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Stat131SG2 - Hank Ibser — Stat 131A — Fall 2009 —...

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Unformatted text preview: Hank Ibser — Stat 131A — Fall 2009 — Study Guide for the Second Midterm The exam'will cover everything in chapters 13—21 of the text EXCEPT anything having to do with Figure 2 on pg 313. The exam will also cover the handout on probability. I also encourage you to do extra problems from the book as well as making up problems and exchanging them with people. You can do the following Special Review Exercises: After Ch 15: pg 268, 18—20. After Ch 23: pg 431—435, 13*18,20,22,25—28,30. The following 12 problems are from my old exams, the first four are about a midterm long, the next five are about a midterm long, and the last three are about half a midterm. The rest of the handout is old problems from Purves' Stat 21 midterms. Answers to all are at the very end. 1) I have two bags of fruit. The first contains 4 apples and 2 oranges, the second contains 5 oranges and 3 bananas. I choose a bag at random and then take 4 pieces of fruit from that bag to eat for lunch. a) What is the chance that the 4 pieces of fruit are the same? b) What is the chance that I get at least one orange? 2) Suppose you and I play the following game. A die is rolled. If the die shows a l or 2, then you pay me 51. If the die shows a 5 or 6, I pay you. Otherwise, no money is exchanged. a) We play this game 400 times. Find the expected value and the SE for the amount of money I win in these 400 times. b) Approximately what is the chance that I win $l0 or more in'these 400 games? o) The percent of times that I win will be about give or take about . d) About what is the chance that I win more than 30% of the 400 games? 3) Jerry, Doug, and Hank play roulette repeatedly at different wheels, betting on red each time. There are 18 red slots on a roulette wheel and 38 total slots. For each bet, each gambler will either win a dollar if red comes up or lose a dollar if not. For each question, answer either 38 or 380 and give a brief reason. (a) Jerry thinks in percentages — if he wins at least 51% of the time he will be pleased. In order to have the best chance of winning at least 51% of the time, should Jerry play 38 times or 380 times? (b) Doug thinks about money — if he wins at least $10 more than he expects, he will be happy.' In order to have the best chance of winning more than $10, should Doug play 38 times or 380 times? (c) Hank doesn‘t care about wins or losses, he just wants everything comes out the way it is supposed to. In other words, Hank wants the outcome to be exactly what the expected value is. Should Hank play 38 times or 380 times? 4) A poll is taken to determine the percent of UC Berkeley undergraduates who support Kerry (suppose there are 22,850 total undergraduates). An interviewer goes to Sproul plaza and interviews students, asking them if they are undergrads and if so, if they support Kerry. Only two responses are allowed, support Kerry or don't support Kerry. About 30% of the undergrads respond, the rest walk away. The interviewer stops after she gets 400 responses; 300 say they support Kerry. a) Say in words what each of the following are, in the context of this problem: population, sample, parameter, and statistic. b) For each of the following, if you know the number, give it, if not, put "unknown." No explanation necessary. population size: sample size: parameter: statistic: c) Is this a probability method? Explain briefly. d) Give two ways in which the results might be biased and explain briefly, using statistical terms from lecture and the text. (Several correct answers are possible.) 5) I roll a fair six—sided die 4 times. a) What is the chance that I get exactly 2 fours? b) What is the chance that not all the rolls are 2 or more? c) What is the chance that I get exactly 2 ones or exactly 2 sixes? 6) A casino offers the following game. A standard deck of cards (52 cards, of which 4 are aces, 4 kings, 4 queens, 4 jacks, and 4 of every number from ten to two) is shuffled an a card is drawn. Chris wins $2 if the card is an ace, king, queen, or jack. On any other draw Chris loses $1. Chris plays 400 times. a) Chris shOuld have a net gain of about , give or take b) What is the chance that Chris wins money? 0) What is the chance that Chris wins exactly 120 times? d) Consider the event that Chris wins more than 25% of the_bets. Is the chance that this occurs bigger with 40 bets or with 400 bets? Choose one and explain briefly. 7) You roll 48 dice all at once. What is the chance that the sum of the even numbers that land face up will be 104 or more? 8) In a simple random sample of 100 Berkeley rental units, the total number of residents is 218. The SD for the number of residents in the sampled units is 1.2. Of the 218 residents, 43 are vegetarians. If possible, find a 95% confidence interval for the percent of vegetarians living in Berkeley rental units. If this is not possible, explain why not. 9) I roll a fair six—sided die 5 times. a) What is the chance that I get at least one six? b) What is the chance I get exactly 3 sixes? c) What is the expected number of sixes I get? d) What is the SE of the number of sixes? 