~FJf: I~FJf
f4 EI : {'F*tiJf~
1. Consider a supply chain consisting of two firms: a manufacturer and a carrier. To
satisfy a known demand, the manufacturer chooses its production amounts for four
plants that produce the same product but have different requirements of resources and
different margins, as summarized in Table l(a). The total demand of the market is
100 units, and overproduction is not allowed. Once the production is finished, the
manufacturer will deliver products to the market arranged by a carrier. The carrier
considers each plant as a node and decides its distribution plan after the manufacturer's
production plan is determined. Table 1(b) summarizes the information of carrier's unit
shipping cost of each path and path capacities. Each of the firms makes decisions by
optimizing the individual profit or cost. (The manufacturer's and carrier's problems
satisfy the properties required for LP models.)
(a) What properties are required for formulating the firms' problems as LP models?
If the market demand follows a uniform distribution, what property is violated?
(10%)
(b) Find the optimal production quantities of the four plants and profit obtained by
the manufacturer. (10%)
(c) From part (2), find the shadow price of each resource, and interpret that increasing
what resource provides the greatest profit improvement? (10%)
(d) Solve the carrier's problem as a minimum-cost flow problem. Please show the
optimal decisions and total cost borne by the carrier. (10%)
(e) If the margin of Plant 3 decreases to $20, does the basis obtained in part' (d) of
the carrier's problem remain optimal? Explain. (10%)
Table 1: Information of resources, margins, and shipping costs
(b) Shipping costs and path capacities
(a) Resource requirements and margins
Plant 1
Resources Plant
A
B
C
Margin
Plant
Plant
Plant
Plant
2
3
2
2
4
2
3
3
2
3
4
3
$40
$30
$40
$20
280
320
300
1
2
3
4
Available
amount
Plant
Plant
Plant
Plant
1
2
3
4
Plaut 2
Plant 3
Plant 4
$7 (00)
$4 (00)
S3 (30)
$4
$3 (40)
$2 (00)
Note: - unavailable path; (-) path capacity
1
Market
$5 (50)
$5 (00)
~J5JT
: I1fJ5lT
f4 § : 1'F*:fiff~
2. Consider the inventory control problem of a product of a store. The inventory of the
product is checked at the end of each day, if the inventory level is less than s, we place a
replenishment order to bring the inventory level up to S. The replenishment will arrive
at the store at the beginning of the next day. Let Dn be random demand at day n
with probability P(Dn = k) = dk , k = 0,1,2, .... D 1 , D2 , •.. be iid random variables.
Assume that unsatisfied demands are lost with a cost of $b per unit of unsatisfied
demand. Besides, there is a setup cost $M of placing a replenishment order and a
holding cost $h per unit charged for each unit of inventory held at the end of the day.
Let Xn be inventory level of the product at the end of day n. {Xn} is a Markov chain.
(a) Find the transition probability matrix of {Xn}.(% 15)
(b) Assume that {Xn} has steady state distribution limn-tooP(Xn = i) =
the following terms in terms of $M, $b, $h, 1ri, i = 0, 1,2, ...
o Average
• Average
• Average
• Average
• Average
1ri,
express
inventory leveL (% 5)
unmet demand per day. (% 5)
replenishment order size.(% 5)
time between placing replenishment orders.(% 5)
system cost per day.(% 10)
(c) If the product considered is perishable, say the product are boxes of milk. Assume
each box of milk has to be sold within three days after it arrives to the store and
we always sell boxes of milk in the shop longest of time first. If a box of milk has
been in store for more than 3 days, we need to discard it. The status of each box
of milk is checked at the end of each day. The inventory system of the product
considered is still a Markov chain. Define the state of this Markov chain and find
the transition probabilities. (% 5)
2
*JifT: I~JifT
f4El : 1:il:~J!R
Part A
~4U. 11M
:
Part A ~4~~ 44' "litth A-l.i.A-4
0
*41?~~1F~"litii'll~.fiit~ 'PartA ~tt50~ 0 1.A A, B rifJ~A£PJTj;~
-'1J'I.1.,lB~M ~JJ~ i.b 0.5 J4 0.3 +~ rifJ~AJ'1l4*-* 4 *~j; -'R:kuT *- 0A, B rifJftA
£~-'~1.A4*~Mi.b~0J4~0,~ftA£.-*~-'~#i~*fi.b~0*.a •
• *~,l~6i.b 500 +B~ 0 tt4~+1t.4*~El{~T' J:A~,tittt:~lt~l~11~1.Aitt1
;nh~(~;t -'RM) 0
A-I:
(I5%);tt-.::r:..ffiit1t}fH!3/&J~1~11k1.A
A, B
rifJ~A£
0
0
2
600
800
1
A
300
800
B
3
700
1000
4
400
600
J
i
I
A-2:(15%);tt .::r:..~1tffl-far, .$-*i!-1T A J'1l~JJt1.::r:..1"F ~ 0 i.bM!st1~.;!tf..f: ~ l!J 1f~~1f
Mt1t}plTl!4~~jH~;. ' 1~. Mt1t}:g~ rttt• .$-*1*~ *TJ¥.1J'I.~. ffl ~14"ffl ~14"-~J¥.
1J'I.,ijfA4;f..i.b$1000' Jill~~·tii.b I1f.' *-1f.tXJ*-1tfflMJt?t1ii.b 0 0 ~~.~~~1~11k
0
• •
e-A.;{,x
.::r:..flgiji-&JiAAA-it1i'
.1~11kifJ~-1f.-i-1.
3 ;:J::.;!tP.f:,
Ji.i*tA-~i'lic.(P(X=x)=-I- ,
x.
~$1i:kuT*PJT~)0~~~~.~#~i4;f..~~~o
x
P
0
0.0498
1
0.1494
2
0.2240
4
0.1680
3
0.2240
A-3:(10%)~t1tT 7'1~"'~'#~flr,,~~tt J-A CR(critical ratio)%t-M *M.#~Jt
~ifi~ .::r:..1"Fit
' Jt it J. ~~d~. 'f
0
.::r:..
