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Name ___Answer Key_____________________ March 5, 2012 AP Statistics Unit 5 Test (Ch.7 & 8) Multiple Choice 1. A basketball player makes 160 out of 200 free throws. We would estimate the probability that the player makes his next free throw to be a. b. c. d. e. 0.16 0.50 0.80 1.2 None of these 2. 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 if 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. None of these 3. A random variable is a. b. c. d. e. a hypothetical list of the possible outcomes of a random phenomenon. any phenomenon in which outcomes are equally likely. any number that changes in a predictable way in the long run. a variable whose value is a numerical outcome of a random phenomenon. None of these 4. In a certain 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 X be the number of households on the sample with an annual income of over $70,000 and assume that the binomial assumptions are reasonable. What is the mean of X? a. b. c. d. e. 1.6 28,000 0.96 2, since the mean must be an integer The answer cannot be computed from the information given. 5. 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 the 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. 0.9984 0.8000 0.5904 0.4096 The answer cannot be computed from the information given. 6. Which of the following random variables should be considered continuous? a. b. c. d. The time it takes for a randomly chosen woman to run 100 meters. The number of brothers a randomly chosen person has. The number of cars owned by a randomly chosen adult male. The number of orders received by a mail order company in a randomly chosen week. e. None of these 7. Twenty percent of all trucks undergoing a certain inspection will fail the inspection. Assume the trucks are independently undergoing this inspection, one at a time. The expected number of trucks inspected before a truck fails this inspection is a. b. c. d. e. 2 4 5 20 The answer cannot be computed from the information given. 8. In a population of students, the number of calculators owned is a random variable X with P(X = 0) =.2, P(X = 1) = .6, and P(X = 2) = .2. The mean of this probability distribution is a. b. c. d. e. 0 2 1 0.5 None of those 9. The financial aid office at a state university conducts a study to determine the total student costs per semester. All students are charged $4500 for tuition. The mean cost for books is $350 with a standard deviation of $65. The mean outlay for room and board is $2800 with a standard deviation of $380. The mean personal expenditure is $675 with a standard deviation of $125. Assuming independence among categories, what is the standard deviation of the total student costs? a. b. c. d. e. $24 $91 $190 $405 $570 10. Two percent of the circuit boards manufactured by a particular company are defective. If the 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. 1.024 x 10^7 5.12 x 10 ^7 0.1829 0 .8171 The answer cannot be computed from the information given. 11. In a particular game, a fair die is tossed. If the number of spots showing is either 4 or 5 you win $1, if the number of spots showing is 6 you win $4, and if the number of spots showing is1, 2, or 3 you win nothing. Let X be the amount that you win. The expected value of X is a. b. c. d. e. $0 $1.00 $2.50 $4.00 None of these 12. 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. b. c. d. e. 0.3020 0.2634 0.2013 0.5 0.1 13. Refer the previous problem. The standard deviation of the number of students in the sample who have experienced math anxiety is a. b. c. d. e. 0.0160 1.265 0.2530 1 0.2070 14. Let the random variable X represent the profit made on a randomly selected day by a certain store. Assume that X is normal with the mean $360 and a standard deviation $50. What is the value of P(X > $400)? a. b. c. d. e. 0.8459 0.7881 0.2881 0.2119 The answer cannot be computed. 15. Which of the following are true statements? I. II. III. a. b. c. d. e. The expected value of a geometric random variable is determined by the formula (1 p)n 1 p . If X is a geometric random variable and the probability of success is .85, then the probability distribution of X will be skewed left, since .85 is closer to 1 than to 0. An important difference between binomial and geometric random variables is that there is a fixed number of trails in a binomial setting, and the number of trials varies in a geometric setting. I only II only III only I, II, and III None of the above gives the complete set of true responses. 16. A small store keeps track of the number X of customers that make a purchase during the first hour that the store is open each day. Based on the records, X has the following probability distribution. X 0 p(X) 0.1 1 0.1 2 0.1 3 0.1 4 0.6 The standard deviation of the number of customers that make a purchase during the first hour that the store is open is a. b. c. d. e. 1.4 2.0 3.0 4.0 None of these. 