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Colgate MATH 102 - Core 143, Sections AX and BX — Exam II

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April 8, 2003Math 102 / Core 143, Sections AX and BX — Exam IIShow all work clearly for partial credit; in particular, write down any difficult computations beforeyou perform them. An unevaluated expression is worth more than the numerical answer withoutexplanation.1. (20 points) A deck of cards for the game of pinochle has only 6 ranks, 9 through ace, buttwo cards of each rank and suit. So, for example, there are two aces of spades, and thereare a total of only 48 cards in the deck. Suppose two cards from a pinochle deck are dealtface-down on the table.(a) What is the probability that the first card is an ace?(b) Given that the first card is a king, what is the probability that the second is also a king?(c) What is the probability that both cards are kings?(d) What is the probability that the first card is either a queen or a heart?(e) Suppose that one card is dealt face up, noted and returned to the deck, and the deck isreshuffled, all a total of 5 times. What is the probability that at least one of the 5 cardsnoted is a spade?2. (8 points) Suppose 20 children are paired off at random into 10 pairs; one of each pair istaught to read using the phonics method, the other by the whole-word metho d. After 10weeks of instruction, all the children take a standardized reading test. Assume the twoteaching methods are equally effective. What is the probability that in at least 8 out of the10 pairs, the child who was taught by the whole-word method gets a higher score on the test,just by chance?3. (15 points) It is possible to make a solid with eight sides, each an equilateral triangle. Supposethat the sides of such a solid are numbered 1 through 8, to form an 8-sided die, and that thisdie is rolled a large number of times . Which of the following statements are likely to be trueas the number of rolls increases?(a) The (absolute value of the) difference between the number of 4’s rolled and one-eighthof the total number of rolls decreases.(b) The (absolute value of the) difference between the fraction of rolls that are 4’s and thenumber one-eighth decreases.(c) The probability histogram for the sum of the rolls more closely approximates the normalcurve (when converted to standard units).(d) The probability histogram for the average of the rolls more closely approximates thenormal curve (when converted to standard units).(e) The histogram for the numbers rolled more closely approximates the normal curve (whenconverted to standard units).4. (15 points) The game of Essel has two variations. Playing the variation Ess, you have 1 outof 5 chances to win and the game pays 3 to 1. Playing the variation El, you have 1 out of 10chances to win and the game pays 7 to 1. Suppose you play the game 100 times, betting $1each time.(a) If you always play variation Ess, you should expect to win a total of $ , give ortake $ .(b) If you always play variation El, you should expect to win a total of $ , give ortake $ .(c) You are more likely to break even if you always play variation (Ess, El orneither).5. (18 po ints) A marketing survey collects responses by 400 households chosen at random froma city with 100,000 households to learn average household income. The average householdincome in the survey is $24,000 with a standard deviation of $10,000.(a) This survey process can be modelled as drawing tickets from a box. How many draws?(b) When estimating the average income, what values should the tickets in the box be?Choose one answer:income levels ones and zeros(c) True or False: We don’t know the standard deviation of household income for the city,but we can approximate it with the standard deviation of the incomes in the sample.6. (16 points) In the survey described in question 5 above (again, with a sample of 400 house-holds), additional data is collected:• The number of households with at least one pet is 180.• The number of people per household averages 2.4 with a standard deviation of 1.4.• The average for the 400 households in the survey spent on pet food each month is $54with a standard deviation of $20.Answer the following questions, if possible, using the above information. If it is not pos-sible given the data above, write the formula you would use and identify what additionalinformation is needed.(a) Estimate (with error estimate) the percentage of households in the city which have pets.(b) Estimate (with error estimate) the number of households in the city with more than 2people.(c) Give a 95% confidence interval for the average amount each household in the city spendson pet food each month.(d) What is the chance of the survey result for the average number of people per householdbeing 2.4 or lower if the city-wide average is really 2.5 with a standard deviation of 1.0?7. (10 points) Relative to, “Poll watchers: The poll less traveled” by Richard Morin and ClaudiaDeane in The Washington Post, September 20, 2000: What types of error (other than theofficial “statistical error”) are there for these political polls? What might have been meantby the comment that a likely voter model “works better closer to the election”?Some Possibly Useful Formulas:n!k!(n − k)!pk(1 − p)n−kEV of sum (count) = n · (AV of box) SE of sum (count) =√n · (SD of box)EV of avg (%) = AV of box SE of avg (%) =SD of box√nSD = (larger − smaller)p(fraction with larger)(fraction with smaller)Math 102 / Core 143 AX and BX — Solutions to Exam II1. (a)848=16(b)747(c)16·747=7282(d)16+14−124=38(e) 1 − P (no spades) = 1 − (34)5≈ 0.7632. C(10, 8)(12)8(12)2+ C(10, 9)(12)9(12) + C(10, 10)(12)10= [10·92·1+ 10 + 1]/210=561024=7128.3. (a) False: It should increase, because the SE of the count increases with the number of rolls.(b) True: The SE of the percentage decreases with the number of rolls.(c) True: Central Limit Theorem.(d) Ditto. Converting from sum to average is a linear change of variable, so it doesn’t changestandard units.(e) False: It gets closer to a “uniform distribution”, flat from 1 to 8.4. (a) EV of the sum is 100((1/5)3 + (4/5)(−1)) = −20, with an SE of(3 − (−1))p(1/5)(4/5)√100 = 4(2/5)(10) = 16.(b) EV of the sum is 100((1/10)7 + (9/10)(−1)) = −20, with an SE of(7 − (−1))p(1/10)(9/10)√100 = 8(3/10)(10) = 24.(c) Because the EV’s are e qual (and negative) and El has a larger SE, you are mo re likelyto break even (or to lose bigger) playing El.5. (a) 400(b) income levels(c) True: That is the process of “bootstrapping”.6. (a) The EV of the average


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