Quiz 1 Name Date 1. This table lists all possible outcomes when two tetrahedral (four-sided) dice are rolled. First Die Second Die 1 2 3 4 1 1, 1 1, 2 1, 3 1, 4 2 2, 1 2, 2 2, 3 2, 4 3 3, 1 3, 2 3, 3 3, 4 4 4, 1 4, 2 4, 3 4, 4 a. Make a probability distribution table for the difference of the numbers on the two dice (first second), and verify that your distribution is indeed a probability distribution. b. Describe the shape of the distribution in part a. c. What are the expected value and standard deviation for the probability distribution in part a? d. If you roll a pair of tetrahedral dice, what is the probability that the difference (first second) is greater than 1? 2. A large university sponsors a raffle and sells 2400 tickets. a. What is the expected value of a ticket if there is one prize worth $500, four prizes worth $100, and ten prizes worth $10? What is the standard deviation? b. Suppose a person buys five tickets. What is the expected total value of her tickets? Can you compute the standard deviation of this total by multiplying the standard deviation from part a by 5? Explain. 3. A batch of fifteen computer chips contains exactly six that are defective. Four chips are selected randomly without replacement. If X represents the number of defective chips selected, explain why X will not have a binomial distribution. 4. The proportion of households in the United States that own a cat is approximately 0.34. Suppose you pick four households at random and count the number of households that own a cat. a. Verify that you can treat your random sample as a binomial situation. b. What is the probability that none of the households in your sample own a cat? c. What is the probability that at least one of the households owns a cat? d. Make a probability distribution table for the number of households that own a cat. 68 Chapter 6 Quiz 1 Statistics in Action Instructor s Resource Book
Quiz 1 (continued) 5. The weight of a large herd of goats was found to have a distribution that is approximately normal with mean 70.8 kg and standard deviation 6.4 kg. a. Suppose two goats are selected at random. Describe the sampling distribution of the sum of their weights. b. What is the probability that the sum of their weights is less than 145 kg? c. What is the probability that the first goat selected is more than 5 kg heavier than the second goat? d. What is the probability that the sum of the weights of ten randomly selected goats is greater than 750 kg? Statistics in Action Instructor s Resource Book Chapter 6 Quiz 1 69
Quiz 2 Name Date 1. A study of community college enrollments in a certain state finds that 60% of all full-time students are younger than 20. a. If a random sample of 4 community college students is selected from this state, find the probability that at least 3 of the students are younger than 20. b. Consider a random sample of 15 community college students from this state. i. What is the expected number of students who are younger than 20? ii. What is the standard deviation of the number of students who are younger than 20? c. For the situation described in part b, is it appropriate to use the normal approximation to estimate the probability of getting at least 12 students who are younger than 20? Explain. 2. The proportion of households in the United States that own both a cat and a dog is 0.20. Suppose you randomly pick one household at a time until you find a household that owns both a cat and a dog. a. Verify that the distribution of the random variable X that counts the number of trials needed until you pick the first household that owns both a cat and a dog can be considered a geometric distribution. b. What is the probability that you pick a household that owns both a cat and a dog on your first trial? c. What is the probability that you pick the first household that owns both a cat and a dog on your fourth trial? 3. Consider again the situation described in Question 2, where the proportion of households in the United States that own both a cat and a dog is approximately 0.20. a. What is the expected number of households you will have to pick to obtain the first household that owns both a cat and a dog? b. What is the standard deviation of the number of households you will have to pick to obtain the first household that owns both a cat and a dog? 70 Chapter 6 Quiz 2 Statistics in Action Instructor s Resource Book
