Name Period AP Statistics Unit 5 Review

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1 Name Period AP Statistics Unit 5 Review Multiple Choice 1. Jay Olshansky from the University of Chicago was quoted in Chance News as arguing that for the average life expectancy to reach 100, 18% of people would have to live to 120. What standard deviation is he assuming for this statement to make sense? a b c d Populations P1 and P2 are normally distributed and have identical means. However, the standard deviation of P1 is twice the standard deviation of P2. What can be said about the percentage of observations falling within two standard deviations of the mean for each population? a. The percentage for P1 is twice the percentage for P2. b. The percentage for P2 is twice the percentage for P1. c. The percentage for P1 is greater, but not twice as great, as the percentage for P2. d. The percentages are identical. 3. Which of the following are true statements? I. The area under the standard normal curve between 0 and 2 is twice the area between 0 and 1. II. The area under the standard normal curve between 0 and 2 is half the area between 2 and 2. III. For the standard normal curve, the interquartile range is approximately 3. a. II only b. I and II c. II and III d. I, II, and III 4. If 75% of all families spend more than $75 weekly for food, while 15% spend more than $150, what is the mean weekly expenditure and what is the standard deviation? a , b , c , d , If a couple getting married today can be expected to have 0, 1, 2, 3, 4, or 5 children with probabilities of 20%, 20%, 30%, 20% 8% and 2% respectively, what is the average number of children, to the nearest tenth, couples getting married today have? a. 1.0 b. 1.8 c. 2.0 d. 2.2 e Given a probability distribution in which the random variable assumes only the values 0, 1, 2, 3, and P which of the following must be true? suppose P, P and a. P 0 P 1 b. P 0 P 1 c. P 0P 1P d. P 0 P e. P 0 P 1P 4

2 7. Companies proved to have violated pollution laws are being fined various amounts with the following probabilities: Fine($) 1,000 10,000 50, ,000 Probability What are the mean and standard deviation for the fine variable? a. 40, 250, 39,118 b. 40, 250, 45,169 c. 23, 400, 31,350 d. 23, 400, 85,185 e. None of the above gives a set of correct answers 8. Suppose the average height of policemen is 71 inches with a standard deviation of 4 inches, while the average for policewomen is 66 inches with a standard deviation of 3 inches. If a committee looks at all ways of pairing up one male with one female officer, what will be the mean and standard deviation for the difference in heights for the set of possible partners? a. Mean of 5 inches with a standard deviation of 1 inch. b. Mean of 5 inches with a standard deviation of 2.6 inches. c. Mean of 5 inches with a standard deviation of 5 inches. d. Mean of 68.5 inches with a standard deviation of 1 inch. e. Mean of 68.5 inches with a standard deviation of 2.6 inches. 9. In a certain large population, 40% of households have a total annual income of over $70,000. A simple random sample is taken of 4 of these households. Let be the number of households in the sample with an annual income of over $70,000 and assume that the binomial assumptions are reasonable. What is the mean of? a. 1.6 b. 28,000 c d. 2, since the mean must be an integer e. The answer cannot be computed from the information given. 10. 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 b c d e. The answer cannot be computed from the given information.

3 11. 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 given information. 12. According to one poll, 12% of the public favor legalizing all drugs. In a random sample of six people, what is the probability that at least one person favors legalization? a b c d e A judge chosen at random reaches a just decision roughly 80% of the time. What is the probability that in randomly chosen cases at least two out of the three judges reach a just decision? a b c d e Which of the following is not true about the variance in a binomial distribution? a. For a fixed p, the variance increases as n increases. b. For a fixed n, the variance is maximum when p = 0.5. c. The variance depends only on n. d. The variance is constant for a specific n and p. e. None of these. 15. If the expected value of successes in binomial experiment of 100 trials is 55, the standard deviation of the number of success in approximately a b c d e. None of these 16. Let H B100,0.3and K B100,0.6 K? I. H K II. H K III. H K IV. H K a. I and III b. II and IV c. II and III d. I and IV e. They are not comparable since they are discrete.. Which of the following best describes a comparison of H and

4 17. Suppose a basketball player has a probability of scoring of 0.79 on each of his free throw attempts. Assuming that this does not change, he would like to compute the probability that he will score on 1 or more of his next 4 attempts. He should use: a. binomial techniques b. geometric techniques c. normal curve techniques d. normal opposed to binomial e. None of these Use the following information for problems A psychologist studied the number of puzzles subjects were able to solve in a five minute period while listening to soothing music. Let be the number of puzzles completed successfully by a subject. had the following distribution: Probability Using the above data, what is the probability that a randomly chosen subject completes at least 3 puzzles in the five minute period while listening to soothing music? a. 0.3 b. 0.4 c. 0.6 d. 0.9 e. The answer cannot be computed from the information given. 19. Using the above data, P 3 a. 0.3 b. 0.4 c. 0.6 d. 0.9 e. The answer cannot be computed from the information given. 20. Using the above data, the mean of is? a. 2.0 b. 2.3 c. 2.5 d. 3.0 e. The answer cannot be computed from the information given.

5 21. Which of the following random variables should be considered continuous? a. The time it takes for a randomly chosen woman to run 100 meters b. The number of brothers a randomly chosen person has c. The number of cars owned by a randomly chosen adult male d. The number of orders received by a mail order company in a randomly chosen week e. None of the above 22. Let the random variable represent the profit made on a randomly selected day by a certain store. Assume that is normal with mean $360 and standard deviation $50. What is the value of P $400? a b c d e. The answer cannot be computed from the information given. 23. 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 (heard 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 well shuffled. 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 = 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 before a defective board is found is greater than 10 is a b c d e. The answer cannot be computed from the information given.

