May 9, 2003Math 102 / Core 143 Section AX and BX — Final Exam IIShow all work clearly for partial credit — an unevaluated expression is worth more than thenumerical answer.1. (24 points) A survey is taken of 1600 households in a large city; it is found that 900 of themheat with electricity, and that the average monthly electric bill (for all 1600) is $250, with anSD of $120.(a) Assuming that the households surveyed were simple random sample of the city, what isa 95% confidence interval for the average monthly electric bill in the city?(b) How many households would the survey have had include to get a 95% confidence intervalonly half as wide (assuming similar average and SD)?(c) The power company claims that 60% of the city’s households use electric heat. Is thesurvey’s result significant evidence that the company’s estimate is too high?(d) Suppose the survey sample was conducted by e-mail, using the client list of an internetservice provider. How might that have biased the results?2. (20 points) A “Magic 8-Ball” is a toy that is almost spherical but with one flat window intoits interior. A dodecahedron (12-sided solid) floats in the black fluid inside the 8-ball; whenthe 8-ball is held with the window up, one side of the dodecahedron floats up to the window,and the user can read what is printed on that side. Four sides of the dodecahedron say “Yes”,four say “No”, three say “Maybe”, and one says “Ask again later”. The manufacturer teststhe the 8-ball by s haking it 240 times and noting results. She ge ts 85 “Yes ”, 70 “No”, 63“Maybe” and 22 “Ask again later”. Is the 8-ball fair?3. (13 points) From a new batch at the Utica Club Brewery, 16 bottles of Erie Canal Hard Waterare sampled for levels of duclamine (an ingredient necessary for that authentic rock-strewnflavor). The specifications for ECHW say it should contain 3 mg of duclamine per ounce;the 16 bottles averaged 2.8 mg per ounce, with an SD of 0.7 mg. We are to decide whetherthe sample’s higher level batch means (with 95% certainty) that the whole batch’s duclaminelevel is too low. Complete as much of the decision process as you can.4. (16 points) Recall that a “straight” deck of cards (i.e., one used for bridge or poker) has 52cards, in 13 ranks and 4 suits.(a) If two cards are chosen at random without replacement, what is the probability thatboth of them are clubs?(b) If a card is chosen at random, what is the probability that it is either a king or a club?(c) What is the probability of getting a king at least once if a card is drawn 8 times withreplacement?(d) What is the probability of getting a club exactly 5 times if a card is drawn 8 times withreplacement?15. (15 points) At Turning Stone Casino, you decide to play roulette 100 times, betting $1 eachtime.(a) What are your expected total winnings, give or take how much, if you bet splits (2winners out of 38 numbers, paying 17 to 1)?(b) What are your expected average winnings, give or take how much, if you bet sections(12 winners out of 38 numbers, paying 2 to 1)?(c) With which game are you more likely to lose more than $10? Explain.6. (15 points) The employment rate (fraction of the work force that is employed) averages 87%in New York villages, with a SD of 10%, and the welfare rate (fraction of the population onwelfare) averages 8% with an SD of 6%. The correlation between the rates is −0.6.(a) What should we guess is the welfare rate in a village with an employment rate of 92%?(b) How far should we expect our guess in (a) to be off?(c) If we found that the welfare rate in that village was really 7%, what is the correspondingresidual?7. (12 points) In a small community, an unusually large number of people seem to have colds,and many of the sufferers are found to frequent the community’s only restaurant. Answereach of the following in a sentence or two:(a) How might going to the restaurant have caused its customers to have colds?(b) How might the fact that they have colds caused people to eat at the restaurant?(c) What other factor may have caused people both to get colds and to eat at the restaurant?8. (10 points) Relative to the article “Monitor after-school programs carefully” by Megan Beck-ett: Describe how this article calls for methods from the current section of this course [sig-nificance tests] and from the first section on experimental design. Why are both conceptsneeded?2Math 102 / Core 143 AX and BX— Solutions to Final Exam II1. (a) $250 ± 2($120/√1600) = $250 ± $6.(b) Four times as many households, or 6400.(c) Taking the power company’s estimate as the null hypothesis, we perform a z-test: Be-cause the SE for percent is (using the figure from the null hypothesis)(1 − 0)q(.6)(.4)/√1600 ≈ 1.2% ,we haveP (% ≤ 900/1600 ≈ 56%) = P (z ≤ (56% − 60%)/1.2% ≈ −3.1) < 5% ;so we reject the NH: the power com pany’s estimate is too high.(d) One p os sible answer: People with e-mail addresses are more likely to be technologicallyinclined, and hence to prefer electric heat to, say, oil heat.2. We expect counts of 80,80,60,20 of a fair 8-ball, and we got 85,70,63,22; so we can do a χ2-testto decide whether there is a significant difference between the two.χ2=(85 − 80)280+(70 − 80)280+(63 − 60)260+(22 − 20)220=2580+10080+960+420=25 + 100 + 12 + 1680≈ 1.91and with 4 −1 = 3 degrees of freedom, we see from the table that a χ2-value that high occursby chance between 70% and 50% of the time, not less than 5%; so we do not reject the nullhypothesis: The 8-ball is fair.3. With the small sample, we conduct a t-test, with the null hypothesis that the batch’sduclamine level is the required 3 mg per ounce, and using the SD+of the sample to esti-mate the SD of the batch: The SE is .7(q1615)/√16 ≈ .18, so t = (2.8 − 3)/.18, which isslightly less than −1. The degree s of freedom is 16 −1 = 15. We don’t have a t-table, but itwould probably show that the probability of a t-value that low or lower is not less than 5%,so we would not reject the null hypothesis and would accept the batch.4. (a) (13/52)(12/51) = 1/17 ≈ 6%.(b) (4/52) + (13/52) − (1/52) = 16/52 = 4/13 ≈ 31%.(c) 1 − (12/13)8≈ 47%(d) C(8, 5)(1/4)5(3/4)3≈ 2%5. (a) The EV of the sum is (238(17) +3638(−1))(100) =−10019≈ −5.3 dollars, and the SE of thesum is (17 − (−1))q238·3638(√100) ≈ 40.2 dollars.(b) The EV of the average is (1238(2) +2638(−1)) =−119≈ −0.053 dollars, and the SE