Test A Name Date 1. If random variables X and Y are independent, then which of these statements is not true? 2 2 2 2 2 2 A. X Y X Y B. 25 X 25 X C. 2X 5 2X 2 2 2 2 2 D. 2X 4 X E. X Y X Y 2. Suppose you buy a raffle ticket in each of 25 consecutive weeks in support of your favorite charity. One of the 1200 raffle tickets sold each week pays $2000. What do you expect to win for those 25 weeks, and with what standard deviation? A. win about $0, give or take about $58 B. win about $25, give or take about $58 C. win about $42, give or take about $58 D. win about $42, give or take about $289 E. win about $210, give or take about $1443 3. Suppose Lynn rolls a fair die until a six appears on top. What is the probability that it will take Lynn more than two rolls to get a six the first time? A. 6 2 1 6 2 5 6 4 B. 5 6 5 6 2 C. 1 5 6 2 D. 1 1 6 2 E. 1 1 6 5 6 1 6 4. A wheel has 38 numbers, 1, 2, 3,..., 38. A player picks a number and bets $20. The wheel is spun, and if the player s number results, he or she is paid $700 (and gets to keep the $20 bet). If another number results, the house keeps the $20. Suppose a typical player bets on 60 spins per hour. What is the expected net gain per hour for the house on each player? A. $63 B. $43 C. $38 D. $20 E. $20 5. Which of these statements are true when comparing the binomial and geometric distributions? I. Both the binomial and geometric distributions have a fixed number of trials. II. Both the binomial and geometric distributions appear mound-shaped when the probability of success in each trial, p, equals 0.5. III. Both the binomial and geometric distributions assume the trials are independent and the probability of success on each trial is the same. A. I only B. II only C. III only D. I and II E. I and III Statistics in Action Instructor s Resource Book Chapter 6 Test A 71
Test A (continued) 6. If Chris rolls two dice and gets a sum of 2 or 12, he wins $20 from the house. If he gets a 7, he wins $5. The cost to play the game is $3, which isn t returned on a win. Explain whether this is a fair game. If not, who has the advantage? 7. Of all U.S. fourth graders, 71% are assigned mathematics homework three or more times per week. If 5 fourth graders are randomly selected to be checked for this frequency of homework assignment during a randomly chosen week, find the probability that a. exactly one fourth grader was assigned homework three or more times that week b. no fourth grader was assigned homework three or more times that week c. at least one fourth grader was assigned homework three or more times that week 8. A survey of drivers in the United States found that 15% never use a cell phone while driving. Suppose that drivers arrive at random at an auto inspection station. a. If the inspector checks 10 drivers, what is the probability that at least one driver never uses a cell phone while driving? b. Suppose the inspector checks 1000 drivers. Use the normal approximation to the binomial distribution to find the approximate probability that at least 13% of these drivers never use a cell phone while driving. c. If the drivers are inspected sequentially as they arrive randomly at the inspection station, what is the probability that the first driver who uses a cell phone while driving is the third driver checked? d. What is the expected number of drivers who must be checked to find the first who never uses a cell phone while driving? e. What is the expected number of drivers who must be checked to find the first driver who uses a cell phone while driving? f. If it costs $5 to question each driver, what is the expected cost and standard deviation of questioning up to and including the first driver who uses a cell phone while driving? g. Will the cost of inspection in part f often exceed $15? Explain. 9. Describe how to use a table of random digits to simulate the situation in Question 8, part a. 72 Chapter 6 Test A Statistics in Action Instructor s Resource Book
Test A (continued) 10. This table presents data about the number of television sets per household in the United States in 2000. Number of Television Sets Proportion of Households 0 0.02 1 0.22 2 0.35 3 0.24 4 0.12 5 or more 0.05 a. Explain how you would simulate a distribution for the number of television sets in a random sample of ten U.S. households. b. Use your process once with these lines from a random digit table to find the mean number of television sets for ten randomly selected households. 37542 04805 64894 74296 24805 10097 32533 76520 13586 34673 c. Compute the mean and standard deviation of this population. Count 5 or more as 5. d. What is the probability that a random sample of 10 U.S. households will have a mean of 3 television sets or more? (Make sure to check conditions and draw a sketch.) Statistics in Action Instructor s Resource Book Chapter 6 Test A 73
