Chapter 8.1 Quiz



Chapter 8.1 QuizName___________________________________AP StatisticsPeriod___________________ AUTHOR \* MERGEFORMAT Mr. Daniels – DOCPROPERTY "Category" \* MERGEFORMAT Binomial DistributionsDate__________________Possible Points: 20Points Earned:% Correct:Letter Grade:Instructions: Choose the BEST response to each prompt. SHOW YOUR WORK. (2pts each)1. A set of 10 playing cards consists of 5 red cards and 5 black cards. The cards are shuffled thoroughly, and we draw 4 cards one at a time and without replacement. Let?X?= the number of red cards drawn. The random variable?X?has which of the following probability distributions?? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"3286988375046804\" VALUE=\"9705258441081249\">" MACROBUTTON HTMLDirect ?A. binomial distribution with parameters?n?= 10 and?p?= 0.5 PRIVATE "<INPUT TYPE=\"radio\" NAME=\"3286988375046804\" VALUE=\"8219939776093736\">" MACROBUTTON HTMLDirect ?B. binomial distribution with parameters?n?= 4 and?p?= 0.5 PRIVATE "<INPUT TYPE=\"radio\" NAME=\"3286988375046804\" VALUE=\"8997146697678105\">" MACROBUTTON HTMLDirect ?C. neither (A) nor (B)2. There are 20 multiple-choice questions on an exam, each having four possible responses, of which only one is correct. Each question is worth 5 points if answered correctly. Suppose that a student guesses the answer to each question, with her guesses from question to question being independent. If the student needs at least 40 points to pass the exam, the probability that she passes is closest to? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6968796823448354\" VALUE=\"6964943183068894\">" MACROBUTTON HTMLDirect ?A. 0.0609. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6968796823448354\" VALUE=\"7887596531607957\">" MACROBUTTON HTMLDirect ?B. 0.1018. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6968796823448354\" VALUE=\"2862032441053378\">" MACROBUTTON HTMLDirect ?C. 0.9591.3. There are 20 multiple-choice questions on an exam, each having four possible responses, of which only one is correct. Each question is worth 5 points if answered correctly. Suppose that a student guesses the answer to each question, with her guesses from question to question being independent. The probability that the student scores lower than a 60 on the exam is? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5735487302606223\" VALUE=\"6616151940719603\">" MACROBUTTON HTMLDirect ?A. 0.0009. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5735487302606223\" VALUE=\"0346556293965268\">" MACROBUTTON HTMLDirect ?B. 0.9998. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5735487302606223\" VALUE=\"8959159148652657\">" MACROBUTTON HTMLDirect ?C. 0.9991.4. There are 20 multiple-choice questions on an exam, each having four possible responses, of which only one is correct. Each question is worth 5 points if answered correctly. Suppose that a student guesses the answer to each question, with her guesses from question to question being independent. The student’s expected (mean) score on this exam is? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5049337891001659\" VALUE=\"4450994741985825\">" MACROBUTTON HTMLDirect ?A. 25. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5049337891001659\" VALUE=\"4774661399347964\">" MACROBUTTON HTMLDirect ?B. 5. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5049337891001659\" VALUE=\"5544983599521045\">" MACROBUTTON HTMLDirect ?C. 50.5. There are 20 multiple-choice questions on an exam, each having four possible responses, of which only one is correct. Each question is worth 5 points if answered correctly. Suppose that a student guesses the answer to each question, with her guesses from question to question being independent. The standard deviation of the student’s score on the exam is? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"8202615172402590\" VALUE=\"5720503520571074\">" MACROBUTTON HTMLDirect ?A. 9.68. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"8202615172402590\" VALUE=\"6140592922542951\">" MACROBUTTON HTMLDirect ?B. 1.94. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"8202615172402590\" VALUE=\"6738013146813036\">" MACROBUTTON HTMLDirect ?C. 93.75.6. In the gambling game of chuck-a-luck, three dice are rolled using a rotating, hourglass-shaped cage. The player chooses one of the 6 possible sides (1, 2, 3, 4, 5, or 6) and receives a payoff the amount of which depends on how many dice turn up on that particular side. Let?X?