1"F
1'F1f-*M
jlj JiA a
1
2
5
12
8
25
3
14
42
4
6
20
5
3
18
6
2
6
~pff: I~pff
f4El :
A-4:(10%);6: "if ffl }t 5~ 1.1 4 J¥-1ft tt t.J.. ACTRES
* JlIJ ' it it r rlJ }t
$~ 1: JILt ~.lJ. ~ if 1£
r", Jl!
l!ff7t.:r..Bi-ra, ~*~ [tt : r Ii tp ~.it~1tJJ'l ~-F-&-1'F 1t ~1'FtatM'&PJf;t,~}t5i,itl
0
*rJT :
f4 § :
~Et€"gf:!E
Part BttJl1t1lJ3 :
1. Part BttJl#:t9J1iIUfJl' ~ t 44... :il"l("!ft: B-l.i.B-4)' 5J1:flil"!(Jlft: B-5.i.B-9)
2. "'ilJl.Jl5~(#20~) , :fl:il4.1fl6~(#30~) Part B~1t50~
0
0
0
3.... il"!.n~.~~~*il4,~il"l~~~4~*#~o
B-1. (-'il4) ~iH~Jifm1§::Itl~JJ%TI~H~~~I¥J~{t ' ~ifliz:~'ifl¥Jifm1§::It{&Ff7tJjUm :
2,000 ». 2,400" llt9f. ' 0PJ~ifllB*I¥J~~1§:il§~}J%ffi5EI¥J*~1'i~{t ' {i5gt~ifiW-=-*I¥J*
fr'jf~J&~Ff7tJjlJm : 1.2,0.7,0.8 " {&~iWillt1ifl · {i5~tl¥JifmllB*l¥Jm1§::It "
(A) 780
(B) 910
(C) 2080
(D) 3120
(E) 3640
B-2. ("':il"l)
l£~~ol¥J~~tW~~I¥J~~
(A) Reverse Engineering
(C) Quality Function Deployment
(E) Service Blueprint
, Tmt{PJ1fW
r~f*J 1¥Jl1l!~iB'imftlli?
(B) Mass Customization
(D) Remanufacturing
~mI!JcpI¥JV~m-W1iJltlI¥J1Ilw9t~" §iWV~l¥JftRmfUm : -tlmf!~
~dHttf!m!»'~m1m! · =;fIm~f4r~W:»'{,*W:m! ' -=-;fIm~Mm!».*m ' llB;fI»'1i;flmtt~
mmm! T51U{PJfmtfflThfI¥Jl1l!~W~Vlttl¥JftRtJHlfrlJ!I!~iB'i~;f§{~?
(A) Product Layout
(B) Fixed-Position Layout (C) Process Layout
(D) Cellular Layout
(E) V-Shaped Layout
B-3. (... il4)
0
B-4. ("':illfl) ~0PJIEl£~~m9f.~~IHijI¥J{3L:hl: ' 0PJ8tk:JElf~H*€I~!t1ifmrzl*{j1ftm~
:hl:~~ " ~ill¥J~*~;2; : (1)~~~* ' (2)jiitr~* ' (3)~{'*{~{tj: , (4)*It5tViq7j(~ ,
(5)t~1ifl~rrlotl! ~~0PJl:illtl¥Jrm* ' !,[UT51Ui¥Hi51J${PJ1fiB'i~:ii§€I?
0
(A) Locational Cost-Pro fit-Volume Analysis
(C) Closeness Rating Method
(E) Factor Rating Method
B-5. (~ilJl)
(B) Center of Gravity Method
(D) Transportation Model
T51U{PJfmm~J:iJJ:)~{ttm1j1H[!I¥J~IJJ(~p~:pflr:U1lUftl[l¥J~{t:!l:)?
(A)B1*~!WJ2fi¥-:)$mn!U(Ft=(Lr=i At.i)/n) • !,[UJ:iJ~ n {[!~:JJO
(B)B1*:JJD~~IJJ2fi¥-:)$m1j1tj(Ft=( a At.!+ (3 At-2+ r At -3) , !'[u~m{[!J:iJIDijjft~
(C)B1*f~J&2fim$m~IJ(Ft=Ft_!+ a (At-I-F t-!) · !,[UJ:iJ~2fim{*,J& a {[!r~/j\
(D)B1*U[~~~~$ffln!U(Ft=a+bt) · !'[fjJ:iJ~~~~z a '([!IDijj~
(E)J:iJ~
: a +{3 + r =2.00
MAD {[!». MSE {[!IDijj~
~~~JJ%~E8m~ A ~~~~~5b B " ~*»'~~1ifltlDf& : m~ A ~ifl¥J~
JE~*~$20,000 7Glt; , ~5b B 1¥J~1JJ~*m$50 7Glftj: , ~~IWim 20 Ij\WiI{tj: , m~ A
~if~~~~IWi~ 2,000 Ij\Wilt;o~~Wi~~I¥JT-J5V¥m 20%' llT- ~~o~j.,:J-¥~JlJj!l:]!
~5b B l¥Jiftf:t~il~ 96 {tj: " Tillt*~illt1PJ1fIEfft?
(A)ji~ifti:t~:!I:zT ' !'[fjm~ A I¥Jrm*;.m 2
(B)jifflGiftfj~:ItZT ' !,[lJif~~~~*m 24,800
(C)ji~ifti:t~;'z T ' ~fjif~~*,r~fflG*m 44,800 7G
(D)ji~iftfjjtltzT ' ;fi1§:f!~J~ 500 5O{tj: , !,[Uif3l;f1J*,~~~ 2,000 7G
(E)Bl£iftfjJl{;'ZT~~f~:6itZP:~ , !,[UlltWi1§:{Jf~ 383.33 7G
B-6, (:fIilJl)
0
ilU JL
~
** W tx *.
~
*pff : I1fpff
101 ~~Il1iJi±FJf~1iJi±1:E~WFJfffi~~~~m
B-7. (~1I")
f4El: §:.lI1fJ!f!
