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fall 2007 - rm 1/‘.1“*’L Paul M Sommers Fall 2007...

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Unformatted text preview: rm .. 1% _/‘:.1“*’L\, - Paul M. Sommers Fall, 2007 Economics 210 ECONOMIC STATISTICS Exam #2 HONOR boos (‘fl'have ' neither given not received unauthorized aid on this exam”): Signature ' Please print your name . _ : ' ‘. 1 You have up to two and a half hours to complete this 100 point exam. Be brief and to‘the point since only the main p'oint(s) asked for in each question will aid your grade while a display of. ignorance can count against you One point (up to a maximum of three points) will be deducted from any accumulated extra credit points for each misspelled word In either the “Honor Code” (which I ask you to write out) or in your answer to a question in which the same word was used (and spelled correctly) in the question. Please do all the questions and show your work. (No credit without an explanation. Correct answers without any work will be penalized.) Good luck, happy landing, and I trust that after the exam you won’t ask yourself the same question Homer Simpson recently asked: “How come things that happen tostupid people keep happening to me?” " ulELL 12911011 THlNK V A REPORT cans- YOU LL 5‘21 A soon You. KNOB) lame. E 61VE“5ATI§FACTORY" ‘T’IHANKS Foa‘ REPORT c.1111; 11115 row. as “111151111591.ch 551115 HERE“? um Fumro swan-m PEANUTS CLASSICS l, _(15 points) Homer Simpson’s “D’oh!” bakery in Springfield, Vermont-purchases large ' quantities of bread for sale during a week. The bread is purchased for 60 cents per loaf and is sold for $2 per loaf. Any loaves of bread not sold by the end of the week can be sold to “Krusty’s Crusts”5 a local discount food store, for $1 per loaf.' The bakery’s weekly demand for loaves of bread is a random variable (x) with the following probability _' distribution: ' Demand Probability x 90:). 40 .10 60 .25. so _ .35 100 . - .20 120 .10 (2) a. If Homer bases weekly orders on the expected value of‘the weekly demand, what should the weekly order quantity be? (3) b. What is the standard deviation in the weekly number of loaves demanded? (3) c. ' If Homer orders 100 loaves, what Is the chance they will all be sold by the end of the week? (7) d. What would be Homer’s expected profit (or loss) if he orders l00 loaves of bread per week? {Hint Firstfind Homer’s expected revenue. Then find his expected cost Finally, find his expected profit, the difference between his expected revenue and his expected cost. } d %éxg 2X 6‘8 disalfigl‘mkifiaoéafig 77 £2in a We fie lei [swig , * * . [40% 1:33 a... l~ @9005}— 742‘ . . . - 4W _ , Sr «(:01 Wzréza'lwm (L‘- .L/Xflé meg “'- if}; meagre lCKe/zog al Eflatéége :g/bfit +EOZZS§+xoOZEaL Efmmg ‘5 77x12 4-4/4» 77£x$ / La.) 2. (12 points) According to the National Bridge Inspection Standard (NBIS), public bridges over 20 feet in length must be inspected and rated every two years. The NBIS ' _ rating scale ranges from 0 (poorest rating) to 9 (highest rating). Suppose that engineers use-a probabilistic model to forecast the inSpection ratings of all major bridges in Vermont. For the year 2008, suppose that the engineers forecast that 18 percent of all major bridges in Vermont will have ratings of 4 or below. (4) a. Use the forecast to find the probability that in a random sample of 12 major bridges in Vermont none of these bridges will have an inspection rating of 4 or below in 2008. {Hint A bridge either has a rating of 4 or below or it does not. The probability (.18) a bridge has a rating of 4 or below is assumed constant from one inSpected bridge to the next. And, the outcome of inspecting , one bridge is independent of the outcome of inspecting another bridge} (4) b. Use the forecast to find the probability that in a random sample of 12 major bridges in Vermont at least 3 (three) will have an inspection rating of 4 or below in 2008. ' (4) c. If x denotes the number of bridges in Vermont with an inspection rating of 4 or below in 2008 and the sample size is 12, find E(x) and the standard deviation of x. y M: (a. graze ”(ET-4 03 (a it ‘ie § ~=l‘ :1 B ‘3 3} N 6,“ 1% 3. (l0 points) As part of a class project targeted at improving the services of “D’oh!” bakery, Lisa Simpson monitored customer arrivals for several Saturdays. Using the . arrival data, she estimated the average number of customer arrivals per 10-minute period on Saturdays to be 6.2. She assumed that arrivals per 10-minute interval followed the Poisson distribution. (4) a. What is the probability that exactly a dozen (namely, 12) customers will enter ‘ the bakery in a 10-minute interval on any given Saturday? (3) b. If x denotes the number of customer arrivals per Ill—minute period on Saturdays, what is the standard deviation of x? (3) 0. Bart claimsthat the probability that six customers will enter the bakery in a 5-minute interval (on any given Saturday) is identical to the probability described in part (a). If p = 6.2 per 10—minute interval, do you agree with Bart? (f, @erS“ 3&2?