10) A drawer contains 20 socks. Ten of the socks are white, five are brown, four are gray, and one is red. a) You take two socks from this drawer at random without replacement. What is the chance that they are both the same color? b) 3 socks are drawn without replacement. What is the chance that exactly 2 white socks are drawn? 11) A researcher takes a simple random sample of 400 California adults in order to determine what percent of California adults support allowing same—sex marriage. The researcher finds that 44% of the adults in the sample support allowing same—sex marriages. If appropriate, construct an 80% confidence interval for the percent of all California adults who support allowing same—sex marriages. If not appropriate, explain why not. 12) Alex is interested in finding out what percent of UC Berkeley students support the war in Iraq. The question asked is, "In view of the developments since we first sent our troops to Iraq, do you think the United States made a mistake in sending troops to Iraq, or not?" Alex gets phone numbers from all UC Berkeley students who have released that information (60% of students release the information) and calls 1000 of them on the phone. Of these, 600 .answer the phone, and 400 are willing to answer the question. a) Describe briefly (one sentence each) two potential reasons for bias, using vocabulary from the class. b) Bert does exactly the same thing, but instead calls 2000 students, getting 1200 students to answer the phone and 800 willing to answer the question. Which researcher do you expect to have larger chance error? Or do you expect them to have about the same chance error? Explain briefly. c) Which researcher do you expect to have larger bias? Or will they be the same? Explain briefly. Statistics 21 Problems from past midterms ' ‘ “Midtown 2 ' ' ' ‘ ' I ' ' ' ' tWo brothers in the . 5 arms Ten en 1e are in a room, waiting to be 1nterv1ewed. There are _ _ . I ' (grovup, biit these? an;3 the only two that are-related. The people are called to, one at a hme, to be interviewed. The choice of who goes next IS done at random. (a) Find the Chance the brothers are the first two people to be interviewed. (b) Find the chance the older brother is first-person to be interviewed and the younger brother A isthefifth. __I _ __________ __ ,_,_ . _ 2. ( 5 points) A tiny town consist of five blocks. Three people-live on each block, so the total population of the town is 15. A simple random sample of five people is taken from the population. Find the chance they all live on different blocks. 3. ( 5 points) Someone shuffles a deck of cards and dealsoutthO cards. Then he does this ' again with asecond deck of cards. Find the chance that the two cards from the 'first deck are red and the two from the second deck are black. 4. (10 points) Seventy-—-fiv-e draws will be made at random, with replacement, from the box: I Some of the numbers drawn will be positive, others negative. Find, approximately, the chance the sum of all the positive numbers is bigger than 80. 5. (10 points) An educational sociologist mails a questionnaire to a simple random sample of 500 teachers taken from the 30,000 members of a state teachers association. Eighty percent of the state’s teachers belong to the association. The questionnaire contains both multiple choice items and Open—ended questions. One of the open-ended questions asks the teachers'to describe their attitude to home schooling. When the sociologist starts to review the returned questionnaires, he finds that some «of the teacher‘s responses to this particular question are brief, and others are much more extensive. He puts aside, for later analysis, the longer responses. There were 97 such responses. Of the remaining 403,, he finds that 238 clearly favored home schooling and 16.5, were opposed to it. Using this 238 out of 403., the sociolOgist intends to calculate a 95% confidence interval for the percentage of the association members-hip who are in favor of home schooiing. Is this calculation appropriate? If it is, explain why and find the confidence interval. If it is not appropriate, explain why not. ' '" (Note: 'Please choose one and only one of the two options. For example, do not write, “I don’t think the calculation-is appropriate, but in case it is, here is how to write the confidence interval” and then go on to do the calculation of the confidence interval.) i 9-. (10 points) A gambling house offers the following game. The player names any card--the queen. of hearts, says—and stakes $1 on it. The dealer shuffles a deck of cards and turns over thetop four cards. If the named card is one of the four, the player gets the $1 back and a prize $10. 'If not, the player loses the stake. ' A gambler plays this 40 times. , Her net gain will be around __ give or take _______ or so. 7. (IOpoinrs) A coin is tossed 49 times. Find, approximately, the chance of getting 25 heads and 24 tails in the 49 tosses. 