*fiijT~1ifl3i:t;F~i1E~ ~~i1E*~l¥J~*~*~rmJj :(1 ).R3i:t;FMYlIIf'Fltti '
liMYIf'F%l¥Jf'F~pg¥i1'f§] (2)liIli7~1t'If'F1l~l¥JMilll~ra' (3)1'~lIZP:f.fIf'Fltti (4)
:g.f'F~¥:n:(Task)l¥JI f'F~ra'1'PJfI}0'etJ *t~ttmt~*3i:t;F~f*~ii*~ , 1E31S0'~fll¥J
ilf'F~T ' ~~T~U1Etiil¥J*~mt
0
0
0
0
0
0
Task Time
Immediate
Task
Followers
(0'fi)
0.2
a
b
0.4
d
b
c
0.3
d
1.3
g
0.1
f
O.S
g
f
g
0.3
h
h
1.2
(A)f'F~¥:n: d JJ!3i:t;F1n~-lII f'Fltti
(B)f'F~¥:n: f J!!3i:t;F1n~-lII f'Fi4!i
(C)~-lIIf'Fi4!i l¥J 1M' illl~ra'~0.20'fi
(D)=11I 1'Ftt~l¥J~m $¥-:J:ill 00%
(E)B~x1JDI~ra~~4600'fi ' JtU~@:*~~x@:1ir.¥U1n lS0ftj:
Po
B-S. (;fill") ~ftj: A EE0~ EI
'Mf~k@i3H'I~~*~pJT~ ~ftj: A l¥J~*lifl.~
tlDi& :
A l¥Jif.~*Fi~ 30,000 ftj: , 1&.~rc;*~mBm*£~ftj: A l¥J~;1§f§]Jl1i5E
mm~i1E~~illl~*~ 5,000 :n:/1;( , mll~ftj: A l¥Jif.f¥~~*1~~~1tj:tt*4~1Jll¥J 5% '
ffi]~~1tj:l¥JttflZP¥-:J1*~flf~~ 2,000 :n:/ftj: ttt1i~@:~ ,
B ~@:~$~~ B m*£~$
l¥J=ffi ~if.tJ 250 lII1'Fx~t. ~a1C1E~tE~i1E:t1t1il¥J3If'FT ' JtUT~U*~mtfPJ~1Etii?
(A)~ftj: A m:t1tl¥J~i1E:tJt1i*1n 2,000 ftj: (B)if.f¥~r&;**1n 70,000:n: ' liif.~~*~if.f¥~~*I¥JMYffi (C)~1tj: AI¥JZP¥-:JJ!ftf¥ ;*1n 1,000 ftj: (D)-if.CP , ~ftj: A ~~i1Ex~f~if.I1'FX~1¥J 33.33% (E)*~~lft~~;3i:~J!ftf¥ ' jl&~1tt.f¥1i~
0
0
0
0
0
°
B-9. (;fill") ~~ftj:*'~ MRP ~~~i&' ~ 1 Wl~~ 6 w]~r¥~*;*;:r;m: 20, 70, 30, 60, 50,
40
Fixed-Period Ordering ~ Fixed-Quantity Ordering fI~t;k:5E1*~1¥J Lot Size JtUTmt
~ 1 :M.¥~ 6:M~ Planned-Order Receipts fPJ~1Etii?
(A)1* Fixed-Period Ordering li~~=:M~*T : 0,0, 120,0,0, 150
(B)1* Fixed-Period Ordering li~~=w]~*T : 120,0,0, 150,0,
(C)1* Fixed-Period Ordering li~~ =:MlI*T : 0, 90, 0, 90, 0, 90
(D)1* Fixed-Period Ordering li~~=w]~*T : 90,0,90,0,90,
(E)1* Fixed- Quantity Ordering li1*~:t1t1im 150 T : 150,0,0, 150,0,
0
*
0
°
°
°
m I
~~mJ1:
Jil#
#f-f.~**
I"VA~ 101 ~1f.JJ{Jj±Ji)l~~][±1.f~Wf)]Jg±~~~~
1. The term "inverse demand curve" refers to
(A) the demand for "inverses."
(8) expressing the demand curve in terms of price as a function of quantity
(C) the difference between quantity demanded and supplied at each price
(D) a demand curve that slopes upward
2. Assume the price of a movie is $10. Jenna demands 2 movies per week, Sam demands 3 movies
per week, and Jordan demands 8 movies per week. From this information we can conclude that
(A) Sam is irrational compared to Jenna or Jordan
(8) Jordan is obviously more wealthy than either Sam or Jeanna
(C) the movie industry is unprofitable.
(D) the market quantity demanded at a price of$1O is at least 13 movies per week
3. A vertical demand curve for a particular good implies that consumers are
(A) not interested in that good
(8) irrational.
(C) not sensitive to changes in the price of that good
(D) sensitive to changes in the price of that good
4. If the demand function for orange juice is expressed as Q = 2000 - SOOp, where Q is quantity in
gallons and p is price per gallon measured in dollars, then the demand for orange juice has a
unitary elasticity when price equals
(A) $0
(8) $1
(C) $2 (0)$4 5. On a linear demand curve, the lower the price,
(A) the elasticity equals -I.
(8) the less elastic is demand.
(C) the elasticity equals zero.
(D) the more elastic is demand
6. Convexity of indifference curves implies that consumers are willing to
(A) settle for less of both "x" and "y"
(8) acquire more "x" only if they do not have to give up any "y"
(C) give up more "y" to get an extra "x" the more "x" they have
(D) give up more "y" to get an extra "x" the less "x" they have.
7. If the utility function (U) between food (F) and clothing (C) can be represented as U
the marginal rate of substitution of clothing for food equals
(A) -F/C
(8) -C/F
(C) - -JCfF
(D) _,JF/C
=
.JF" C,
7
Ji)
~~~lL~*W~*~
~PJT: I~pJT'
fi)""r:r( 10 1 ~!rf:~iiJi±1!)f~iiJi±1£11!W1!)f~B±'~~~Jtm
8.
f4E1:
*~~~(1)
An inferior good exhibits
(A) a decline in the quantity demanded as income rises
(8) a downward sloping Engel curve
(C) a negative income elasticity
(0) All of the above
9. What is the primary difference between the substitution and the income effect of a price change?
(A) The substitution effect holds income constant and the income effect holds utility constant
(8) The substitution effect is always negative and the income effect is always positive
(C) The substitution effect holds utility constant and the income effect holds prices constant
(0) The substitution effect is always positive and the income effect is always negative
