Binomial Probability Analysis in AP Statistics Quiz

Quiz 8.1A AP Statistics Name: 1. A manufacturer produces a large number of toasters. From past experience, the manufacturer knows that approximately 2% are defective. In a quality control procedure, we randomly select 20 toasters for testing. We want to determine the probability that no more than one of these toasters is defective. (a) Is a binomial distribution a reasonable probability model for the random variable X? State your reasons clearly. (b) Determine the probability that exactly one of the toasters is defective. (c) Define the random variable. X = . Then find the mean and standard deviation for X. (d) Find the probability that at most two of the toasters are defective. (Include enough details so that it can be understood how you arrived at your answer.) 2. Draw a card from a standard deck of 52 playing cards, observe the card, and replace the card within the deck. Count the number of times you draw a card in this manner until you observe a jack. Is a binomial distribution a reasonable probability model for the random variable X? State your reasons clearly. Chapter 8

Quiz 8.1B AP Statistics Name: In problems 1 and 2, indicate whether a binomial distribution is a reasonable probability model for the random variable X. Give your reasons in each case. 1. The pool of potential jurors for a murder case contains 100 persons chosen at random from the adult residents of a large city. Each person in the pool is asked whether he or she opposes the death penalty. X is the number who say “Yes.” 2. Joey buys a Virginia lottery ticket every week. X is the number of times in a year that he wins a prize. 3. A fair coin is flipped 20 times. (a) Determine the probability that the coin comes up tails exactly 15 times. (b) Find the probability that the coin comes up tails at least 15 times. (Include enough details so that it can be understood how you arrived at your answer.) (c) Find the mean and standard deviation for the random variable X in this coin-flipping problem. (d) Find the probability that X takes a value within 2 standard deviations of its mean. Chapter 8

Quiz 8.1C AP Statistics Name: Ladies Home Journal magazine reported in 1993 that 66% of all dog owners greet their dog before greeting their spouse or children when they return home at the end of the workday. Suppose that 12 dog owners are selected at random. 1. Show that the four requirements for a binomial setting are satisfied. 2. Define the random variable: X = 3. Find the probability that exactly 7 of the 12 dog owners greet their dog first when they arrive home. 4. Find the probability that at least 5 of the 12 dog owners greet their dog first when they arrive home. 5. What is the expected number of dog owners who greet their dog first when they arrive home? 6. Find the mean and standard deviation for the random variable X in this problem. Chapter 8

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Quiz 8.1D AP Statistics Name: “What do you think is the ideal number of children for a family to have?”” A Gallup poll asked this question of 1006 randomly chosen adults. Almost half (49%) thought two children was ideal. Suppose that p = 0.49 is exactly true for the population of all adults. Let X = number of adults who thought that two children was ideal. 1. Is X binomial? Check the four conditions for a binomial setting. 2. Find the mean and standard deviation of X. 3. Find the probability that X = 493. 4. Find the probability that X is within two standard deviations of the mean. 5. Describe the distribution of the binomial random variable X. 6. Would it make sense to compare your results in Question 4 with the empirical rule? Explain. 7. Use the Normal approximation to find the area between X =461 and X = 525. 8. The results in Questions 4 and 7 should be extremely close. Why would this be so? Chapter 8

Quiz 8.2A AP Statistics Name: 1. In parts (a) and (b), indicate whether a geometric distribution is a reasonable probability model for the random variable X. Give your reasons in each case. (a) Suppose that one of every 100 people in a certain community is infected with HIV. You want to identify an HIV-positive person to include in a study of an experimental new drug. How many individuals would you expect to have to interview in order to find the first person who is HIV-positive? (b) In high-profile discrimination court cases in the past, 76% of prospective jurors have been found eligible to serve on juries (that is, no objection by either the prosecution or the defense). We have 25 people in the pool of potential jurors, and we want to know if we will be successful in finding 12 people to serve on the jury from the pool. Specifically, we want to determine the probability that the 12th acceptable juror is found by the time that the 25th prospective juror is interrogated. 2. When a computerized generator is used to generate random digits, the probability that any particular digit in the set {0, 1, 2, ..., 9} is generated on any individual trial is 1/10 = 0.1. Suppose that we are generating digits one at a time and are interested in tracking occurrences of the digit 0. (a) Determine the probability that the first 0 occurs as the fifth random digit generated. (b) How many random digits would you expect to have to generate in order to observe the first 0? (c) Construct a probability distribution histogram for X = 1 through X = 5. Use the grid provided. Chapter 8