8 . A die is rolled three times. Find the chance that: (a) The three numbers rolled are either all the same or all different. (1)) Two of the numbers are the same and the third is different. 9. if 5 points) Two draws are made at random with replacement from the box: IIIIIEIHEE Find the chance the second number is bigger than twice the first one. 10. (1.0 points) A county contains 100,000 households. One quarter of one percent of these households have an annual income of one milliOn dollars or mOre. A polling organization is about to take a simple random sample of 400 households from the county. Of course, very few, if any, of the million dollar or more households will appear in the sample. Someone suggests using the normal curve to figure thec‘hance that no such households show up. (a) What answer would this lead to? (b) The answer in (a) is: too low too high - about right Choose one option and explain. 1 l. (5 points) In the preceding exercise (Problem #10), find the exact chance that no million dollar or-morehouseholds show up in the sample. Do not work out the arithmetic. r ,_ I" Z (10 points) The following procedure isrepeated 4 times: , One hundred draws are made at random, with replacement from the Box: 1 1 5 7. s 58' l and the sum of the 100 numbers is calCulated. 7(a) Find, approximately, the chance the first sum is bigger than 515-. . ' (1)) Find, approximately, the chance the 4 sums add up to a number bigger than 2060. 13. 14. 15. 16. (5 points) Two draws will be made at random with replacement from the box: El” (a) Find the chance the sum of the two draws turns out to be 4. (b) Sketch the probability histogram for the sum of the two draws. '(5 points) A box contains two red, two white, and one blue marble. Tomorrow, two marbles will be drawn at random, with replacement from the box. Find the chance the second marble is a different color than the first one. {10 points) A gambling house offers the following game: The player puts down a stake of $20. A glass cylinder is loaded with 20 plastic balls, 19 green and 1 red. The halls are blown around inside the cylinder for a while, and then a miniature crane draws out one ball. If that ball is green, the player gets his $20 back, along with a $1 dollarlprize; but if it is red, the player loses the $20. One hundred gamblers resolve to play the game once and then stop. They reason as follows: “On one play, it’s almost certain the ball will be green, and I’ll pick up the dollar.” Suppose the 100 gamblers all play at different times. i (a) The net gain of the house from these gamblers will be around give or take or so. ' (b) Out of the 100 gamblers, the number who win will be around give or take or so. (c) Over a five year period, it might happen that 100,000 gamblers play the game once. Is the house in much danger of losing money to these gamblers? (5 points) A person you trust writes a percentage on a slip of paper, puts the paper in an envelope, and seals the t-envelope. The only thing she tells you is that the percentage is under 10 percent. Your are offered two choices (a) A die will be rolled .60 times. If the percentage of times [3 comes up is bigger than the percentage in the envelope, you win a dollar. (b) As in (a), except the die will be rolled 90 times. Once you make your choice, the envelope will be opened and the die rolled. Which is better: 60 rolls or 90 rolls? Or do the two options offer the same chance of winning? Answer and explain briefly. ‘ ' 17. (10 points) Sixty draws are made at random with replacement from the box: Find,approximately, the chance that shows up 12 times, no more or no less, in the sixty draws. — ___. 1g; (5 points) The box and the number of draws are the same as in a preceding exercise (Problem #17). _ . (a) Find the expected value for the-percentage of times [El shows up. 03) Find ‘the'SE for the percentage-of times [3 shows up. fH'. (IOpoints) A group of $0,000 tax forms shows an average gross income of $37,000 with an SD of $20,000. Furthermore, 25% of the forms show a gross income over $46,000. A simple. random sample of 300 forms is chosen for audit. Find, approximately, the chance that there are either 75 or 76 forms in the sample which show a gross income over $46,000. 