17. Suppose there are 3 balls in a box. On one of the balls is the number 1, on another is the number 2, and on the third is the number 3. You select two balls at random and without replacement from the box and note the two numbers observed. The sample space S consists of the three equally likely outcomes {(1,2), (1,3), (2,3)}. Let X be the total of the two balls selected. Which of the following is the correct set of probabilities for X? (a) _X_____1_____2_____3__ P(X) 1/3 1/3 1/3 (b) _X_____3_____4_____5__ P(X) 1/3 1/3 1/3 (c) _X_____1_____2_____3__ (PX) 1/6 2/6 3/6 (d) _X_____3_____4_____5__ P(X) 1/6 2/6 3/6 18. 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. b. c. d. e. 16.20 1.62 4.02 16.00 The answer cannot be computed from the information given. 19. The weight of medium-sized tomatoes selected at random from a bin at the local supermarket is a random variable with mean µ= 10 ounces and standard deviation σ = 1 ounce. Suppose we pick two tomatoes at random from the bin. The difference in the weights of the two tomatoes selected is a random variable with a mean and standard deviation (in ounces) of a. b. c. d. e. f. 0, 1 20, 1 0, 1.41 20, 1.41 0, 2 20, 2 20. In a certain population, 40% of households have a total annual income of over $70,000. A simple random sample is taken of 4 of these households. What is the probability that 2 or more of the households in the survey have an annual income of over $70,000? a. b. c. d. e. 0.3456 0.4000 0.5000 0.5248 The answer cannot be computed from the information given. Free Response Please show all work and computations!!!! 21. The CFO of a trucking firm believes their fleet of trucks, on cross-country hauls, has a mean of 12.4 miles per gallon with a standard deviation of 1.2 miles per gallon. If this is a normal distribution a) What is the probability that one of the trucks averages fewer than 10 mpg? Answer: z 10 12.4 2.0 1.2 P( X 10) 0.0228 b) What is the probability that one of the trucks averages between11.6 and14 mpg? 11.6 12.4 0.6667 1.2 14 12.4 z 1.3333 1.2 z Answer: P(11.6 X 14) 0.6563 22. Describe the similarities and differences between a binomial and geometric distribution. Similarities Independent Observations Constant probability of success Only 2 possible outcomes Difference Binomial: fixed # of observations Geometric: # of observations varies 23. ACT scores for the 1,171,460 members of the 2004 high school graduating class who took the test closely followed the normal distribution with mean 20.9 and standard deviation 4.8. Choose two students independently and at random from this group. a) What is the expected sum of their scores? Expected (Mean) Sum: X X X X 20.9 20.9 41.8 b) What is the expected difference of their scores? Expected (Mean) Difference: X X X X 20.9 20.9 0 c) What is the standard deviation of the difference in their scores? Standard Deviation (Difference): 2 X X 2 X 2 X 4.8 2 4.8 2 46.08 X 46.08 6.7882 d) Find the probability that the sum of their scores is greater than 50. Show your method. Answer: z 50 41.8 1.208 6.7882 P( sum 50) 0.1135 24. A headache remedy is said to be 80% effective in curing headaches caused by simple nervous tension. An investigator tests the remedy on 100 randomly selected patients suffering from nervous tension. (a) Define the random variable being measured. X = #of adults who experience headache relief (b) What kind of distribution is this? Justify your answer. Binomial: 2 outcomes – relief, no relief p is constant = 0.80 fixed # ob observations = 100 independent observations (c) Calculate the mean and the standard deviation of X. np(1 p) np Answer: 100(0.8) 80 100(0.80)(0.20) 4 (d) Determine the probability that at least 87 subjects experience headache relief with this remedy. Answer: 100 100 100 (0.80) 87 (0.20)13 (0.80) 88 (0.20)12 ... (0.80)100 (0.20) 0 P( X 87) 87 88 100 P( X 87) 0.0469 (e) What is the probability that the number of subjects who will obtain relief is within 1.5 standard deviations of its mean. Justify your method of solution. Answer: 100 100 100 (0.80) 74 (0.20) 26 (0.80) 75 (0.20) 25 ... (0.80) 86 (0.20)14 P(74 X 86) 74 75 86 P(74 X 86) 0.8973 (f) Find the probability that the number of subjects who experience headache relief with this remedy is between74 and 86 using the normal approximation method. P(74 X 86) 0.8663 np 10 n(1 p) 10 74 80 Answer: 100(0.80) 10 100(0.20) 10 z 1.5 4 80 10 20 10 86 80 z 1.5 4 25. 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 class are overweight. Answer: 30 30 30 P( X 18) (0.63)18 (0.37)12 (0.63)19 (0.37)11 ... (0.63) 30 (0.37) 30 18 19 30 P( X 87) 0.7055 (b) How many Americans do you expect to weigh before you encounter the first overweight person? Answer: 1 1 1.587 p 0.63 (c) What is the probability that an overweight person is found on the 3rd attempt? P( X 3) (1 p) n1 p Answer: P( X 3) (0.37) 2 (0.63) P( X 3) 0.0862 (d) What is the probability that it takes more than 5 attempts before an overweight person is found? Answer: P( X 5) (1 p) n P( X 5) (0.37) 5 0.0069 On my honor I have neither given nor received, in any form, information about this exam. I understand this means, but is not limited to, visual exchanges, coded exchanges, text messaging or discussing it with or around others who have not taken the exam. Signature