Quiz 8.2B AP Statistics Name: There is a probability of 0.08 that a vaccine will cause a certain side effect. Suppose that a number of patients are inoculated with the vaccine. We are interested in the number of patients vaccinated until the first side effect is observed. 1. Define the random variable of interest. X = 2. Verify that this describes a geometric setting. 3. Find the probability that exactly 5 patients must be vaccinated in order to observe the first side effect. 4. Construct a probability distribution table for X (up through X = 5). 5. How many patients would you expect to have to vaccinate in order to observe the first side effect? 6. What is the probability that the number of patients vaccinated until the first side effect is observed is at most 5? Chapter 8

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Quiz 8.2C AP Statistics Name: Suppose that 20% of a herd of cows is infected with a particular disease. 1. What is the expected number of cows that you would have to inspect until you find a cow that has the disease? 2. Identify the random variable of interest. X = Verify that X is a geometric random variable. 3. What is the probability that the first diseased cow is the 3rd cow tested? 4. What is the probability that 6 or more cows would need to be tested until a diseased cow was found? 5. Construct a probability distribution table for X (up through X = 5). Chapter 8

Quiz 8.2D AP Statistics Name: Government data reveal that approximately 65% of all divorce cases cite incompatibility as the underlying cause. Let X be the number of divorce cases that cite incompatibility. 1. You randomly select divorce cases to study. How many cases would you expect to see until you see one that cites incompatibility as the cause? 2. What is the probability that it takes more than 4 cases to find one that cites incompatibility? 3. Construct a probability distribution table (out to n = 5) for the number of cases examined until you find one that cites incompatibility. 4. Sketch a probability histogram (out to n = 5) for your table in Question 3. Chapter 8

Test 8A AP Statistics Name: Directions: Work on these sheets. Part 1: Multiple Choice. Circle the letter corresponding to the best answer. 1. Ina large population of college students, 20% of the students have experienced feelings of math anxiety. If you take a random sample of 10 students from this population, the probability that exactly 2 students have experienced math anxiety is (a) 0.3020. (b) 0.2634. (c) 0.2013. (d) 0.5. (e) 1. 2. Refer to the previous problem. The standard deviation of the number of students in the sample who have experienced math anxiety is (a) 0.0160. (b) 1.265. (c) 0.2530. (d) 1. (e) 0.2070. 3. Ina certain large population, 40% of housecholds have a total annual income of at least $70,000. A simple random sample of 4 of these households is selected. What is the probability that 2 or more of the households in the survey have an annual income of at least $70,000? (a) 0.3456 (b) 0.4000 (c) 0.5000 (d) 0.5248 (e) The answer cannot be computed from the information given. 4. A factory makes silicon chips for use in computers. It is known that about 90% of the chips meet specifications. Every hour a sample of 18 chips is selected at random for testing. Assume a binomial distribution is valid. Suppose we collect a large number of these samples of 18 chips and determine the number meeting specifications in each sample. What is the approximate mean of the number of chips meeting specifications? (a) 16.20 (b) 1.62 (c) 4.02 (d) 16.00 (e) The answer cannot be computed from the information given. Chapter 8 1 Test 8A