10. Isoquants that are downward-sloping straight lines exhibit
(A) a decreasing marginal rate oftechnical substitution
(8) an increasing marginal rate of technical substitution
(C) a marginal rate of technical substitution that cannot be determined
(0) a constant marginal rate of technical substitution
11. Let the production function be q=ALaKb. Returns to scale are equal to
(A) a + b
(8) La + Kb
(C) a
*b
(0) A
*L
12. Suppose the short-run production function is q
10
* L. If the wage rate is $10 per unit oflabor,
then Me equals
(A) q/lO
(8) q
(e) 1
(0) 10/q
13. Assuming that wand r are both positive, if the long-run expansion path is horizontal, then
(A) MRTS is a function of capital only
(8) w
=r
(C) MP K
0
(0) All of the above
14. If all conditions for a perfectly competitive market are met
(A) firms demand curves are horizontal
(8) the market demand curve is horizontal
(C) firms face sunk cost when entering the market
(0) the firms' demand curves are downward-sloping
15. Producer surplus equals
(A) total revenue minus total variable cost
(8) profit plus fixed cost
(e) total revenue minus the sum of all marginal cost
(0) All of the above
16. Which of the following is not a potential result of a price floor?
(A) Price greater than free-market equilibrium price
(8) Excess supp Iy
3
~~ IiI 11: ~
}i{(;it
** f-f. 1X * *
1iJ",,~ 101 ~iFltm±Ji)f~m±:tE~WJi)f*Et.~~~Jt~
(C) Lower quality inputs are used which increases marginal cost
(D) All of the above
17. If a society only cares about efficiency and not equity, then
(A) all points on the contract curve yield the same level of social welfare
(B) it will not rely on competitive markets to allocate goods
(C) an equitable outcome is impossible
(D) it will maximize the utility of its worst-off member
18. If a monopoly discovers that the demand for its output has become more elastic at the original
output level, then it will respond by
(A) setting a lower price
(B) producing more while leaving price unchanged
(C) producing more and setting a higher price
(D) setting a higher price
19. The producer surplus to a monopolist must be
(A) positive, otherwise why would the monopoly produce?
(B) at least as great as the producer surplus in a competitive market.
(C) less than zero or the firm is in violation of anti-trust statutes.
(D) the same as for a competitive market.
20. A perfect-price-discriminating monopoly's marginal revenue curve
(A) varies for each consumer.
(B) is the demand curve.
(C) lies below the demand curve.
(D) is the same as the monopolist's marginal revenue curve.
21. Airlines offer lower prices to vacationers than to business travelers because
(A) of government regulations requiring them to do so.
(B) business travelers are less flexible in their travel plans than vacationers are.
(C) airlines know that business travelers enjoy flying more than vacationers do.
(D)
busine~s
travelers do not care at all about costs.
22. A typical firm in a cartel will hold which of the following attitudes?
(A) If everyone cheats, I'm better off, and so is everyone in the cartel.
(B) Ifl suspect others are planning to cheat, I'll do best for myself by deciding not to cheat.
(C) I can never do better for myself than following agreed-upon cartel rules.
(D) Ifl alone cheat, I'm better off; if everyone cheats, I'm worse off.
23. Suppose a market with a Cournot structure has five firms and a market price elasticity of demand
equal to -2. What is a Cournot firm's Lerner Index?
(A) 1
(B) 0.2
(C) 0.1
(D) 0.5
24. A strategy is dominant if
(A) the player cannot gain by changing strategy, assuming that no other player changes strategy
?
}i{1
~~~ft~**f-J.~*¥
itl.A.~ 101 ~!tf.f:tm±rJI~m±1:ElfflU,rJIm±.~~:a:~~
*plT:
I ~.PJT
' m!il#.PJT
: *~~!¥(l)
(8) it yields a greater payoff than any other player receives.
(C) it is part of a Nash equilibrium.
(D) it yields a payoff at least as large as that from any other strategy. regardless of the actions of
other players.
25. If a Cournot duopolist announced that it will double its output
(A) the other firm does not view the announcement as credible.
(B) it becomes the leader.
(C) the other firm will double output also.
(D) the other firm will shut down.
*ffi~~m=mHtm
NPIiI '
9 jd~IiJjl1Efm · 4ij/N1Ii 2 5t ' m:tEi§~~l:~~p..r-fm:ttf'Fi§ · {ltJ:i§~
::fg~flJ3JJj(!Zg&~Ji~~
f'Fi§IfIH{9lj:
(1) i§*...
0
(::fm~fJllJJj(!?>I&~Ji~~) _ _ __
(2) 1t~ ...
(25)
U ...
1. Tlilmte~, rp~*!l!
' ~.&~2001
, ,
¥~ 2010 S"J~!7q~ii:=5~(GOP)JVG~$S"JJE~ ,
20] 0 1fS"J{tl!EE~¥Uft\:5t]jUm A *7~ 8 *~ C *~, .& D ~ ~P.lr~~~[9{~*~5t]jUml!JJ~f~~*S"J GDP
0
JVG~$JE~? A *~~ml, B ~~~, C **mill, D *~~0 (~~llP.lB.~~*~ffl})
0
25.00% , . - - - - - - - - - - - - . - - - - - - ­
20.00% + - - - - - - - - - - - - - - 3 i e : . . . . . - - - - 3 " ' " = : - - - - ­
15.00% -l----_£..--...:~~=-----~-__:.,...
10.00%
~-~-------- ----~~____I
-A
-B
-c
-0
-10.00%
- L ­ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __
' m{!ft$G*~ 500 JC • -!jflMl~1Jlm 1,000,
fg~~~1'H .RJlte 50 {!f · !jfl~~/J\:ffi~fifrE~~tfX~.., ~~2.T .RtlTj;J 300 S"J1Jlf~~A~ 150
2. *[9S"J/J\:ffi*1Nt:E~~~D 200 f!ft$G::&@lir~ts
*?fT:
I~?fT
f4 §I:
*Ji!~~(1)
'
{4~(f~.tB
• ~~/J\1Eit~.B"JEft¥&FJT1~ , ~NFJ:t'/J\1Ett9~1Smlt1W!f~a~[jJ[7g~~=§M(GDP)
tt9~MilN1Jo7m •
3. 1£ai!l1f1t1m 65 ~ , 1J1JoAl@lI1l15f1:1t1m 61iBt)LZI1l~ , Mzt-JiJ~1fI1l~mfU~~r5'M1'J{,*1Jtl
m3 f,*z~jJEff ' -arjPJFJTJIHllitEftIDfJ:t?:M~~r5 ' mB ~JUIX NT$6,000 ~)'jj(A9EL~ B
tXf~ 2012.1.3 !:f:1:9Cffr±i¥j®TfYj¥&~:
0
0
.~~.4~~*L~I.~*~*ff.~'~~+.I~~'~.*M~
i:;"
7,000 .ito
P
~ ~~ ~ ,fJf
i~
68
~
A..~.t.
*1*"~j+*"tJt, ~ I*Jl'!;IJaAi~;jif1±.
2 ~ 20 a It=-3\.A1* '
0
~~~AA.~.I·~~±~'.~1±.~.12~21a~~*L~.I~.