= the number of times the dice have to be rolled until we see “three of a kind” (of?any?type). Which of the following probability distributions does?X?have?? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0829600990195449\" VALUE=\"3669778175863261\">" MACROBUTTON HTMLDirect ?A. geometric with?p?= 1/216 PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0829600990195449\" VALUE=\"4556916500471162\">" MACROBUTTON HTMLDirect ?B. geometric with?p?= 6/216 PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0829600990195449\" VALUE=\"1927197445604508\">" MACROBUTTON HTMLDirect ?C. binomial with?n?= 3 and?p?= 6/2167. In the old children’s game of “rock?scissors?paper,” two players simultaneously use their hands to show one of three objects: a rock (a closed fist), a pair of scissors (two fingers extended in a V-shape), or a piece of paper (an open palm). The winner is chosen in the following ways:Rock beats scissors (rock can crush scissors).Scissors beats paper (scissors can cut paper).Paper beats rock (paper can wrap around rock).If the players each show the same object, then the game is played again, with additional repeats as needed until one player wins. Assume that each player selects an object independently and that each player is equally likely to choose any of the three objects. Let?X?= the number of games that must be played in order to decide a winner. (We assume that the identity of the winner is unimportant). Then?X?is a geometric random variable with probability of success?p equal to? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0042378148024285\" VALUE=\"1078105794701150\">" MACROBUTTON HTMLDirect ?A. 1/3. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0042378148024285\" VALUE=\"5827167862218673\">" MACROBUTTON HTMLDirect ?B. 2/3. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0042378148024285\" VALUE=\"2764825507032413\">" MACROBUTTON HTMLDirect ?C. 1/9.8. In a certain large population, 70% are right-handed. You need a left-handed pitcher for your softball team and decide to find one by asking people chosen from the population at random. (We assume that once you do find a left-hander, he or she will be happy to join your team and will not say no.) The probability that the first left-hander you find is the fourth person you ask is approximately? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6899418394788842\" VALUE=\"0860721699442913\">" MACROBUTTON HTMLDirect ?A. 0.1029. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6899418394788842\" VALUE=\"3821510411442264\">" MACROBUTTON HTMLDirect ?B. 0.019. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6899418394788842\" VALUE=\"1872621532809502\">" MACROBUTTON HTMLDirect ?C. 0.072.9. In a certain large population, 70% are right-handed. You need a left-handed pitcher for your softball team and decide to find one by asking people chosen from the population at random. (We assume that once you do find a left-hander, he or she will be happy to join your team and will not say no.) The probability that you will have to ask more than three people before finding your first left-hander is approximately? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9677326354116547\" VALUE=\"3501708395459112\">" MACROBUTTON HTMLDirect ?A. 0.027. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9677326354116547\" VALUE=\"1540596699949441\">" MACROBUTTON HTMLDirect ?B. 0.240. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9677326354116547\" VALUE=\"7687873916142033\">" MACROBUTTON HTMLDirect ?C. 0.34310. In a certain large population, 70% are right-handed. You need a left-handed pitcher for your softball team and decide to find one by asking people chosen from the population at random. (We assume that once you do find a left-hander, he or she will be happy to join your team and will not say no.) The probability that you will have to ask at most two people to find your first left-hander is approximately? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9108856373254149\" VALUE=\"0445024115447585\">" MACROBUTTON HTMLDirect ?A. 0.51. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9108856373254149\" VALUE=\"4477023150787508\">" MACROBUTTON HTMLDirect ?B. 0.49. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9108856373254149\" VALUE=\"2601057616788360\">" MACROBUTTON HTMLDirect ?C. 0.91.Solutions:BBBAABBABA* Page already viewedYates TPS 3e Chapter 08 completedTotal score: 4 out of 12, 33%?Top of Form1. A set of 10 playing cards consists of 5 red cards and 5 black cards. The cards are shuffled thoroughly, and we draw 4 cards one at a time and without replacement. Let?X?= the number of red cards drawn. The random variable?X?has which of the following probability distributions?? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"3286988375046804\" VALUE=\"9705258441081249\" CHECKED>" MACROBUTTON HTMLDirect ?A. binomial distribution with parameters?n?= 10 and?p?= 0.5 PRIVATE "<INPUT TYPE=\"radio\" NAME=\"3286988375046804\" VALUE=\"8219939776093736\">" MACROBUTTON HTMLDirect ?B. binomial distribution with parameters?n?= 4 and?p?= 0.5 PRIVATE "<INPUT TYPE=\"radio\" NAME=\"3286988375046804\" VALUE=\"8997146697678105\">" MACROBUTTON HTMLDirect ?C. neither (A) nor (B)0 out of 1Incorrect. Since 4 cards were selected, the number of trials (draws) is 4, not 10. This could not possibly be a binomial distribution with parameter?n?= 10.?2. There are 20 multiple-choice questions on an exam, each having four possible responses, of which only one is correct. Each question is worth 5 points if answered correctly. Suppose that a student guesses the answer to each question, with her guesses from question to question being independent. If the student needs at least 40 points to pass the exam, the probability that she passes is closest to? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6968796823448354\" VALUE=\"6964943183068894\" CHECKED>" MACROBUTTON HTMLDirect ?A. 0.0609. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6968796823448354\" VALUE=\"7887596531607957\">" MACROBUTTON HTMLDirect ?B. 0.1018. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6968796823448354\" VALUE=\"2862032441053378\">" MACROBUTTON HTMLDirect ?C. 0.9591.0 out of 1Incorrect. The random variable?X?= “number correct out of 20” has a binomial distribution with?n?= 20 and?p?= 0.25. To get at least 40 points, the student must get at least 8 of the 20 questions correct. You have calculated the probability of the student getting exactly 8 questions correct.?3. There are 20 multiple-choice questions on an exam, each having four possible responses, of which only one is correct. Each question is worth 5 points if answered correctly. Suppose that a student guesses the answer to each question, with her guesses from question to question being independent. The probability that the student scores lower than a 60 on the exam is? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5735487302606223\" VALUE=\"6616151940719603\" CHECKED>" MACROBUTTON HTMLDirect ?A. 0.0009. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5735487302606223\" VALUE=\"0346556293965268\">" MACROBUTTON HTMLDirect ?B. 0.9998. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5735487302606223\" VALUE=\"8959159148652657\">" MACROBUTTON HTMLDirect ?C. 0.9991.0 out of 1Incorrect. The random variable?X?= “number correct out of 20” has a binomial distribution with?n?= 20 and?p?= 0.25. Getting a score of 60 corresponds to getting 12 of the 20 questions right, so in terms of?X, the event whose probability we seek is?X?< 12, that is,?X?≤ 11. You have found the probability of the complement of this event.?4. There are 20 multiple-choice questions on an exam, each having four possible responses, of which only one is correct. Each question is worth 5 points if answered correctly. Suppose that a student guesses the answer to each question, with her guesses from question to question being independent. The student’s expected (mean) score on this exam is? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5049337891001659\" VALUE=\"4450994741985825\" CHECKED>" MACROBUTTON HTMLDirect ?A. 25. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5049337891001659\" VALUE=\"4774661399347964\">" MACROBUTTON HTMLDirect ?B. 5. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5049337891001659\" VALUE=\"5544983599521045\">" MACROBUTTON HTMLDirect ?C. 50.1 out of 1Correct. The random variable?X?= “number correct out of 20” has a binomial distribution with?n?= 20 and?p?= 0.25. The mean of?X?is μX?=?np?= (20)(0.25) = 5. The student’s score on the exam can be written as the linear function 5X, so the student’s mean score is μ5X?= 5μX?