“ 12! 51mm [lama/a r 5. z . ,3 6323. (5; - 4. (15 points) Suppose that x is a continuous random variable that denotes annual family income in thousands of dollars. There are three branches to the probability density function (pdt) for x: r - _ ' _ 1 “Low Income” families 0 S x S 50 p(x)= x , 3250 ' 1 “Middle income” families 50 < x S 80 p(x)= E5- 1 “High income” families 80 < x S 100 p(x) = W_(100 — x) . - I300 . , (3)7 I a. What fraction of all families are “low income”? (3) b. What fraction of all families are “middle income”? (l) C. Is the median income more or less than-$50,000? Briefly explain your answer. ' (4) d. What is the median family income (to the nearest thousand)? (4) e. What fraction of these families (described by the three branches of the pdt) I earn less than the annual comprehensive fee at Middlebury College of $46,910? a“ idea )6?be (x8 ”53?; (5-5-3 “545. (go/53493 ' " %é{45’0§ 1“” 25935543; 3 :3345 CQQMBKWS ' * =,4A/5‘ ééw‘l‘ 444/ I We: -- - am alWég M“ - lg: flag/615a: 91a MZW; /, mg 754% flail/Mme file!” @0529 / . ' 5b we. 3’ e Elf (3g%+n?wgl " 4’ ~ ' . fll» ”52% =5 (##50§(Z§> Mm M/éwmafifim -' , a; M=5¢5 (gr £§Z§m§ , a . . filfle X 2%,41 we. Ufie film gist—Emma affie; 6/)4 _ I' I. 36454151§L§56¥é i 13133315 " _ . firaém2fr%,55f6wfi a If! C.“ 5. (15 points) Before negotiating a long-term construction contract, building contractors must carefully estimate the total cost of completing the project. For one particular Middlebury College construction contract, assume that total cost, x, is normally _ distributed with mean $850,000 and standard deviation $170,000. The revenue, R, promised to the contractor is $1,000,000. (6)~ a. The contract will be profitable if revenue exceeds total cost. What Is the _ probability that the contract will be profitable for the contractor? That Is, ‘ what 13 p(x < $1,000,000)? (2) b. What is the probability that the project will result in a loss for the contractor? (7) 0. Suppose the contractor has the opportunity to renegotiate the contract with Middlebury‘ College. What value of R should the contractor strive for In order to have a 95 percent pr bability of making a profit? .4. falcons) -- £53600 ._ is??? g 5;: #9005: Mb 1: 'gg : 25,5" + doAE-CflZZS 6. (8 points) The Value Line Survey, a service for common stock investors, provides its subscribers with up—to—date evaluations of the prOSpects and risks associated with the purchase of a large number of common stocks. Each stock is ranked “I” (highest) to “5” (lowest) according to Value Line’s estimate of the stock’s potential for price appreciation during the next 12 months. SUppose you plan to purchase stock in three electrical utility companies from among eight that possess rankings of “2”'for price _ appreciation. Unknown to you, two of the companies will experience serious difficulties with-their nuclear facilities during the coming year. If you randomly select the three companies from among the eight (that is, you are sampling without replacement), what is the probability that you select: (4) a. None of the companies with prospective nuclear difficulties. (4) b. Both of the companies with prospective nuclear difficulties. ”/94 GM will ' ' 4 M £2 ’1 3 Eli/l MW aware ' - ' . Jl‘fiijfl/érf‘ @f/fi’ég _ 7. (10 points) Suppose that “Neil & Otto’s Pizza Cellar” here in Middlebury has mailed out 250 coupons to local residents good for $5 off their Earge pizza if redeemed within , a month. From past experience, they expect about 22 percent of all coupons to be used. (2) a. What is the expected number of redeemed coupons? (2) b. What is the standard deviation? - (6) c. Use the normal approximation to the binomial distribution to find the probability that between 50 and 65 coupons (including 50 and 65) will be redeemed. That Is approximate pB(50_ < x < 65). 4’ , 51598 "7‘ I???” 932-5362;: e 575- b (alga/[S VHZI’KFTl-‘b a" é 54¢? 