This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 swanm PEANUTS CLASSICS l, _(15 points) Homer Simpson’s “D’oh!” bakery in Springﬁeld, Vermontpurchases 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 $1 per loaf.' The bakery’s
weekly demand for loaves of bread is a random variable (x) with the following probability
_' distribution: ' Demand Probability x 90:).
40 .10
60 .25.
so _ .35
100 .  .20
120 .10 (2) a. If Homer bases weekly orders on the expected value of‘the weekly demand,
what should the weekly order quantity be? (3) b. What is the standard deviation in the weekly number of loaves demanded? (3) c. ' If Homer orders 100 loaves, what Is the chance they will all be sold by the
end of the week? (7) d. What would be Homer’s expected proﬁt (or loss) if he orders l00 loaves of
bread per week? {Hint Firstﬁnd Homer’s expected revenue. Then ﬁnd his
expected cost Finally, find his expected proﬁt, the difference between his
expected revenue and his expected cost. } d %éxg 2X 6‘8 disalﬁgl‘mkiﬁaoéaﬁg 77 £2in a We fie lei [swig , * * .
[40% 1:33 a... l~ @9005}— 742‘ . . . 
4W _ ,
Sr «(:01 Wzréza'lwm (L‘ .L/Xﬂé meg “' if}; meagre lCKe/zog al Eﬂatéége :g/bﬁt +EOZZS§+xoOZEaL
Efmmg ‘5 77x12 44/4» 77£x$ / 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
usea 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 ”(ET4
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 10minute
period on Saturdays to be 6.2. She assumed that arrivals per 10minute interval
followed the Poisson distribution. (4) a. What is the probability that exactly a dozen (namely, 12) customers will enter
‘ the bakery in a 10minute 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 5minute 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?; (553 “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 longterm 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" + doAECﬂZZS 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 withtheir 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???” 9325362;: 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 arrivingat 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
21
26
27
28
.29
30 AREAS 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 Bum6 6151mm. 77:81.: 67176666: 1176861111773 Func
tions. Applied Mathemaﬁes Series 23. 0.5. Depanmem of Comm. 1953. ...
View
Full Document
 Fall '09

Click to edit the document details