= 5(5) = 25 by the rule for the mean of a linear function of a random variable.?5. There are 20 multiple-choice questions on an exam, each having four possible responses, of which only one is correct. Each question is worth 5 points if answered correctly. Suppose that a student guesses the answer to each question, with her guesses from question to question being independent. The standard deviation of the student’s score on the exam is? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"8202615172402590\" VALUE=\"5720503520571074\" CHECKED>" MACROBUTTON HTMLDirect ?A. 9.68. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"8202615172402590\" VALUE=\"6140592922542951\">" MACROBUTTON HTMLDirect ?B. 1.94. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"8202615172402590\" VALUE=\"6738013146813036\">" MACROBUTTON HTMLDirect ?C. 93.75.1 out of 1Correct. The random variable?X?= “number correct out of 20” has a binomial distribution with?n?= 20 and?p?= 0.25. The variance of?X?is σ2X?=?np(1 -?p) = (20)(0.25)(0.75) = 3.75. The student’s score on the exam can be written as the linear function 5X, so the variance of the student’s score is σ25X?= 25σ2X?= 25(3.75) = 93.75 by the rule for the variance of a linear function of a random variable. The standard deviation of the score is therefore √93.75 = 9.68.?6. In the gambling game of chuck-a-luck, three dice are rolled using a rotating, hourglass-shaped cage. The player chooses one of the 6 possible sides (1, 2, 3, 4, 5, or 6) and receives a payoff the amount of which depends on how many dice turn up on that particular side. Let?X?= the number of times the dice have to be rolled until we see “three of a kind” (of?any?type). Which of the following probability distributions does?X?have?? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0829600990195449\" VALUE=\"3669778175863261\" CHECKED>" MACROBUTTON HTMLDirect ?A. geometric with?p?= 1/216 PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0829600990195449\" VALUE=\"4556916500471162\">" MACROBUTTON HTMLDirect ?B. geometric with?p?= 6/216 PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0829600990195449\" VALUE=\"1927197445604508\">" MACROBUTTON HTMLDirect ?C. binomial with?n?= 3 and?p?= 6/2160 out of 1Incorrect. This is an example of a geometric setting, since the results of individual rolls of the dice are independent and the variable of interest is the number of trials required to obtain the first success (“three of a kind”). However, the value of the probability of a success, p, is not correct.?7. In the old children’s game of “rock?scissors?paper,” two players simultaneously use their hands to show one of three objects: a rock (a closed fist), a pair of scissors (two fingers extended in a V-shape), or a piece of paper (an open palm). The winner is chosen in the following ways:Rock beats scissors (rock can crush scissors).Scissors beats paper (scissors can cut paper).Paper beats rock (paper can wrap around rock).If the players each show the same object, then the game is played again, with additional repeats as needed until one player wins. Assume that each player selects an object independently and that each player is equally likely to choose any of the three objects. Let?X?= the number of games that must be played in order to decide a winner. (We assume that the identity of the winner is unimportant). Then?X?is a geometric random variable with probability of success?pequal to? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0042378148024285\" VALUE=\"1078105794701150\" CHECKED>" MACROBUTTON HTMLDirect ?A. 1/3. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0042378148024285\" VALUE=\"5827167862218673\">" MACROBUTTON HTMLDirect ?B. 2/3. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0042378148024285\" VALUE=\"2764825507032413\">" MACROBUTTON HTMLDirect ?C. 1/9.0 out of 1Incorrect. You need to consider the sample space of all possible pairs of objects that the players can display, rather than simply counting all the objects that an individual player can display.?8. In a certain large population, 70% are right-handed. You need a left-handed pitcher for your softball team and decide to find one by asking people chosen from the population at random. (We assume that once you do find a left-hander, he or she will be happy to join your team and will not say no.) The probability that the first left-hander you find is the fourth person you ask is approximately? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6899418394788842\" VALUE=\"0860721699442913\" CHECKED>" MACROBUTTON HTMLDirect ?A. 0.1029. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6899418394788842\" VALUE=\"3821510411442264\">" MACROBUTTON HTMLDirect ?B. 0.019. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"6899418394788842\" VALUE=\"1872621532809502\">" MACROBUTTON HTMLDirect ?C. 0.072.1 out of 1Correct. We want?P(X?= 4), where?X?= “number of people you ask to get your first left-hander” is geometric with?p?= 0.3. By the geometric probability formula, this probability is (1 - 0.3)4-1(0.3) = (0.7)3(0.3) = 0.1029.?9. In a certain large population, 70% are right-handed. You need a left-handed pitcher for your softball team and decide to find one by asking people chosen from the population at random. (We assume that once you do find a left-hander, he or she will be happy to join your team and will not say no.) The probability that you will have to ask more than three people before finding your first left-hander is approximately? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9677326354116547\" VALUE=\"3501708395459112\" CHECKED>" MACROBUTTON HTMLDirect ?A. 0.027. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9677326354116547\" VALUE=\"1540596699949441\">" MACROBUTTON HTMLDirect ?B. 0.240. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9677326354116547\" VALUE=\"7687873916142033\">" MACROBUTTON HTMLDirect ?C. 0.3430 out of 1Incorrect. Recall that?p?= the probability of a success and 1 -?p?= the probability of a failure in the geometric setting. What have you identified as a “success” and a “failure”??10. In a certain large population, 70% are right-handed. You need a left-handed pitcher for your softball team and decide to find one by asking people chosen from the population at random. (We assume that once you do find a left-hander, he or she will be happy to join your team and will not say no.) The probability that you will have to ask at most two people to find your first left-hander is approximately? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9108856373254149\" VALUE=\"0445024115447585\" CHECKED>" MACROBUTTON HTMLDirect ?A. 0.51. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9108856373254149\" VALUE=\"4477023150787508\">" MACROBUTTON HTMLDirect ?B. 0.49. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"9108856373254149\" VALUE=\"2601057616788360\">" MACROBUTTON HTMLDirect ?C. 0.91.1 out of 1Correct. We want?P(X?≤ 2), where?X?= “number of people you ask to get your first left-hander” is geometric with p = 0.3. By the geometric probability formula and the addition rule for probabilities,?P(X?≤ 2) =?p(1) +?p(2) = 0.3 + (0.7)(0.3) = 0.51.?11. In a certain large population, 70% are right-handed. You need a left-handed pitcher for your softball team and decide to find one by asking people chosen from the population at random. (We assume that once you do find a left-hander, he or she will be happy to join your team and will not say no.) The mean and variance of the number of people you will have to ask to find your first left-hander are? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0887778227147706\" VALUE=\"4855717108871083\" CHECKED>" MACROBUTTON HTMLDirect ?A. mean = 3.33, variance = 0.612. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0887778227147706\" VALUE=\"2503252188639073\">" MACROBUTTON HTMLDirect ?B. mean = 3.33, variance = 7.78. PRIVATE "<INPUT TYPE=\"radio\" NAME=\"0887778227147706\" VALUE=\"5589932373041609\">" MACROBUTTON HTMLDirect ?C. mean = 0.3, variance = 7.78.0 out of 1Incorrect. You have interchanged the values of?p?and 1 -?p?in the formula for the variance of the geometric variable.?12. For which of the following choices of?n,?p?can we?not?use the normal approximation to the binomial distribution?? PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5809594105207911\" VALUE=\"5355995656259867\" CHECKED>" MACROBUTTON HTMLDirect ?A.?n?= 25,?p?= 0.6 PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5809594105207911\" VALUE=\"7053462487320844\">" MACROBUTTON HTMLDirect ?B.?n?= 40,?p?= 0.4 PRIVATE "<INPUT TYPE=\"radio\" NAME=\"5809594105207911\" VALUE=\"0326833416182764\">" MACROBUTTON HTMLDirect ?C.?n?= 60,?p?= 0.90 out of 1Incorrect. Review the conditions for validity of this approximation.?Bottom of FormPerception licensed to Bedford, Freeman & Worth Publishing Group ................
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