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5. Which of the following are true statements? [. The expected value of a geometric random variable is determined by the formula (1-p" . II. If X is a geometric random variable and the probability of success is 0.85, then the probability distribution of X will be skewed left, since 0.85 is closer to 1 than to 0. [1I. An important difference between binomial and geometric random variables is that there is a fixed number of trials in a binomial setting, and the number of trials varies in a geometric setting. (a) I only (b) IT only (c) III only (d) I, II, and I1I (e) None of the above are true. 6. Ina group of 10 college students, 4 are business majors. You choose 3 of the 10 students at random and ask their major. The distribution of the number of business majors you choose is (a) binomial with n = 10 and p = 0.4. (b) binomial with n = 3 and p = 0.4. (c) not binomial and not geometric. (d) geometric with p = 0.4. (e) geometric with p = 0.4 and n = 10. 7. Government statistics tell us that 2 out of every 3 American adults are overweight. Let X = number of Americans that are overweight. How large would an SRS of American adults need to be in order for it to be safe to assume that the sampling distribution of X is approximately Normal? (a) 3 (b) 9 (c) 15 (d) 18 (e) 30 8. A basketball player makes 70% of her free throws. She takes 7 free throws in a game. If the shots are independent of each other, the probability that she makes the first 5 and misses the last 2 is about (a) 0.635. (b) 0.318. (c) 0.015. (d) 0.49. (e) 0.35. Chapter 8 2 Test 8A

Part 2: Free Response Answer completely, but be concise. Write sequentially and show all steps. 9. Describe the four conditions that are required for a binomial setting. 10. A quarterback completes 44% of his passes. (a) Explain how you could use a table of random digits to simulate this quarterback attempting 20 passes. (b) Explain how you could use a TI-83/84/89 to simulate this quarterback attempting 20 passes. (c) Using your scheme from either (a) or (b), simulate 20 passes. If you use the random digit table, begin on line 149. If you use the TI-83/84/89, first enter 149—rand to seed your random number generator, and indicate which one you use. List the numbers generated and circle the “successes.” Calculate the percent of passes completed. Line 149 71546 05233 53946 68743 72460 27601 45403 88692 Line 150 07511 88915 41267 16853 84569 79367 32337 03316 Chapter 8 3 Test 8A

(d) What is the probability that the quarterback throws 3 incomplete passes before he has a completion? (e) How many passes can the quarterback expect to throw before he completes a pass? (f) Use two methods to determine the probability that it takes more than 5 attempts before he completes a pass. (g) Construct a probability distribution table (out to n = 6) for the number of passes attempted before the quarterback has a completion. (h) Sketch a probability histogram (out to n = 6) for the table you constructed in the previous problem. I pledge that I have neither given nor received aid on this test. Chapter 8 4 Test 8A

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Test 8B AP Statistics Name: Directions: Work on these sheets. Part 1: Multiple Choice. Circle the letter corresponding to the best answer. 1. A dealer in the Sands Casino in Las Vegas selects 40 cards from a standard deck of 52 cards. Let Y be the number of red cards (hearts or diamonds) in the 40 cards selected. Which of the following best describes this setting? (a) Y has a binomial distribution with n = 40 observations and probability of success p = 0.5. (b) Y has a binomial distribution with n = 40 observations and probability of success p = 0.5, provided the deck is shuffled well. (c) Y has a binomial distribution with n = 40 observations and probability of success p = 0.5, provided after selecting a card it is replaced in the deck and the deck is shuffled well before the next card is selected. (d) Y has a normal distribution with mean p = 0.5. (e) Y has a geometric distribution with n = 40 observations and probability of success p = 0.5. 2. A cell phone manufacturer claims that 92% of the cell phones of a certain model are free of defects. Assuming that this claim is accurate, how many cell phones would you expect to have to test until you find a defective phone? (a) 2, because it has to be a whole number (b) 8 (c) 12.5 (d) 92 (e) 93 3. The probability that a three-year-old battery still works is 0.8. A cassette recorder requires four working batteries to operate. The state of batteries can be regarded as independent, and four three- year-old batteries are selected for the cassette recorder. What is the probability that the cassette recorder operates? (a) 0.9984 (b) 0.8000 (c) 0.5904 (d) 0.4096 (e) The answer cannot be computed from the information given. 4. Twenty percent of all trucks undergoing a certain inspection will fail the inspection. Assume that trucks are independently undergoing this inspection, one at a time. The expected number of trucks inspected before a truck fails inspection is (a) 2. (b) 4. (c) 5. (d) 20. (e) The answer cannot be computed from the information given. Chapter 8 1 Test 8B

