetitions to estimate the expected number of combo meals one would need to purchase to win
5.95 Simulate law of large numbers Using the Random Numbers web app (or other software), simulate the flipping of balanced and biased coins.a. Report the proportion of heads after (i) 10 flips,(ii) 100 flips, (iii) 1000 flips, and (iv) 10,000 flips.(Select the Coin Flips tab in the app. You can
5.96 Illustrate probability terms with scenariosa. What is a sample space? Give an example of a sample space for a scenario involving (i) a designed experiment and (ii) an observational study.b. What are disjoint events? Give an example of two events that are disjoint.c. What is a conditional
5.97 Short term versus long run According to countrymeters.info, the American population in 2016 consisted of 49.4%males and 50.6% females. To illustrate how short-term aberrations do not affect the long run, suppose that you select 10 Americans and you get 10 males.a. Find the cumulative
5.98 Risk of space shuttle After the Columbia space shuttle disaster, a former NASA official who faulted the way the agency dealt with safety risk warned (in an AP story, March 7, 2003) that NASA workers believed, “If I’ve flown 20 times, the risk is less than if I’ve flown just once.”a.
5.117 Simulating matching birthdays Do you find it hard to believe that the probability of at least one birthday match in a class of 25 students is 0.57? Let’s simulate the answer.Using the Random Numbers web app accessible from the book’s website, each student in the class should simulate 25
6.33 SAT versus ACT SAT math scores follow a normal distribution with an approximate m = 500 and s = 100.Also ACT math scores follow a normal distrubution with an approximate m = 21 and s = 4.7. You are an admissions officer at a university and have room to admit one more student for the upcoming
6.17 Probability in graph For the normal distributions shown below, use Table A, software, or a calculator to find the probability that an observation falls in the shaded region. a. P b. +0.670 -0.50 +0.5
6.18 Empirical rule Verify the empirical rule by using Table A, software, or a calculator to show that for a normal distribution, the probability (rounded to two decimal places) withina. 1 standard deviation of the mean equals 0.68.b. 2 standard deviations of the mean equals 0.95.c. 3 standard
6.19 Central probabilities For a normal distribution, use Table A to verify that the probability (rounded to two decimal places) withina. 1.64 standard deviations of the mean equals 0.90.b. 2.58 standard deviations of the mean equals 0.99.c. Find the probability that falls within 0.67 standard
6.20 z-score for given probability in tails For a normal distribution,a. Find the z-score for which a total probability of 0.04 falls more than z standard deviations (in either direction)from the mean, that is, below m - zs or above m + zs.b. For this z-score, explain why the probability of values
6.21 Probability in tails for given z-score For a normal distribution,a. Show that a total probability of 0.01 falls more than z = 2.58 standard deviations from the mean.b. Find the z-score for which the two-tail probability that falls more than that many standard deviations from the mean in either
6.23 z-score and central probability Find the z-score such that the interval within z standard deviations of the mean(between m - zs and m + zs) for a normal distribution containsa. 50% of the probability.b. 90% of the probability.c. Sketch the two cases on a single graph.