38. ( 6 points) Before each play, a gambling machine is loaded with five bills: 1_ fifty dollar bill, I twenty dollar hill, 2 ten dollar bills, and 1 five dollar bill. The machine whirls the bills around and then ejects one (and only one) hill. If the ejected bill is anything except the fifty, the player can keep it. But if it is the fifty, the player must return that fifty and in addition, pay the gambling house another $50. - ' ' After 16 plays, the net gain of the house from this game will be around ‘ give or take or so. 2: . (4 paints) You and a friend are going to play the game described in the preceding problem You agree on the following contract: If the twenty comes out, the friend _ . keeps it. If either of the tens or the five come out, you keep that. But if it is the fifty, you both pay $25 (to make up the $50 payment required by the house). Suppose‘the two of you play the game 10 times. (a) Your net gain is like the sum of - draws from the box Fill in the blanks. . (b) Your friend’s net gain is like the sum of draws from the box ' . Fill in the blanks. - ' .179. (5 points) A tiny town consists of 2 blocks. The total population is 10:7 people live on one block and 3 on the other. Every year, a different survey organization takes a simple random sample of 3 people from the town. ' t (a) Fin-d the chance the 1996 sample includes at least one person from-each block.- (b) It is possible that the first person in the town chosen for the sample is the mayor. Find the _ chance this happens only once-inthe-nextfiye years. ' ‘f ‘ .23 ( 5 points) One part of a large survey involves taking a sample of size 100' from the 10,000 - '- households in a certain district of a 'city. The organizers of the survey divide this district into ‘ five sections, each containing 2,000 households. Then they obtain the sample in two stages. First, they draw two sections at random from the five sections in the district. Second, they - , draw 50 households at random from each of these two sections. (All draws are made without replacement.) . (a) Is this sample .of 100 households drawn by a probability method? . (b) Is it a simple random sample of 100 households from the 10,000 households in the district? ' - ' For both parts, answer yes or no and explain your answer. _ 24’ ‘. (IOpoinIs) In a certain game, a player picks a number from 1 to 6 and bets on it. Then a die 15 rolled 4 times. If the player’s number shows up one or more times, the player wins. Otherwise, the player loses. Six people are playing the game: Oliver bets on 1, Tanya on 2, Thad on 3, Felice on 4, Filene on 5, and Sam on .6. (a) Find the chance-that Sam wins. (1)) Find the chance neither Oliver nor Tanya win. (c) Find the chance both Oliver and Tanya win but no one else does. 15’. '( 5 points) Two cards are dealt from a deck of cards. Find Ihechance that neither card is a diamond. (A deck of cards contains 52 cards: 13 spades, l3 hearts,,l3 diamonds, and 13 clubs.) ll; . (1-0 points) A city has 24,300 rental units. The planning department takes a simple random sample of 225 of these units. The distribution of rents in the sample is shown in the table below. The first line of the table, for example, says that 6 out of the 225 units in the sample rented for somewhere between $100 and $249 (inclusive). Range (in dollars) Number of Units ______________________.____-—-—————-———- .100 - 249 6 250 - 499 12 500 - 749 108 750 - 999 71 1,000 and up 28 _‘_.______________—__————-———-"———“__.-_‘-.—_-—— (3) Estimate the percentage of rental units in the city which rent for somewhere between $500 and $999 (inclusive). ‘ (b) Attach an SE to the estimate in (a). is a 95% confidence interval for [0 (c) The range from M. 2-7 (10 points) There are 300 people in a room. Each of them is going to toss aeoin, and anyone ' ' who gets tails will leave the room. Then, all those left behind will toss a com one more time, and again, anyone who gets tails will leave the room. After the two sets of tosses, the number of people remaining in the room will be around give or take or so. Find the chance that the first and fifth cards are hearts, and these are the only hearts Ill the five cards. (You do not have to Work out the arithmetic.) ( A deck of cards contains 52 cards: 13 spades, 13 hearts, 13 diamonds, and 13 clubs.) 7,1. "(5p0ints) First one ticket, and then a second one, are drawn at random from the’box shown below. The draws are made without replacement. tnflnmm Find the Chance the letters on the two tickets are next to each other in the alphabet. ’30 -. (.5 points) A large number of draws will be made at random, with replacement, from the box: IIE Here are two offers: (i), You win a dollar if the number of A's in the draws is 10 or more above the expected number. {ii} You win a dollar if the number of B’s in the draws is 10 or more above the expected number. Check (N!) the most reasonable option below: (i) gives a better chance of winning than '(ii). (i) and (ii) gives the same chance of winning. mfii) gives better chance of winning than (i). there is not enough information'to decide. State the reasoning behind your choice. 3), (5 points) A pairtof dice is rolled 100 times. As far as the chances are concerned, the number of times a total of 10 shows uplike the sum of - draws from the box: Fill in the first‘b‘lank with a number and the second with a box. Your box should show tickets marked with numbers. . 12, (10 points) A roulette wheel will be spun 100 times. Find, approximately, the chance that red comes up on less than half of the spins. (Note: A roulette wheel has 38 slots: 18 red, 18 blacks, and 2 green. When the wheel is spun, the ball is equally likely to end ...
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