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Chapter 8 Everything

Chapter 8 Everything - Quiz 8.1A AP Statistics Name 1 A...

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Unformatted text preview: 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. ((1) 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 0f52 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 ofa 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.) (6) 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% ofall 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 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. I. 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 1. AP Statistics Name: 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 ifwe 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. 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 of0.08 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 ofintercst. 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 Quiz 8.2C AP Statistics Name: Suppose that 20% ofa herd of cows is infected with a particular disease. 1. What is the expected number ofcows 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% ofall divorce cases cite incompatibility as the underlying cause. Let X be the number ofdivorcc 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 ofcases 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 Directions: 1. Chapter 8 1 AP Statistics Name: Work on these sheets. Part 1: Multiple Choice. Circle the letter corresponding to the best answer. In a 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. 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. ((1) 1. (e) 0.2070. In a certain large population, 40% of households have a total annual income of at least \$70,000. A simple random sample of4 of these households is selected. What is the probability that 2 or more of the households in the survey have an annual income ofat least \$70,000? (a) 0.3456 (b) 0.4000 (0) 0.5000 (d) 0.5248 (c) The answer cannot be computed from the information given. 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 (c) The answer cannot be computed from the information given. Test 8A 5. Which of the following are true statements? I. The expected value ofa geometric random variable is determined by the formula (1 —p)""p. II. IfX is a geometric random variable and the probability of success is 0.85, then the probability distribution ofX will be skewed left, since 0.85 is closer to 1 than to 0. III. 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) 11 only (c) III only (d) I, II, and III (e) None of the above are true. 6. In a 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 andp = 0.4. (b) binomial with n = 3 andp = 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 ofX is approximately Normal? (a) 3 (b) 9 (e) 15 (d) 18 (c) 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% ofhis 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. Ifyou use the TI-83/84/89, ﬁrst enter 14 9—trand 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? (0) How many passes can the quarterback expect to throw before he completes a pass? (0 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 Ilia] I have neither given nor received aid on this test. Chapter 8 4 Test 8A Test 8B Directions: 1. Chapter 8 1 AP Statistics Name: Work on these sheets. Part 1: Multiple Choice. C [role the letter corresponding to the best answer. A dealer in the Sands Casino in Las Vegas selects 40 cards from a standard deck of 52 cards. Let Y be the number ofer 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. (C) Y has a geometric distribution with n = 40 observations and probability of success p = 0.5. A cell phone manufacturer claims that 92% of the cell phones ofa 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 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 (c) The answer cannot be computed from the information given. 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. Test 83 5. 8. Chapter 8 2 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 ofcircuit boards inspected until a defective board is found is greater than 10 is (a) 1.024 x 107. (b) 5.12 x107- (e) 0.1829. (d) 0.8171. (c) The answer cannot be computed from the information given. 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 (c) 0.0041 . 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. ((1) 0.591. (e) 0.99999. Which of the following is NOT an assumption of the binomial distribution? (a) All trials must be identical. (b) All trials must be independent. (c) Each trial must be classified as a success or a failure. ((1) The number of successes in the trials is counted. (e) The probability of success is equal to 0.5 in all trials. Test 813 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 ofdistribution does X have? (c) Calculate the mean and standard deviation ofX. (d) Determine the probability that exactly 80 subjects experience headache reliefwith 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. Ifwe 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 SB (c) Construct a cdf (cumulative distribution function) table for X. ((1) 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? ((1) 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 4 Test 813 Test 8C Directions: 1. Chapter 8 1 AP Statistics Name: Work on these sheets. Part 1: Multiple Choice. C [role the lelrer corresponding to the best answer. 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) 0.1 18 (c) 0.882 (d) 0.039 (c) 0.079 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) Which ofthc 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 of52 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 find a 7 (d) The number of 73 in a row of40 random digits (0) All of the above. 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. (0) none ofthc above. IfX has a binomial distribution with n = 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. ((1) 0.6201. (c) 0.6320. Test 8C 6. 10. Chapter 8 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. (c) 0.00128. . In order for the random variable X to have a geometric distribution, which of the following conditions must X satisfy? I np 210 and n(l —p) z 10 II 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) 11, III, IV, and V (e) I and 111 (d) I, III, and V (e) 11 and III A set of 10 cards consists of 5 red cards an...
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