5. Two percent of the circuit boards manufactured by a particular company are defective. If circuit boards are randomly selected for testing, the probability that the number of circuit boards inspected until a defective board is found is greater than 10 is (a) 1.024 x 10 - (b)5.12 x 10”. (c) 0.1829. (d) 0.8171. (e) The answer cannot be computed from the information given. 6. A random sample of 15 people is taken from a population in which 40% favor a particular political stand. What is the probability that exactly 6 individuals in the sample favor this political stand? (a) 0.6098 (b) 0.5000 (c) 0.4000 (d) 0.2066 (e) 0.0041 7. Experience has shown that a certain lie detector will show a positive reading (indicates a lie) 10% of the time when a person is telling the truth and 95% of the time when a person is lying. Suppose that a random sample of 5 suspects is subjected to a lie detector test regarding a recent one-person crime. Then the probability of observing no positive reading if all suspects plead innocent and are telling the truth is (a) 0.409. (b) 0.735. (c) 0.00001. (d) 0.591. (e) 0.99999. 8. Which of the following is NOT an assumption of the binomial distribution? (a) All trials must be identical. (b) All trials must be independent. (¢) Each trial must be classified as a success or a failure. (d) The number of successes in the trials is counted. (e) The probability of success is equal to 0.5 in all trials. Chapter 8 2 Test 8B

Part 2: Free Response Answer completely, but be concise. Write sequentially and show all steps. 9. A headache remedy is said to be 80% effective in curing headaches caused by simple nervous tension. An investigator tests this remedy on 100 randomly selected patients suffering from nervous tension. (a) Define the random variable being measured. X = (b) What kind of distribution does X have? (c) Calculate the mean and standard deviation of X. (d) Determine the probability that exactly 80 subjects experience headache relief with this remedy. (e) What is the probability that between 75 and 90 (inclusive) of the patients will obtain relief? Justify your method of solution. 10. The Ferrells have three children: Jennifer, Jessica, and Jaclyn. If we assume that a couple is equally likely to have a girl or a boy, then how unusual is it for a family like the Ferrells to have three children who are all girls? Let X = number of girls (in a family of three children). (a) Construct a pdf (probability distribution function) table for the variable X. (b) Construct a pdf histogram for X. Chapter 8 3 Test 8B

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(c) Construct a cdf (cumulative distribution function) table for X. (d) Construct a cdf histogram for X. (e) What is the probability that a family like the Ferrells would have three children who are all girls? 11. A survey conducted by the Harris polling organization discovered that 63% of all Americans are overweight. Suppose that a number of randomly selected Americans are weighed. (a) Find the probability that 18 or more of the 30 students in a particular adult Sunday School class are overweight. (b) How many Americans would you expect to weigh before you encounter the first overweight individual? (c) What is the probability that it takes more than 5 attempts before an overweight person is found? (d) Construct the cumulative distribution table (stop at n = 6) for the number of Americans weighed before an overweight person is found. I pledge that I have neither given nor received aid on this test. Chapter 8 K Test 8B

Test 8C AP Statistics Name: Directions: Work on these sheets. Part 1: Multiple Choice. Circle the letter corresponding to the best answer. 1. It has been estimated that about 30% of frozen chickens contain enough salmonella bacteria to cause illness if improperly cooked. A consumer purchases 12 frozen chickens. What is the probability that the consumer will have more than 6 contaminated chickens? (a) 0.961 (b) 0.118 (c) 0.882 (d) 0.039 (e) 0.079 2. Refer to the previous question. Suppose that a supermarket buys 1000 frozen chickens from a supplier. The number of frozen chickens that may be contaminated that are within two standard deviations of the mean is between A and B. The numbers A and B are (a) (90, 510) (b) (290.8, 309.2) (c) (0, 730) (d) (271, 329) (e) (255, 345) 3. Which of the following random variables is geometric? (a) The number of phone calls received in a one-hour period (b) The number of cards I need to deal from a deck of 52 cards that has been thoroughly shuffled so that at least one of the cards is a heart (c) The number of digits I will read beginning at a randomly selected starting point in a table of random digits until I finda 7 (d) The number of 7s in a row of 40 random digits (e) All of the above. 4. Suppose X has a geometric distribution with probability 0.3 of success and 0.7 of failure on each observation. The probability that X = 4 is (a) 0.0081. (b) 0.0189. (c) 0.1029. (d) 0.2401. (e) none of the above. 5. If X has a binomial distribution with » = 400 and p = 0.4, the Normal approximation for the binomial probability of the event {155 <X <175} is (a) 0.6552. (b) 0.6429. (c) 0.6078. (d) 0.6201. (e) 0.6320. Chapter 8 1 Test 8C