8. (3 points) In one or two sentences, what is the biggest problem with using “covariance” (rather than the correlation coefficient) as a measure of the strength of the linear relationship between two variables, say, height x and weight y? - TWpEflpflagfi/H’Wéefié 1%.ka ' 0M3 flzmewcmmf‘ {<2} X and ’ E47114 g MW # WWQMD 2/ $412 @3521!" f8 m Wilt/6g lM xe Maj—d 4 Z: _ [Ml/g (1F magma?“ 6F K dirt/j JE’ ' @Wé/ @1997? (599(44)::er (4) 10. (12 points) Because delays at major airports are now more frequent than ever, it helps to know which airports are likely to throw off your schedule. In addition, if your plane ' is late arriv-ingat a particular airport where you must make a connection, it would help to know if the departure will also be late (thereby inereasing your chances of making the connection). Data were collected for a sample of 13 airports around the US. in August of a recent year. These data included the percentage of late arrivals (for example. in Chicago it was 30 percent) and late departures (in Chicago it was 29 percent). Let )4 denote the percentage of late arrivals and y denote the percentage of' late departures. (That is, note that 'x = 30 means “30 percent” and y = 29 means “29 percent") Then for the 1'3 airports, we find that: _Zx,. =273 Zyi=265 2x? =5899 2y} #5553 2w. .= 5707 a. Find the sample correlation coefficient between it and y. Please remember to show your work. {Hintz If you are unable to compute rxy , then for partial credit, find the covariance. That is, tell me whether the relationship between x and y is direct or indirect. No credit for a correct guess. Please use the numbers presented here to support your conclusion} E). Find the estimated regression equation relating the percentage of late departures (y) to the percentage of late arrivals (x). - . c. Suppose the percentage of late arrivals at the Philadelphia airport (not in the original sample of 13) for August (in that same recent year) was 22 percent. What is an estimate of the percentage of late departures (at the Philadelphia airport)? I. 0.0 0.1 .-0.2 0.3 ' 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4‘ 1.5 1.6 1.7 .1.8 1.9 2.0 . 23 ; 22 _23 2A 2-1 26 27 28 .29 30 AREA-S UNDER. THE STANDARD NORMAL CURVE This table shows the area between zero (the mean of a standard normal variable) and 2. For example if z— = 1. 50, this rs the shaded area shown below which equals ..4332 .00 .0000 .0398 .0793 .1179 .1554 .1915 .2257' .2580 .2881- .3159 -.3413_ .3643 .3849 .4032 .4192 .4332 .4452 .4554 .464 1 .4713 I .4772 .4821 .4861 .4893 .4918 .4938 .4953' .4965 .4974_ A981 .4987 .01 .0040 .0438 .0832 ‘.1217 .1591‘ .1950 .2291 .2611 .2910 .3186 .3438 ' .3665 .3 869 .4049 .4207 I .4345 .4463 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920 -4940 .4955_ .4966 .4975 .4982 .4987 .02 .0080 .0478 .0871‘ .1255 .1628 '.l.985 .2324 .2642 .2939 .32 12 .3461 .3686, .3888 .4066 .4222 .4357 .4474 .4573 .4656 .4726 .4783 .4830 L4868' .4898 - .4922 .4941 .4956 .4967 .4976 .4982 .4987V .03. .0120 .0517 .091 0. .1293 .1664 .2019 ' .2357 .2673 .2961 .3238 .3485 ' .3708 .3907 . .4082 .4236 .4370 ' .4484 .4582 .4664 ' $4732 .4788 .4834 .4871 . .4901 .4925 .4943 " .4957 .4968 .4977 .4983 .4988_ .04 .0160 1557 .0948 .1331 .1700 .2054 .2389 .2704 .2995 .3264 .3508 .3729 .3925 .4099 .425 1 .4382 .4495 - .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927 .4945 .4959 .4969 .4977 .4984 .4988 .05 .0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3023 .3289 .3531 .3749 .3944? .4115 .4265 .4394 .4505 .4599 .4678 .4744 .4798 .4842 .4878 - .4906 .4929 .4946 .4960 .4970 .4978 .4984 .4989 .06 .0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .305 1 .33 15 .3554 -3770 .3962 .413 1 .4279 .4406 .4515 .4608 .4686 .4750 .4803 .4846 .488 l .4909 .493 l . .4948 .496 1 .497 1 .4979‘ .4985 .4989 .07 .0279 .0675 .1064 .1443 .1303 .2157 1 .2486 .2794 ' .3078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .4525 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932. .4949 A962 .4972 .4979 .4985 .4989 .08 .0319 .0714 .1103 .1480 .1844 .2190 1.2517 -2823 .3106 .3365 .3599 .3810 .3997- .4162 34306 .4429 .4535 .4625 .4699 .4761 .4812 .4854 .4887 .4913 ' .4934 .4951 .4963 .4973 .4980 .4986 .4990 .09 .0359 .0753 .1141 ' .1517 -1879 .2224 . .2549 .2852 .3133 .3389 .3621 .3830 .4015 .4177 .4319' .4441 .4545 .4633 .4706 .4767 .48 17 .4857 .4890 .491 6 .4936 .4952 .4964 .4974 .4981 . .4986 .4990 " 'S¢mrce.' This table is named from “11.101131 Bum-6 6151mm. 77:81.: 67176666: 117-6861111773 Func- tions. Applied Mathemafies Series 23. 0.5. Depanmem of Comm. 1953. ...
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