Test B Name Date 1. If random variables X and Y are independent, then which of these statements is not true? 2 2 2 2 2 2 A. X Y X Y B. 9 X X C. 5X 10 5X 2 2 2 2 2 D. 9X 9 X E. X Y X Y 2. Suppose you buy a raffle ticket in each of ten consecutive weeks in support of your favorite charity. One of the 1500 raffle tickets sold each week pays $1000. What do you expect to win for those 10 weeks, and with what standard deviation? A. win about $0, give or take about $26 B. win about $7, give or take about $26 C. win about $7, give or take about $82 D. win about $10, give or take about $26 E. win about $10, give or take about $82 3. Suppose Alex rolls a fair die until either a one or three appears on top. What is the probability that it will take Alex more than three rolls to get either a one or three the first time? A. 6 2 1 3 2 2 3 4 B. 1 1 3 3 C. 1 2 3 3 D. 2 3 2 3 2 2 3 3 E. 1 1 3 2 3 1 3 2 3 2 1 3 4. A wheel has 38 numbers, 1, 2, 3,..., 38. A player picks a number and bets 10 chips. The wheel is spun, and if the player s number results, she is paid 300 chips (and gets to keep the 10 chip bet). If another number results, the house keeps the 10 chips. Suppose a typical player bets on 30 spins per hour. What is the approximate expected net chip gain per hour for the house on each player? A. 65 B. 55 C. 38 D. 10 E. 10 5. Which of these statements are true when comparing the binomial and geometric distributions? I. Both the binomial and geometric distributions include trials that have two outcomes. II. Both the binomial and geometric distributions are skewed right. III. Both the binomial and geometric distributions assume the trials are independent and the probability of success on each trial is the same. A. I only B. II only C. III only D. I and II E. I and III 74 Chapter 6 Test B Statistics in Action Instructor s Resource Book
Test B (continued) 6. If Chris rolls two dice and gets a sum of 7, he wins $10 from the house. If he gets a 2, 3, or 12, he wins $20. The cost to play the game is $5, which isn t returned on a win. Explain whether this is a fair game. If not, who has the advantage? 7. According to a Nielsen NetRatings survey, about 59% of the population of the United States and Canada (adults and children older than 2) had accessed the Internet recently. If four people from the United States or Canada had been selected at random, find the probability that a. exactly two accessed the Internet recently b. no one accessed the Internet recently c. at least one accessed the Internet recently 8. A recent report from the Postal Rate Commission Office of the Consumer Advocate concluded that 33% of three-day Priority mail was not delivered by the end of the third day. Suppose an impartial inspector checks the on-time delivery of a random sample of Priority packages at the time of the report. a. If the inspector randomly checks ten packages, what is the probability that at least one package did not arrive on time (by the end of the third day)? b. Suppose the inspector checks 1000 Priority packages. Use the normal approximation to the binomial distribution to find the approximate probability that at least 25% of these packages don t arrive on time. c. What is the probability that the first Priority package delivered on time is the fourth package checked? d. What is the expected number of packages that need to be checked to find the first that doesn t arrive on time? e. What is the expected number of packages that need to be checked to find the first that does arrive on time? f. If it costs $3 to check the delivery of each Priority package, what is the expected cost and standard deviation of checking up to and including the first package that did not arrive on time? g. Will the cost of inspection in part f often exceed $25? Explain. 9. Describe how to use a table of random digits to simulate the situation in Question 8, part a. Statistics in Action Instructor s Resource Book Chapter 6 Test B 75
Test B (continued) 10. This table presents data about the number of telephone lines per household in the United States. Number of Telephone Lines Proportion of Households 0 0.04 1 0.18 2 0.54 3 0.20 4 or more 0.04 a. Explain how you would simulate a distribution for the number of telephone lines in a random sample of ten U.S. households. b. Use your process once with this random digit table to find the mean number of telephones for ten randomly selected households. 11805 05431 39808 27732 50725 83452 99634 06288 98083 13746 c. Compute the mean and standard deviation of this population. Count 4 or more as 4. d. What is the probability that a random sample of 10 U.S. households will have a mean of 2.5 telephone lines or greater? (Make sure to check conditions and draw a sketch.) 76 Chapter 6 Test B Statistics in Action Instructor s Resource Book