~*M*ff.~'**~~~~**~4.,;jif*~t*~~ft •• ~.~~&
~~~'~.~.~*~~~~'~f~~o
,It=-3\.{t}*-I!!.Jft1TMf #i~: PlTff{l~
11- ' ;Jut ~PlT1~~i& so ;it .itJ-:AJ:.*' ~ 1JMJt~~ I*Jl,!; ; ~ftJ,t 7J iii '.;jtl~ tOt!:. •
I •• '.A..PIT~±~&&£~.'.~~~500~.it~J:.*'~M~~~I*
M;~a~ •• Ak.A..PIT~ ~%d~~-&£'.§~~~400;it.ito
1t.itt.fl7t*.:L~~ t"*.:L1*d!fx'AJ;fitiJt15i.~
(a) ~~FJ:t'~M~~r51£~Mtk5llWEB
6,000 1JD~¥U 7,000 5t ' ~J!~f#Hfi
*llt~1JO~1&i&Jf.fnflfi~ttt1Joa~r#jm~ru
(b) ~~~~r5 19951f~~.flt!i~e~MIl~
0
~NrJ:t'ftH~®TM¥&~If9I!z
0
3,000
'
f$:*m~~~1f{¥!!1J[]u.~
1,000
' L~~~
¥U 6,000 i¥j 8 W!m 2007 8 B ' ~~e~!fo/.J11U~I!z~ 102 20121f1f.WB"JnW.ff~?!Jfilml!z*¥U
108 • ~NFJ:t'~~1JO~B~ft(~~?!JfilnX*$~1'Jaf5j~ffiJ?Fil1I~1JO 1,000 7C ' ~UJ!!mB ~)'jj(~MJ!!~
0
~¥U~o
20 II 1f 6 B tBjlB"J52rJN*~~~~¥&15-{¥!!
1~tB ' ai!l~Mt*5Itt9~.WJ~ti ' gt1£gt*W~m~~~~§ 20081f~Mt*5Im~~ , ,~~Mt1£
~ff~!1iE[§fJ~m7!<~-JE~:tE*J~FJT1~tt9it~~Jf§:{iil!zill¥1J 6 {tit) -r flU~fri&lIiG1:lltJ1ll r *
~!&:R~~ J ' ~ 2008 ~ 2010 i:J7Il~~~~§)t5jU~ 6.05 • 6.34 • 6.19 {ii
(a) f'Ji&IIiG1:HJ1ll r %~!&:R~1tf J !:f:1B"J~~~~§~flH'm:~~ X%If9ZP:flo)*JkI-ar:R~FJT1~~
4.
~~~Jf§:jPJ*~~*JE*5Ie~m~JjM
0
1E1J1t~Nf'J~
0
0
0
M~~~tt9X%*JkIZP:flo)-ar:R~Mmo
Xm~~?w
(b) t)-rWff~-ar~~~~nXlli 20 1f*i:J~~~~Jf§:~iffJ*tt9~~: (2j (~NtI i, ii ~ iii)
i) ~~Att9FJT1~!tl1JoB"Jtt~Attt1JDtt9'l*
ii) ~~Ae~FJT1~tW1JO
iii)
0
' 1El.~AB"JFJT1~~t~~~~
~~Att9FJT1~ffiU~~~ , ~AB"JpJTf~::f!i'JBZ~
0
0
(c) ~N~ili-Jji&Jf.f§lWIE1£mj}J-ar~g~WJ~~JJ\~j;~~§tt9i&m m~
10 ~t)pg): mn
~~f@JL~~f-+tx*~
iV"l:!(
10 1 ~!cpNm±fi)I:a:m±1:E$X.fi)Im~~~~~ ~k:;f;~llJit4!1iii!i!&
•
300
1;t~it.~
it.)5.*~~'~:I:~ • •
~RJ~3k~A
;.7~, t~IlJ:I:»Jf;f~-t
§.'*••
6.47%M:;f;'J $- -t If.j it:M?fk ~ , 1:1:. II
*::.k.~.;t"tt~t~t.nli
•
ffi
1.421%'
30
1:t~it.
(39
~*.~1~7.U*M~o~llJ~
~
14 8 ~ @J;f~ -t B1-ij!] 6.29%l! ~
.f9.:t~-re-tt~&~~
ij!] 1.47 -re­
0
0
~~~.~~A~~.~~4&ffi.~I~*~:ra~~~~'
••M
.~k~M~ij!]~h~~~~@J.'~_~~~~3k,~a~ffiM** •• ~.~
~*n-1t&
'
M~ 11/Jm-~iiH£~
0
J
i:::lt.B1-MntrB, 6 B1- 14 ft' ~~;f~1.1!ij!]£'~ 10 • • M:;f;U$-ts 6.65%'
~ JiJl ~1. FII, ij!] 7i:;f;'J $-1: * i'J 4.62 11i1a ft.~
;fn1!~
0
(a) ~flI~T{l~~flJ$Bf.J5ER (~N 10 @~tJpg) @
(b) ~*fIJBf.J0{l~flJ$tt~~Bf.J~r%ij 4.62% ' :jt~1&R~TfiiJfmJlID.~! (~N 5 {~~tJpg)? 02)
0
6. ~rmt.&*~~s*Bf.J~~PJTi{J~$t~DT
y (~~,PJTi{J)=10,000
c (~rB'mjf)=6,000
T Cffl:i&)= 1,500 G (i&fff:Rte) =1,700 1(~rel']15t:~) == 3,300 - 10000r, NX (~teO)= -500 ~fIf~t~tJTBf.J~S*~~~I1z:
(a) ~rB'WifIf(private saving) = fITI
(b) i&fff~lf(public saving) = Ci.~
(c) If;;$:f¥7m:te(net capital outflow)= Ct~)
(d) ~ftit§titi 1*= ~
(e) ~fti.~flj$ r*= fU)
7.
""6J~-ifiI25:lC ' ~~PJoPJ~-mr 0.8 ~7C §M~fft:i'~3t~7CBf.J!Ii$~~ 30
(a) @i&~)l~jiltrN.~~nx;;$: , ~1ffFo5~~DfiiJ1:Em~r~~J;lr.""6J~ffiHlfIJ (~flfBl i ~ ii)? (8)
0
i)
ii)
0
, ~~Jllli
, t:i'liiIfJllli
0
(b) ~lWro5:ffHJJJ'l'~::tJZfS{I@~(Purchasing Power Parity), ffl PJ~Bf.Jfl:f~'pff~t~lliBf.J-g.flJ!*'ft:i'm3t
~7C!Ht$~~~j;? 0'9)
(c) ;Gt:i'i!li&fffg;g~tlf:IJD~~f~~;l:m{:g , 1:E~ml1z£~t)&ft~~::tJZfS{I@~8f.J@~Z'"f ' *'f
t:i'~3t~7C!ii$l!~H~~~Q1
0
J-tl'fJl
~.A.~
m
101
**
f4 ~ * ¥
~1fll1i][±fJl~1i][±1£~~fJlffi~~~g~m
A~.#
•
~
••••• *~a.~T.+~~
~
~
}r%pff:
I~pff
f4§:
*~~~(l)
'
~.pff
•• A*ffi~'M*~ ••
~~Dr.~T&.~JA~'~~~.~~ft~~~~.~~.~*£;o-~
.a'A*~.~ • • m.Sfu'~ • • ili.~~.i.a.;
itt::t1bJtfi (QE3)
~lld~-*.{ff ill
~.a
JJ '
¢
0
itftllniit it~1.*-
• • • A~ft*.~
7f-~$i5w.r.i~;fll.~ it
+ Aim
;f~'fittft rRBC Capital MarketsJ A~itJ$j~.~~i.ttJ];f'l
(Torn Porcelli) :f~ili '~1--t~ljn.it r~*! r
J
'~1~;f'J.a.A*-~tjljJ.i;ff~A
~o
(a) ~Ft:l'11i1~*i'l~IHj.~L$(Dfi'iJ¥Jt1jii1tji[~il&m?
