1. What is the difference between sampling with replacement versus sampling without replacement?

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1. What is the difference between sampling with replacement versus sampling without replacement? 2. For each of the following, assume you are selecting cards from a standard deck of 52 playing cards. a.
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1. What is the difference between sampling with replacement versus sampling without replacement? 2. For each of the following, assume you are selecting cards from a standard deck of 52 playing cards. a. What is the probability that a randomly selected card is a jack? b. What is the probability that a randomly selected card is a king or a jack? c. What is the probability that a randomly selected card is a four or an ace? d. What is the probability that a randomly selected card is an eight or a heart? e. What is the probability that a randomly selected card is a seven and a queen? f. What is the probability that a randomly selected card is a diamond and a six? g. What is the probability that a randomly selected card is a diamond, a king, or a spade? 3. Use this information to answer the questions that follow: Jack has a bag of magic beans. His bag has contains 5 red beans, 2 blue beans, 10 green beans, 7 black beans, and 6 sparkled beans. Based on this information, answer the following questions, and be sure to incorporate jack's prior actions selections into your answer for each successive question. a. Jack reaches into his bag, selects a bean, looks at it, and replaces it. What is the probability this was a blue bean? b. Jack reaches back into his bag, selects a bean, looks at it, and eats it. What is the probability this was a red bean? c. Assume that the bean that jack selected in b was green. Jack reaches back into his bag, selects a bean, looks at it, and then replaces it. What is the probability that jack selected a black bean? d. Jack reaches back into his bag, selects a bean, looks at it, and eats it. What is the probability this was a green bean? e. Assume that the bean that jack selected in d was green. Jack reaches into his bag and pulls out a bean, looks at it, and eats it. What is the probability that this bean was a sparkled bean or a black bean? Use this information to answer Exercises 4 18: A political scientist interviewed 500 professors at university to study the relationship between area of scholarship and attitudes toward embryonic stem cell research. The following contingency table was observed: Attitude toward Stem Cell Research Area of Scholarship Favors Opposes No Opinion Totals: Natural Sciences Behavioral Sciences Theology Humanities Totals: n T = Are levels of the independent variable Attitude toward stem cell research mutually exclusive? 5. What is the probability that a randomly selected individual favors stem cell research? 6. What is the probability that a randomly selected individual has no opinion on stem cell research? 7. What is the probability that a randomly selected individual studies the natural sciences? 8. What is the probability than a randomly selected individual studies theology? 9. What is the probability than a randomly selected individual studies the behavioral sciences and has no opinion on stem cell research? 10. What is the probability that a randomly selected individual favors stem cell research given the individual is a theologian? 11. What is the probability that a randomly selected individual is a theologian given the individual favors stem cell research? 12. Why are the probabilities in 10 and 11 different? 13. Are being a natural scientist and opposing stem cell research independent? Why or why not? 14. What is the probability that a randomly selected individual studies the behavioral sciences and also favors stem cell research? 15. What is the probability that a randomly selected individual is a theologian and opposes stem cell research? 16. What is the probability that a randomly selected individual studies in the humanities or favors stem cell research? 17. What is the probability that a randomly selected individual is a behavioral scientist or has no opinion on stem cell research? 18. What is the probability that a randomly selected individual favors stem cell research or opposes stem cell research? Use the following information to complete Exercises 19 32: A political scientist interviewed 500 people to study the relationship between political party identification and attitudes toward public sector unionization. The following contingency table was observed: Attitude Toward Public Sector Unionization Political Party identification Favors Opposes Totals Democrat Republican Independent Totals N = What is the probability than an individual favors public sector unionization? 20. What is the probability than an individual opposes public sector unionization? 21. What is the probability than an individual is a Democrat? 22. What is the probability than an individual is a Republican? 23. What is the probability than an individual is an Independent? 24. What is the probability that an individual favors public sector unionization given the individual is a Democrat? 25. What is the probability that an individual is a Democrat given that the individual favors public sector unionization? 26. Are being a Democrat and opposing public sector unionization independent? Why or why not? 27. What is the probability that an individual is a Republican who favors public sector unionization? 28. What is the probability that an individual is a Republican who opposes public sector unionization? 29. What is the probability that an individual opposes public sector unionization given they are a Republican? 30. What is the probability that an individual is an Independent who opposes public sector unionization? 31. What is the probability that an individual is a Republican or favors public sector unionization? 32. What is the probability that an individual is an Independent or opposes public sector unionization? Use the following information to complete Exercises 33 41: I asked n = 1000 students the following questions: (a) are you an underclassman (freshman/sophomore) or are you an upperclassman (junior/senior); and (b) do you own the latest Tim McGraw album. Use this contingency table to answer the following questions: Own the Latest Tim McGraw Album? College Year Yes No Underclassman Upperclassman Is the variable Own the Latest Tim McGraw album mutually exclusive? Why or why not? 34. What is the probability that a randomly selected student owns the Tim McGraw album? 35. What is the probability that a randomly selected student is an underclassmen? 36. What is the probability that a randomly selected student owns the Tim McGraw album given that he/she is an upperclassman? 