6.24 U.S. Air Force To join the U.S. Air Force as an officer, you cannot be younger than 18 or older than 34 years of age. The distribution of age of Americans in 2012 was normal with m = 38 years and s = 22.67 years. What proportion of U.S. citizens are not eligible to serve as an officer due to
6.25 Blood pressure A World Health Organization study(the MONICA project) of health in various countries reported that in Canada, systolic blood pressure readings have a mean of 121 and a standard deviation of 16. A reading above 140 is considered high blood pressure.a. What is the z-score for a
6.26 Coffee Machine Suppose your favorite coffee machine offers 12 ounce cups of coffee. The actual amount of coffee put in the cup by the machine varies according to a normal distribution, with mean equal to 13 ounces and standard deviation equal to 0.6 ounces. For each question below, sketch a
6.27 Lifespan of phone batteries Most phones use lithium-ion(Li-ion) batteries. These batteries have a limited number of charge and discharge cycles, usually falling between 300 and 500. Beyond this lifespan, a battery gradually diminishes below 50% of its original capacity.a. Suppose the
6.28 Birth weight for boys In the United States, the mean birth weight for boys is 3.41 kg, with a standard deviation of 0.55 kg. (Source: cdc.com.) Assuming that the distribution of birth weight is approximately normal, find the following using a table, calculator, or software.a. A baby is
6.30 Quartiles and outliers For an approximately normally distributed random variable X with a mean of 200 and a standard deviation of 36,a. Find the z-score corresponding to the lower quartile and upper quartile of the standard normal distribution.b. Find and interpret the lower quartile and upper
6.31 April precipitation Over roughly the past 100 years, the mean monthly April precipitation in Williamstown, Massachusetts, equaled 3.6 inches with a standard deviation of 1.6 inches. (Source: http://web.williams.edu/weather/)a. In April 1983, the wettest April on record, the precipitation
6.32 Automatic filling machine A machine is programmed to fill packets with 500 grams of nuts. It is known from previous experiences the net weight of nuts in the packets are normally distributed with m = 502 grams and s = 3 grams. A packet is considered conformant with the weight specifications if
6.16 Probabilities in tails For a normal distribution, use Table A, software, or a calculator to find the probability that an observation isa. at least 1 standard deviation above the mean.b. at least 1 standard deviation below the mean.c. within 1 standard deviation of the mean.For each part,
6.15 TV watching A social scientist uses the General Social Survey (GSS) to study how much time per day people spend watching TV. The variable denoted by TVHOURS at the GSS Web site measures this using the discrete values 0, 1, 2, c , 24.a. Explain how, in theory, TV watching is a continuous random
6.14 Uniform distribution A random number generator is used to generate a real number between 0 and 1, equally likely to fall anywhere in this interval of values. (For instance, 0.3794259832c is a possible outcome.)a. Sketch a curve of the probability distribution of this random variable, which is
5.119 Which tennis strategy is better? A tennis match can consist of the best of three sets (that is, the winner is the first to win two sets) or the best of five sets (the winner is the first to win three sets). Which would you be better off playing if you are the weaker player and have
5.118 Simulate table tennis In a table tennis game, the first person to get at least 11 points while being ahead of the opponent by at least two points wins the game. In games between you and an opponent, suppose successive points are independent and that the probability of your winning any given
5.120 Saving a business The business you started last year has only $5000 left in capital. A week from now, you need to repay a $10,000 loan or you will go bankrupt. You see two possible ways to raise the money you need. You can ask a large company to invest the $10,000, wooing it with the $5000
6.1 Rolling dicea. State in a table the probability distribution for the outcome of rolling a balanced die. (This is called the uniform distribution on the integers 1, 2,c, 6.)b. Two balanced dice are rolled. Show that the probability distribution for X = total on the two dice is as shown in the
6.2 Dental Insurance You plan to purchase dental insurance for your three remaining years in school. The insurance makes a one-time payment of $1,000 in case of a major dental repair (such as an implant) or $100 in case of a minor repair (such as a cavity). If you don’t need dental repair over
6.3 San Francisco Giants hitting The table shows the probability distribution of the number of bases for a randomly selected time at bat for a San Francisco Giants player in 2010 (excluding times when the player got on base because of a walk or being hit by a pitch). In 74.29% of the at-bats the
6.4 Basketball shots To win a basketball game, two competitors play three rounds of one three-point shot each. The series ends if one of them scores in a round but the other misses his shot or if both get the same result in each of the three rounds. Assume competitors A and B have 30% and 20% of