(~~~N
10 *t)[7g)
(b) ~P!Ft:l~~7ii{tJ[f~t)j}'~fll~W;1tfiE, 2 @~~.:tIH~fUlpJt)ij [~$!~flJ*~~l¥J1J$?~2?),
9. tJ~fi'iJ1fW20114~~m*~~~~1~.:t T. Sargentl¥JUH7'G1J$EdiJfJi:m*~/f;f§~: c251
i, ii, iii
~
iv)
i) t)JI'lifflWJl¥JfI1.tE~*ft~tim*~~flml ii)
iii)
ft:ffl fti)iI: El ~~~j( vector auto-regressjon)*{!5~ttim*~~f*ml CPI¥J~~
f1!{fflWJfU a~il&JNil&mj~J!{li5&1¥J iv) ~ffJ~ltl[M~~-f~ji@J' *JlJT :
' imiJJlJT
f4 El : WXf.lltjJ-(2)
tJiJE1m~
1.
201m'
5
:7t
Please solve the inequality Ix -II-Ix - 312! 5
.
2. Find the value of a such that the I1m
3X2
x..... -2
3. For a function f(x) = x 2 -
X -
+ ax + a + 3
exists.
+x-2
4, please find a number 8 such that if Ix - 21 < 8 then
If(x) + 21 <1.
4. Find the limit lim(_l_ - 2 1
)
H2 x-2
x -3x+2 5. Please find the normal line of the tangent for equation x 2
+ xy + y2 = 3 at point (1, 2),
+ 1)
' d the I"1mIt va1ue 0 f I'1m sin(x
6 , FIn
2
x..... -l x -2x-3
•
7. A boat is pulled into a dock by a rope attached to the bow of the boat and pass through a pulley
on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at the rate of 1
mis, how fast is the boat approaching the dock when it is 8 m from the dock?
8. Ifj(2)=8 and / (x) 2! 5 for 2:S; x :s; 6, how small can the f(6) possible be?
9. Using Newton's method to find a root of the equation
10, Find the limit value of lim(xe 1/x
x-+ao
-
x) .
ao
11. Find the sum of the series 1+ ~
(-1
en)
t --;;!
[
12. Find the radius of convergence of the series
f (n!)2
(2n)! xn
n=l
ril'l 2
13. Find
sin x
dx
Jo 1 + cosx 2
14. If f(x)=x+x 2 +e x and g(x)=f- 1 (x),findg'(I).
15. Find lnl0eX~
J
16.Find
JIo~
~
17. y'+y
=~e-x ,yeO) =3, findy(x)
o
eX +8
dx
X
S
= 5x -
2. Calculate two iterations.
mJJ:~~f-t1X*~
~pff
:
101 ~4N~±F)lJl~±1£IMWF)lm~~~Jt~Jt~
18. Find
JL (x
and y
2
+ y2 )3/2 dA ,where D is the region in the first quadrant bounded by the lines y = 0
=J3x
and the circle 2 + y2
x
=9
19. Find the maximum rate of change ofJat the given point and the direction in which it occur.
J(x,y,z) = In(xy2 z 3), (1-2,-3)
az
ax
az
20. yz = In(x+ z), find- and-.
ay
~f5fT
: I'iff5fT '
ji'if~
flj. § : !FfC~t¥( 1)
1. The probability of a customer arrival at a grocery service counter in any 1 second is equal to
0.15. Assume that customers arrive in a random stream and hence that the arrival any I second
is independent of any other.
(a) Find the probability that the first arrival will occur during the third I-second interval. (5%)
(b) Find the probability that the first arrival will not occur until at least the third I-second
interval. (5%)
2. Let the moment-generating function for Y be met)
= !e + ~e21 + 3 e 31 . Find the distribution
l
6
6
6
of Yand its expected value and variance. (l 0%) 3. As a measure of intelligence, mice are timed when going through a maze to reach a reward of
food. The time(in seconds) required for any mouse is a random variable Y with density
function given by
bl
y>b
f(y)= ly2
­
{ o elsewhere
where b is the minimum possible time needed to traverse the maze.