37. What is the probability that a randomly selected student is an underclassman given that he/she owns the Tim McGraw album? 38. Is the probability that a randomly selected student owns the Tim McGraw album independent of being an upperclassman? Why or why not? 39. What is the probability that a randomly selected student owns the Tim McGraw album and is an upperclassman? 40. What is the probability that a randomly selected student is an underclassman, or owns the Tim McGraw album, or both? 41. What is the probability that a randomly selected student is an underclassman or is an upperclassman? 42. Two events, A and B, are mutually exclusive. The probability of event A is and the probability of event B is 0.500, what is the probability of at least one of the two events occurring? 43. The probability of event B is 0.410, and the probability of event A given event B is What is the probability of both events A and B occurring? 44. Two events, A and B, are independent. The probability of event A is 0.35 and the probability of event B is What is the probability of both event A and event B occurring? 45. What is the probability of flipping a heads on an unbiased, two-sided coin and then rolling a 6 on an unbiased, six-sided die? 46. What is the probability of flipping a heads on an unbiased, two-sided coin three times in a row? 47. The probability of some event A is 0.800, the probability of some other event B is 0.700, and the probability of event B given event A is Use Bayes Theorem to solve for the probability of even A given event B. 48. The probability of some event C is 0.060, the probability of some other event D is 0.950, and the probability of event D given event C is Use Bayes Theorem to solve for the probability of even C given event D. 49. The probability of some event C is 0.060, the probability of some other event D is 0.950, and the probability of event D given event C is Use Bayes Theorem to solve for the probability of even C given event D. 50. A patient goes to see his doctor complaining of blurred vision and thinks he has dreaded yellow fickle-berry disease (DYFBD). Based on the patient s history, the doctor knows the probability of this patient having DYFBD is The doctor orders a test, which comes back positive for DYFBD with a probability of The doctor also knows that if the patient does have DYFBD, the probability of the test being positive is The test comes back positive for DYFBD. Using Bayes Theorem, what is the probability that the patient has DYFBD, given the positive test result? 51. A patient goes to see his doctor complaining of blurred vision and thinks he has dreaded yellow fickle-berry disease (DYFBD). Based on the patient s history, the doctor knows the probability of this patient having DYFBD is The doctor orders a test, which comes back positive for DYFBD with a probability of The doctor also knows that if the patient does have DYFBD, the probability of the test being positive is The test comes back positive for DYFBD. Using Bayes Theorem, what is the probability that the patient has DYFBD, given the positive test result? 52. Compute the following permutations and combinations: a. 5P 2 b. 5C 2 c. 6P 2 d. 4C 4 e. 4P 3 f. 6C 3 g. 4P 4 h. 3C 2 i. 3P You conduct a study that involves showing participants six clips. You are concerned that the order in which you present the movie clips could affect the outcome, so you decide to present them to each person in a different order. In how many ordered sequences can the movie clips be presented? 54. From #53, you decide that you can only show three movie clips to each subject. How many different ordered sequences of three movie clips of six movie clips can be presented? 55. You are interested in the effects of five independent variables on a dependent variable, but, you can study only two independent variables. How many combinations of two variables could you study? 56. From #55, how many combinations of three independent variables are possible? 57. A company has developed ten hot sauces, but only wants to market the three best-tasting sauces. To identify the best-tasting sauces, the company has people taste the hot sauces and give ratings. But, the company only wants to have each person taste four of the ten sauces. How many combinations of four hot sauces can be administered across people? 58. A team of three people from a widget company to assess the widget needs of a client is formed from a group of 3 managers, 10 analysts, and 15 technicians. a. What is the probability that the team is composed of only analysts? b. What is the probability that the team is composed of only technicians? c. What is the probability that the team is composed of only managers? d. What is the probability that the team is composed of two managers and the third person is either an analyst or a technician? e. What is the probability that the team is composed of two analysts and one technician? ANSWER 1. Sampling with replacement means selecting an individual for a sample and returning that individual to the population, to possibly be selected later. Sampling without replacement means after selecting an individual for a sample, they are not returned to the population and, hence, are selected only once. 2. a. 4/52 = b. 8/52 = c. 8/52 = d. 16/52 =.308 e. 0/52 = 0 f. 1/52 = g. 28/52 = a. 2/30 = b. 5/30 = c. 7/29 = d. 9/29 = e Yes, because a person can belong to only one of these three categories, and not more than one. That is, you cannot simultaneously oppose and favor something /500 = /500 = /500 = /500 = /500 = /150 = /200 = In both cases we are interested in the same combination of variables (Theologians who favor stem cell research), but the reference groups differ between 10 and They are not independent, because the probability of being a natural scientist is not equal to the probability of being a natural scientist given a person opposes stem cell research. That is: p(natural Science) = 150/500 = and p(natural Science Opposes) = 20/200 = /500 = /500 = /500 = /500 = 0.420 /500 = /500 = /500 = /200 = /290 = They are not independent, because the probability of being a Democrat is not equal to the probability of being a Democrat given that you oppose public sector unionization. That is: p(democrat) = 200/500 = and p(democrat Opposes) = 40/210 = /500 = /500 = /200 = /500 = Yes, the variable own the latest Ministry CD is mutually exclusive, because you cannot simultaneously own and not own the CD. Also: p(yes U No) = 1 and p(yes) = and p(no) = 0.300, thus: p(yes U No) = p(yes) + p(no) /1000 = /1000 = /500 = /700 = The probability that a randomly selected student owns the Ministry CD is not independent of being an upperclassman, because: p(yes) = 700/1000 = and p(yes Upper) = 400/500 = 0.800, thus: p(yes) p(yes Upper) /1000 = a. 20 b. 10 c. 30 d. 1 e. 24 f. 20 g. 24 h. 3 i a b c d e
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