6. A college basketball player makes 80% of her free throws. Suppose this probability is the same for each free throw she attempts. The probability that she makes all of her first four free throws and then misses her fifth attempt this season is (a) 0.32768. (b) 0.08192. (c) 0.0064. (d) 0.0032. (¢) 0.00128. 7. In order for the random variable X to have a geometric distribution, which of the following conditions must X satisfy? I np=10andn(l —p)=10 I The number of trials is fixed. III Trials are independent. IV The probability of success has to be the same for each trial. V' The n observations have to be a random sample. (a) III and IV (b) I, IIT, IV, and V (c) I'and III (d) I, III, and V (e) II and III 8. A set of 10 cards consists of 5 red cards and 5 black cards. The cards are shuffled thoroughly and you turn cards over, one at a time, beginning with the top card. Let X be the number of cards you turn over until you observe the first red card. The probability that X is greater than 2 is (a) 0.125. (b) 0.5. (c) 0.25. (d) 0.945. (e) none of (a) to (d). 9. Suppose we select an SRS of size » = 100 from a large population having proportion p of successes. Let X be the number of successes in the sample. For which value of p would it be safe to assume the sampling distribution of X is approximately Normal? (a) 0.01 (b) 1/9 (c) 0.975 (d) 0.9999 (e) All of these 10. Suppose we roll a fair die 10 times. The probability that an even number occurs exactly the same number of times as an odd number on the 10 rolls is (a) 0.1667. (b) 0.2461. (c) 0.3125. (d) 0.5000. (e) none of these. Chapter 8 2 Test 8C

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Part 2: Free Response Answer completely, but be concise. Write sequentially and show all steps. 11. The Internal Revenue Service estimates that 8% of all taxpayers filling out long forms make mistakes. Suppose that a random sample of 10,000 forms is selected. What is the probability that more than 800 forms have mistakes? 12. A survey conducted by the Harris polling organization discovered that 63% of all Americans are overweight. Suppose that a number of randomly selected Americans are weighed. (a) Find the probability that the fourth person weighed is the first person to be overweight. (b) Find the probability that it takes more than 4 people to observe the first overweight person. (c) Find the mean and variance of the number of Americans that would have to be weighed in order to find the first person that was overweight. Chapter 8 3 Test 8C

13. Amarillo Slim, a professional dart player, has an 80% chance of hitting the bull’s eye on a dartboard with any throw. Suppose that he throws 10 darts, one at a time, at the dartboard. Let X = number of bull’s eyes. (a) Does X describe a binomial setting or a geometric setting? Justify your answer. (b) Compute the mean and variance of the number of bull’s eyes in 10 throws. (c) Find the probability that he hits the bull’s eye at least four times. (d) Now let’s suppose that Slim keeps throwing darts until he hits the bull’s eye. What’s different about this setting and the setting described above? (e) Find the probability that Slim’s first bull’s eye occurs on the fourth throw. (f) Find the probability that it takes Slim more than 2 throws to hit the bull’s eye. I pledge that I have neither given nor received aid on this test. Chapter 8 4 Test 8C