(a) Show thatf(y) has the properties of a density function. (5%)
(b) Find F(y). (3%)
(c) Find P(Y> b + c) for a positive constant c. (2%)
4. A soft drink machine can be regulated so that it discharges an average of f.l ounces per cup. If
the ounces offill are normally distributed with standard deviation of 0.3 ounce, give the setting
for f.l so that 8-ounce cups will overflow 1% of the time. (10%)
5. The length oflife(measured in hundreds of hours) Yfor fuses ofa certain type is modeled by
the exponential distribution, with
y>O
elsewhere
(a) If two such fuses have independent lengths of life, Y1 and Y2 , find the joint probability
density function for Yl and Y2 . (5%)
(b) One fuse in (a) is in a primary system and the other is in a backup system that comes into
use only if the primary system fails. The total effective length of life of the two fuses is then Y,
+ Y2 . Find P(YI + Y2 ::: 1). (5%)
*pfT : I~pfT ' j(~*
~13 : *1t~~(I)
6. (a) 1.;y 7~~fjr:p;!ft-£~¥.Hi ' fj£~fjr:ptH3mz7 25 ~t~~ , 4ij~mzff 5 @t~* ' f~¥O
25 ~f~~B'j¥~m1.;y 31.0 ' ~~~1.;y 2.0 f~~jtllt£jI!f~ti{1ir1~m~5t~c ' EtiKf9)5E
l!t¥~m~B~m- §
30.0 ' ~jif*tB.ftgf 0.27%~~~~( type I error)B'j~Wm(critical
valure) (10%)
(b) EJ.!t~~C1j1!f¥.itEmz~~~£J, 1.0 ' 4'¥J::xmzff 5 f~f~* ' 1~¥U-¥~m£J, 31.0 ' ~rp5
:(f 1%B'j~~~~( type I errorff ' f9)5E~1jJ.!t~(tH3mzJ.!t 5 f~tf*~)B'j~C1jt¥.itimz~
~m~:a~m-§~m 30.0? (15%)
0
0
7. 1.;y 7~mHRM{JtJj~C1~~tiB'j~. ' Ifjgffi~tt1~*4r:pB'g-1~M1Jt ' llJf%=f1~~ctt19tl
A ' B ' C ' ~~
.~1i::XB'j~6*3tDT*' ~jifrp~::f~B'j~~ctt19JJ~:aWt~.J.!t£jt~'ti
mZ311~m? ~~gjtWHIIt~*4r~~~~~5t;ffrB'j~JjlH~~jt
8. 1[~m'HIT~F~B'j~
, a=0.01 (15%)
0
~~C1B'g~?iJi!!' ;!ft~~§fP}&Jj~J!~D¥Jjf$B'jtt$~~WMB'j
, [ZSJ
1.;yf~~J'tgJj~ 1000 A=XB'g~6* '
382 AWM~J!~DB'jfjJf$ , 416 A::fWM ' J'tijtB'j
5~1[~~ ~J;,Z~~t~B'jW~ , f~11fl1t¥&~B'jJ1.§$~:a~1i ' a=0.05 (10%)
0
0
TABLE 3
NomIaJ. CW'IIe Areas
TABLE ...
$
Critical Values of r
~
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
'.7
0.8
0.9
1.0
.()(J
.oj
4')1 F::~
t.
.02
.03 .04
.OJ
.D6
.07
.0199 .0596 .0987 .1368 .1736 .2088 .Q8 .
.09
.0000 .0398 .0793 :1119 .1554 .1915 .0040 .0080
.0438 ..0478 .0832 .0871 .1217 .1255 .1591 .1628 .1950 .198~ .0120 .0517 .0910 .1293 .1664 .2019 .0160 .0557 .0948 .1331 .1700 .2054 .0239 .0636 .1026 .1406 .1172 .2123 .0279 .0615 .1064 .1443 .1808 .2151 .0319 .0714 .1103 .1480 .1844 .2190 .0359 .0153 .1141 .1511 .1879 .2224 .2257 .2580 .2881 .3159 .3413 .wi
.2351 .2613 .2961 .3238 .3485 .2389 .2422 .2454 .2104
.2734 .2764 .3023 .3051 .3264 .3289 .3315 .3508 .3531 .3554 .3729 .3749 .3770 .3925 .3944 .3962 .4099 .4115 .4131 .4151 .4265 .4219 .4382 .4394 .4406 .4495 .4505 .4515 .4591 .4599 .460\1 .4671 .4678 .4686 .4738 .4744 .4150 .4793 .4198 .~3
,2486 .2794 .3078 .3340 .3577 .1517 .2823 .3106 .3365 .3599 .1549 .2852 .3133 .3389 .3621 .2611 .2910 .3186 .3438 8&
.2324 .2642 .2939 .3212 .3461 1.1 .3643 .3665 .3686 .3108 1.2 .3849 .3869 .388~ .3901 1.3 .4032 .4049 .4066 .4082 1.4 .4192 '.4207 .4222 .4236 1.5. .4332 .4345 .4357 ,4,110
1.6 .4452 .4463 .4414 .4484 1.7 .4554 .4564 .4573 .4582 1.8 .4641 .4649 .4656 .4664 1.9 .4713 .4719 .4726 .4732 2;9 .41.7) .4n8 .4783 .4188 21 .4821 .4826 .4830 .4834 2.2 .4861 .4864 .4868 .4871 23 .4893 .4896 .4898 ~
24 .4918 .4920 .4922 .4925 25 .4938 .4940 .4941 .4943 26 .4953 .4955 .4956 .4951 27 ...965 .4966 .4967 .4968 28 .4974 .4975 .4976 .4977 29 .4981 .4982 .4982 .4983 3.0 .4981 .4981 .4987 .4988 .m5
.3790 .3980 .4141 :4292 .4418 ..4515 .4616 .4693 .4156 .4808 .3810 .3830 :W .4015 .4162 .41n
.4306 .4319 .4429 .4441 .4857 .4890 .4916 .4936 .4952 .4838 .4815 .4904 .4921 .4945 .4842 .4846 .4878 .4881 .4906 .4909 .4929 .4931 .4946 . .4948 .4850 .4884 .4911 .4932 .4949 .4535 .4615 .4699 .4761 .4812 '.4854 .4887 .4913 .4934 .4951 .4959 .4969 .49n
.4984 .4988 .4960 .4970 .4918 .4984 .4989 .4962 .4972 .4979 .4985 .4989 .4963 .4973 ..4980 .4986 .4990 .4961 .4971' .4979 .4985 .4989 .4545 .4633 .4706 .4767 .4817 .4964 .4974 .4981 .4986 .4990 This table isabridged rrom Table 1 or Slalis/ical T(J"'~s QlId Formula", by A. Hald (New York: John
Wiley & Sons. Inc.• 1952). Reproduced by permission Or A. Hald and the publishers, John Wiley &
Sons,lnc.
.
r. 100 r.•,o ·
3.078
1.886
1.638
1.533
6.314 2920 2353 2132 1A16
1.440
1.415
1.391 · 1.383 1.372
1.363
1.356
1;350
1.345
1.341
· 1.331
· 1.333
1.330
1.328
·1.315
1.323
1.321
1.319
1.3,8
1.316
1.315
1.314
1.313
1.311
1.282
2015 '
1.943
1.895
1.860
1.833
2571 2.447 2365 2306 2262 1.812
1.'796 .
1.182
1.1'11
1.161
1.753
t.OlS
d.f.