6 Free Response Use the following information for problems The probability that 0, 1, 2, 3, or 4 people will seek treatment for the flu during any given hour at an emergency room is shown in the distribution. 25. What does the random variable count measure? P() What is the mean of (remember that you must show setup)? 27. What is the variance and standard deviation of (remember that you must show setup)? 28. If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If the person gets a 7, he wins $5. The cost to play the game is $3. Find the expectation of the game. Use the following information for problems A box contains 5 pennies, 5 dimes, 1 quarter, and 1 half dollar. You reach into the box (without looking) and select a single coin. 29. Identify the random variable. is 30. Construct a probability distribution for this data. 31. If you reach into the box and randomly select one coin, what is the probability you will get something between 5 cents and 35 cents?

7 Use the following information for problems Here is the probability distribution function for a continuous random variable. 32. P P Px Px2 36. P1 3 Use the following information for problems Suppose that the discrete random variable has the following probability distribution P() 1/4 1/4 1/2 37. Find of (remember that you must show setup). 38. Find 2 of (remember that you must show setup). 39. Define the new random variabley 3 1. Use the properties of the mean of linear functions of random variables and your results in the previous problems to find the mean of Y. 40. Use the properties of the variance of linear functions of random variables to calculate the variance and standard deviation of the new random variable Y.

8 Use the following information for problems 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. 41. Define the random variable,, being measured 42. What kind of distribution does have? 43. Determine the probability that exactly 80 subjects experience headache relief with this remedy. 44. What is the probability that between 75 and 90 (inclusive) of the patients will obtain relief? 45. The Ferrells have three children: Jennifer, Jessica, and Jaclyn. If we assume that a couple is equally likely to have a girl or boy, then how unusual is it for a family like the Ferrells to have three children who are all girls? Let = number of girls in a family of three children.

9 Use the following information for problems 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. 46. Find the probability that 18 or more of the 30 students in a particular adult Sunday School class are overweight. 47. How many Americans would you expect to weigh before you encounter the first overweight individual? 48. What is the probability that it takes more than 5 attempts before an overweight person is found? 49. As reported in Chances: Risks and Odds in Everyday Life by James Burke, a research team at Cornell University conducted a study in which they concluded that 10% of all businessmen who wear ties wear them so tight that they reduce blood flow to the brain, diminishing cerebral functions. a. Assuming the 10% figure is correct, use simulation (and the random table below) to determine the approximate probability that at a board meeting of ten businessmen, all of whom wear ties, at least two are wearing their ties too tight. Show your work b. Calculate the probability from part (a) exactly (This means from formulas, not from your simulation). Show your work.

10 50. Vocabulary: random variable discrete random variable Y B np, probability distribution expected value probability distribution function probability histogram Law of Large Numbers cumulative distribution function density curve variance binomial coefficient continuous random variable standard deviation "n choose k" uniform distribution binomial setting geometric distribution normal distribution binomial random variable Calculator functions: binompdf(n, p, ) geometpdf(p, ) binomcdf(n, p, ) geometcdf(p, ) 51. What is a discrete random variable? binomial distribution 52. If is a discrete random variable, what information does the probability distribution of give? 53. In a probability histogram what does the height of each bar represent? 54. What is a continuous random variable? 55. If is a continuous random variable, how is the probability distribution of described? 56. What is the area under a probability density curve equal to? 57. What is the difference between a discrete random variable and a continuous random variable? 58. If is a discrete random variable, do P 2and P 2have the same value? Explain. 59. If is a continuous random variable, do P 2and P 2have the same value? Explain. 60. Explain the difference between the notations x and. 61. What is meant by the expected value of? 62. How do you calculate the mean of a discrete random variable?

11 63. Explain the Law of Large Numbers. 64. Suppose 5 and Y 10. According to the rules for means, what is Y? 65. Suppose 2. According to the rules for means, what is 3 4? 66. Explain how to calculate the variance of a discrete random variable using the formula x 2 p 2 i x i 67. Given the variance of a random variable, explain how to calculate the standard deviation. 68. Suppose 2 2 and 2 Y 3and and Y are independent random variables. According to the rules for 2 variances, what is Y? What is Y? 69. Suppose 2 4. According to the rules for variances, what is 2 3 2? What is 3 2? 70. What are the four conditions for the binomial setting? 71. In the binomial distribution, what do parameters n and p represent? 72. What is meant by B n, p? 73. What is the difference between a probability distribution function and a cumulative distribution function? 74. What are the mean and standard deviation of a binomial random variable? 75. What are the four conditions for the geometric setting? 76. Explain the difference between the binomial setting and the geometric setting. n 77. If has a geometric distribution, what does 1 1 p p represent? 78. What is the expected value of a geometric random variable?

12 Selected Answers 1. A 2. D 3. A 4. C 5. B 6. C 7. C 8. C 9. A 10. D 11. C 12. C 13. E 14. C 15. C 16. B 17. A 18. B 19. C 20. B 21. A 22. A 23. C 24. D 25. The number of people who seek treatment for the flu during any given hour at the ER E ( ) , E ( ) The amount of money selected P() P P 33. P P P P 37. E ( ) Y , The number of patients that have headaches cured by the treatment 42. B100, P P P (w/Normal distribution methods) 44. (w/binomial methods), 45. P P b. P 47. E ( ) Y P Y

AP Statistics Review Ch. 6

AP Statistics Review Ch. 6 Name 1. Which of the following data sets is not continuous? a. The gallons of gasoline in a car. b. The time it takes to commute in a car. c. Number of goals scored by a hockey