AP Practice Quiz Name Date 1. Which of these statements is not true of discrete probability distributions? The sum of the probabilities is 1. The graph of the distribution must exhibit symmetry. The value of the standard deviation can be less than, equal to, or greater than the value of the mean. Each probability in the distribution must be greater than or equal to 0. All of these are true statements. 2. A probability distribution of earnings from a $1000 investment in an Internet company for a term of three years is Earnings 1000 0 1000 2000 3000 Probability 0.13 0.15 0.24 0.35 0.13 How much do you expect to earn from this three-year investment? $800 $1000 $1200 $2000 $2200 3. If X and Y are random variables, which of these statements must be true? X Y X Y XY 3X 2Y 3 X 2 Y X Y X Y only if X and Y are independent. X Y is a random variable with a mean equal to the greater of X and Y. None of these must be true. 4. In the game of roulette, there are 18 red numbers, 18 black numbers, and 2 green numbers. A player bets that a red number will come up. If it does, the house pays the player $1. If it doesn t, the player pays the house $1. What is the expected value of a player s winnings each time he or she plays this game? 0.50 0.10 0.05 0.01 none of the above 5. The probability of success on any one trial is 0.4, and you have a maximum of 10 trials in which to get a success. Which of these expressions will calculate the probability that you will get a success on one of your first four attempts? 10 4 (0.4) 4 (0.6) 4 10 10 (0.4)10 (0.6) 0 0.4 0.4(0.6) 0.4 (0.6) 2 0.4 (0.6) 3 0.6 0.4(0.6) 0.4 (0.6) 2 0.4 (0.6) 3 P(0.5 X 4.5) where X is normally distributed with mean 4 and standard deviation 10(0.4) (0.6). 6. A blood bank knows that only about 10% of its regular donors have type B blood. a. The technician will check 10 donations today. What is the chance that at least one will be type B? What assumptions are you making? b. The technician will check 100 donations this month. What is the probability that at least 10% of them will be type B? c. This bank needs 16 type B donations. If the technician checks 100 donations, does the blood bank have a good chance of getting the amount of type B blood it needs? What recommendation would you have for the blood bank managers? d. What is the probability that the technician will have to check at least 4 donations before getting the first that is type B? Statistics in Action Instructor s Resource Book Chapter 6 AP Practice Quiz 77
Chapters 2 6 AP Practice Quiz Name Date 1. Which of these statements is not true about the variance in a binomial distribution B(n, p)? For a fixed p, the variance increases as n increases. For a fixed n, the variance is maximum when p 0.5. The variance depends only on n. The variance is constant for a specific n and p. None of these are true. 2. The scores of a number of students on a physical fitness test are given in this cumulative percentile plot. About what percentage of students have scores below 30? Percentile 100 80 60 40 20 0 10 20 30 40 50 60 70 80 Score 5 10 15 48 50 3. Consider this game: In each turn of the game, you flip a coin three times. If you get three heads, you win 7 points. If you get the sequence head, tail, head, you win 3 points. If you get any other sequence, you receive no points for that turn. What is your expected value per turn for this game? 10 points 1.25 points 2.5 points 4 points none of these 4. A simple random sample of current CEOs were asked their number of years as a CEO and the dollar value of their benefits. These data were organized into pairs (time in years, benefits in $1000s). The scatterplot appears exponential, and the transformation (x, y) _ (x, ln y) is applied to the data. A graphing calculator yields the linear regression equation y a bx, where a 0.3079, b 0.464, and r 2 0.922. What are the estimated benefits for a CEO employed 12 years? $5,876 $63,995 $75,519 $356,345 $751,450 5. To study the effects of location and music on studying, a researcher selects 100 college students at random and has them study 2 hours for a standardized test. Half of the students study in a familiar location (their dorm room), and the other half study in an unfamiliar location (a study carrel at the library). Within each group, music is played for half of the students and no music is played for the other half. After 2 hours of study time, all of the students take the standardized test and their scores are compared. Which of these terms best describes the combination of being in an unfamiliar location and having no music playing? experimental unit factor level response variable treatment 78 Chapters 2 6 AP Practice Quiz Statistics in Action Instructor s Resource Book
Chapters 2 6 AP Practice Quiz (continued) 6. Suppose you roll a fair four-sided (tetrahedral) die and a fair six-sided die. a. How many possible outcomes are there? b. Show all of them in a table or in a tree diagram. c. What is the probability of getting doubles? d. What is the probability of getting a sum of 3? e. Are the events getting doubles and getting a sum of 4 disjoint? Are they independent? f. Are the events getting a 2 on the tetrahedral die and getting a 5 on the six-sided die disjoint? Are they independent? Statistics in Action Instructor s Resource Book Chapters 2 6 AP Practice Quiz 79