Test 8D AP Statistics Name: Directions: Work on these sheets. Part 1: Multiple Choice. Circle the letter corresponding to the best answer. 1. A basketball player makes 70% of her free throws. She takes 7 free throws in a game. If the shots are independent of each other, the probability that she makes 5 out of the 7 shots is about (a) 0.635. (b) 0.318. (c) 0.015. (d) 0.329. (e) 0.245. 2. It has been estimated that as many as 70% of the fish caught in certain areas of the Great Lakes have liver cancer due to the pollutants present. A sample of 130 fish is caught and inspected for signs of liver cancer. The number of infected fish within two standard deviations of the mean is (a) (81, 101). (b) (86, 97). (c) (63, 119). (d) (36, 146). (e) (75, 107). 3. In atriangle test a tester is presented with three food samples, two of which are alike, and is asked to pick out the odd one by tasting. If a tester has no well-developed sense of taste and can pick the odd one only by chance, what is the probability that in five trials he will make four or more correct decisions? (a) 0.045 (b) 0.004 (c) 0.041 (d) 0.959 (e) 0.955 4. A set of 10 cards consists of 5 red cards and 5 black cards. The cards are shuffled thoroughly and you turn cards over, one at a time, beginning with the top card. Let X be the number of cards you turn over until you observe the first red card. The random variable X has which of the following probability distributions? (a) The Normal distribution with mean 5 (b) The binomial distribution with p = 0.5 (c) The geometric distribution with probability of success 0.5 (d) The uniform distribution that takes value 1 on the interval from 0 to 1 (e) None of the above Chapter 8 1 Test 8D

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5. Seventeen people have been exposed to a particular disease. Each one independently has a 40% chance of contracting the disease. A hospital has the capacity to handle 10 cases of the disease. What is the probability that the hospital’s capacity will be exceeded? (a) 0.965 (b) 0.035 (c) 0.989 (d) 0.011 (e) 0.736 6. Refer to the previous problem. Planners need to have enough beds available to handle a proportion of all outbreaks. Suppose a typical outbreak has 100 people exposed, each with a 40% chance of coming down with the disease. Which is not correct? (a) This scenario satisfies the assumptions of a binomial distribution. (b) About 95% of the time, between 30 and 50 people will contract the disease. (c) Almost all of the time, between 25 and 55 people will contract the disease. (d) On average, about 40 people will contract the disease. (e) Almost all of the time, less than 40 people will be infected. 7. There are 10 patients on the neonatal ward of a local hospital who are monitored by 2 staff members. If the probability of a patient requiring emergency attention by a staff member is 0.3, what is the probability that there will not be sufficient staff to attend all emergencies? Assume that emergencies occur independently. (a) 0.3828 (b) 0.3000 (c) 0.0900 (d) 0.9100 (e) 0.6172 8. In 1989 Newsweek reported that 60% of young children have blood lead levels that could impair their neurological development. Assuming that a class in a school is a random sample from the population of all children at risk, the probability that more than 3 children have to be tested until one is found to have a blood level that may impair development is (a) 0.064. (b) 0.096. (c) 0.64. (d) 0.16. (e) 0.88. Chapter 8 2 Test 8D

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10. Part 2: Free Response Answer completely, but be concise. Write sequentially and show all steps. Would most wives marry the same man again, if given the chance? According to a poll of 608 married women conducted by Ladies Home Journal (June 1988), 80% would, in fact, marry their current husbands. Assume that the women in the sample were randomly selected from among all married women in the United States. Does the number X in the sample who would marry their husbands again have a binomial probability distribution? Explain. What kinds of Web sites do males aged 18 to 34 visit? About 50% of male Internet users in this age group visit an auction site such as eBay at least once a month. Interview a random sample of 12 male Internet users aged 18 to 34. (a) What is the distribution of the number who have visited an online auction site in the past month? (b) If you interview 12 at random, what is the mean of the count X who have visited an auction site? What is the mean of the proportion p in your sample that have visited an auction site? (c) What is the probability that at least 8 of the 12 have visited an auction site in the past month? (d) Repeat the calculations in (b) for samples of size 120 and 1200. What happens to the mean count of successes as the sample size increases? What happens to the mean proportion of successes”? Chapter 8 3 Test 8D

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11. Suppose that an elite, small, selective college admits, on average, 1 of every 7 males that apply. Let X be the number of male applicants that are admitted. (a) Show that the conditions for a geometric setting are satisfied. If you need to assume that one or more of these conditions are satisfied, state this. (b) What are the mean and standard deviation of X? Interpret the mean in the context of this problem. (c) Construct a probability distribution table and histogram, out to n = 5. (d) What is the probability that the Admissions Committee has to look at exactly 5 applicants until it finds a male student to admit? At most 5 applicants? I pledge that I have neither given nor received aid on this test. Chapter 8 4 Test 8D

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