6.5 WhatsApp reviews 71% of WhatsApp users have given it a five-star rating on Google Play. Of the remaining users, 15%, 6%, 3%, and 5% have given ratings of four, three, two, and one stars, respectively to the application.a. Specify the probability distribution for the number of stars as rated by
6.6 Selling houses Let X represent the number of homes a real estate agent sells during a given month. Based on previous sales records, she estimates that P102 = 0.68, P112 = 0.19, P122 = 0.09, P132 = 0.03, P142 = 0.01, with negligible probability for higher values of x.a. Explain why it does not
6.7 Playing the lottery The state of Ohio has several statewide lottery options. One is the Pick 3 game in which you pick one of the 1000 three-digit numbers between 000 and 999. The lottery selects a three-digit number at random.With a bet of $1, you win $500 if your number is selected and nothing
6.13 Selling at the right price An insurance company wants to examine the views of its clients about the prices of three car insurance plans launched last year. It conducts a survey with two sets of plans with different prices and finds that:• If plan A is sold for $150, plan B for $250, and plan
6.12 Buying on eBay You are watching two items posted for sale on eBay and bid $30 for the first and $20 for the second item. You estimate that you are going to win the first bid with probability 0.1 and the second bid with probability 0.2, and you assume that winning the two bids are independent
6.11 Profit and the weather From past experience, a wheat farmer living in Manitoba, Canada, finds that his annual profit (in Canadian dollars) is $80,000 if the summer weather is typical, $50,000 if the weather is unusually dry, and $20,000 if there is a severe storm that destroys much of his
6.10 Ideal number of children Let X denote the response of a randomly selected person to the question, “What is the ideal number of children for a family to have?” The probability distribution of X in the United States is approximately as shown in the table, according to the gender of the
6.9 More Roulette The previous exercise on roulette described two bets: one bet on the single number 23 with winnings of either $350 or - $10 and a different bet on black with winnings of either $10 or - $10. For both types of bets, the expected winning is - $0.53. Which of the two bets has the
Find the value of z—say c—such that .95 of the area is within 6c standard deviations of the mean
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 12. P(-2.58 < < 2.58)
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 10. P(z
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 11. P(z > 1.96)
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 13. To the left of 1.6
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 14. To the left of 1.83
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 15. To the right of -1.83
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 16. To the left of 4.18
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 17. To the right of -1.96
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 9. P(z
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 8. P(z > 2.81)
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 7. P(z
Let x be a normally distributed random variable with a mean of 10 and a standard deviation of 2. Find the probability that x lies between 11 and 13.6.
Studies show that highway gas mileage for compact cars sold in the United States is normally distributed, with a mean of 35.5 miles per gallon (mpg) and a standard deviation of 4.5 mpg.What percentage of compacts get 40 mpg or more?
Refer to Example 6.9. An automobile manufacturer wants to produce a car that has substantially better fuel economy than the competitors’ cars. Specifically, he wants to develop a compact car that outperforms 95% of the current compacts in fuel economy. What must the highway gas mileage rate for
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 1. P(z
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 2. P(z1.16).
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 3. P(-2.33 < < 2.33)
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 4. P(z
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 5. P(z > 5)
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 6. P(-3 <
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 18. Between -1.4 and 1.4
Let z be a standard normal random variable with mean m =0 and standard deviation s =1. Use Table 3 in Appendix I to find the probabilities 19. Between -1.43 and .68
Let z be a standard normal random variable with mean m =0 and standard deviation s =1.Find the value c that satisfies the inequalities 31. P(-c < < c) = .8262
Let z be a standard normal random variable with mean m =0 and standard deviation s =1.Find the value c that satisfies the inequalities 32. The area to the left of c is .9505
Let z be a standard normal random variable with mean m =0 and standard deviation s =1.Find the value c that satisfies the inequalities 33. The area to the left of c is .05.
Let z be a standard normal random variable with mean m =0 and standard deviation s =1.Find the value c that satisfies the inequalities 34. P(-c
Let z be a standard normal random variable with mean m =0 and standard deviation s =1.Find the value c that satisfies the inequalities 35. P(-c

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