'.OIG
'.001
31.821
6.965
4.541
3.141
63.657
9.915
5.841
4.604
1
2
3
4
3.365
3.143 2998 2896 2821 4.032
3.499
3.355
3.2S0
5
6
7
8
9
2228 2201 2179 . 2160 ' 2145 loBI 2164 2718 . 2681 2650 2.624 2602 3.169
3.106
3.055
-3.012
29n
2947 10 11 12 13 14 15 1.746
1.140
1.734
1,729
1.115
2120 2110 2101 2093 2086, 2583 2567 2552 2539 2528 2921 2898 2818 2861 2845 16 17 18 19 1.721
1.711
1.714
1.711
1.108
·2080 2074 2069 2064 2060 2518 2S08
2500 2492 2485 2831 2819 2807 2797 2787 1.106
2056 2052 2048 2045 1.960
2479 2473 2467 2462 2326 2n9
2771 2.763 2756 1.103
1.701
1.~99
1.645
12106 4.303
3.182
2n6
3.107
2~76
20 21 22 23 24 .~
26 27 28 29 iot From "Table of Percentage Points of the ,.Distribulion." Computed by Muine Mcrrington,B/ometrjlca, Vol. 32 (1941).
· p. 300. Reproduced by permission of Professor E. S. Pearson.
-ttr
~.
~~
mi'»
lI@~
~~
H:i->+
:­m
i1d
Hf
~
~
~
61 ~ ~ lIIJ ~ :m: H
I
-
I\J.J
~ JJll}l
'lIS ~
.....
'-'
)n(
~
JJll}l ~
~
$
"',
~
TABLE 6 Percentage Points ofthe F Distribution: GI = .05
~
0
", (d.f.)
10
Fa
., (df.J
¥.
2
(d.f.)
1
2
3
4
5
6
7
8
9
10
,11
12
·13
14
f5
16
17
18
19
20
21
22
23'
24
25
26
27
28
29
30
40
60
120
00
161.4
199.5
18.51
19.00
10.13 . 9.55
7.71
6.94
,6.61
5.79
5.99,
5.14
4.74
5.59
5.32
4.46
.4.26
,5.12
4.96'
4.10
3.98
·4.84
4.75
3.89
4.67
3.81
4.6Q
3.74
4.54
4.49
4.45
4.41
4.38
4.35
4.32
4.30
4.28
4.26
. ,424
4.23
4.21
4.20
4.18
4.17
4.08
4.00
3.92
3.84
3.68
3.63
3.59
3.55
3.52
3.49
3.47
3.44
3.42
3.40
,-3.39
3.37
3.35
3.34
3.33
3.32
3.23
3. IS
3.07
3.00
1
,4
215.7
19.16
9.28
6.59
-5.41
4.76
4.35
4.07
3.86
224.6
19.2S
9.12
6.39
'S
230.2
19.30
9.01
6.26
$
TABLE,S (Continued)
6
7
'8
9
234.0
236.8
240.5
238.9
19.35
19.38
19.33
19.37
8.94
8.89
8.81
US . 6.04 6.16.
6.09
6.00
5.19
5.OS
4.95
'4.88
4.82 4.77
,4.28
4:53
4.39
4.21
4. IS 4.10
4.12
3.97
3.87
3.79
3.73 3.68
' 3.58 '
3.84
3.69
3.50
3.44 3.39
3.18
3.63
3.48
3.37
3.29
3.23 3.22 ' 3.14
'~".71
3.33
3.02
3.07
3.48 '
3.20 • 3.09
3.59
l36
3.01 2.95
2.90
' 3.26 ' ,3.11
3.49
3.00 , 2.91 2.80
2.85
,3.41
3.18
3.03
2.83
2.71
2.77
2.92
3.11
2.96
2.85
2.76­
2.65
3.3:4
2.,70
3.29
2.90
2.71
2.59
3.06
2.64
2.79
2.85
2.74
3.24
3.01
2.66
2.54
2.S9
2.96 . 2.81
2.70
3.20
2.61 2.55
2.49
. 2.58 2.93
2.77
3.16
2.51
2.46
2.~
2.74'
2.63
3.13
2.90
2.54
2.48
2.42
2.45 ' 2.39
2.71
2.60 " 2.51
2.87
3.tO
. 3.in
2.37
2.84
2.68
2.49
2.42
2.5~
' 2.46
2.82,
3.05.
2.40
2.66
2.55
2.34
2.64
2.32
3.03
2.44
2.80
2.S3
2.37
2.30
3.01
2.78
2.62
2.42
2.51
2.36
2.76 , 2.60
2.49
2.40
2.34
2.99
2.28
2.,47
2.98 ' 2.74
2.59
2.39
2.32 - 2.27
2.37
2.73
2.57
2.40
i31
2.25
2.96
• 2.S6
2.45
2.95
2.36
2.29
2.24
271
2.93
2.70
i43
2.35
2.22
2.28
2.S5
i21
2.53
2.42
2.92
269
2.33
2.27
' 2.45
2.34
2.18
2.12
2.25
2.84
2.61
2.37
2.04.
2.76
2.S3
2.25
2.17
2.10
2.29 . 2.17
2.02
1.96
2.68
2.45
2.09
. 2.JO ' 2.01
1.88
2.60
2.37
221
1.94
12
IS
20
24
.w
4()
6(J
/20
§R • 00
'
".
(d.
I
2SO.1
251.1 252.2 253.3 254.3
. 241.9 243.9 245.9 248.0 249.1
2
19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.5O
19.40 . 19.41
3
8.57
8.55
8.74
8.64
8.59
8.53
8.70
8.66
8.62
8.79
4
5.66
5.63
5.91
5.86
S.80
5.77
5.75 .. 5.72 ' 5.69
5.96
4.62' 4.56
4.36
4.46
4.43
4.68
4.SO
4.40
4.53
4.74
3.114 . , 3.81
3;74
3.77
3.67
3.70
4.00
3.94
3.87
4.06
I
3.57
3.51
3.44
3.41
:U8 3.34 3.30 3.27 3.23
3.64
2.97
3.22
3.12
3.04
S
2.93
3.28
3.15
3.08
3.01'
3.35
2.94' 2.90 2.86
2.75
S
2.83
2.71
3.07
3.01
2.79
3.14
1(
2.66
2.91
2.'14
2.62
2.54
2.85
2.70
2.58
2.98
~77
2.72­
2.S3
2.45
2.40
11
2.61
2.49
2.79
2.65
2.57
2.85
2.38'
1.
2.62
2.51
2.47
2.43
2.34
2.69
2.54
2.30
2.75
,2.21
1:
2.53
2.34
2.30
2.60
2.46
2.42
2.38
2.2S
2.67
14
2.27
2.18
2.46
2.39
2.35
2.31
2.22
2.13
2.60 ' 2.53
2.25 c 2.20.. 2.16 ' 2.11
1~
2.07
2.48
2.40
2.33
2.29
2.54
2.19 .' 2.15
2.06.
2.01 ' If
2.11
2.42
2:35 , 2.28
2.24
2.49
2.06.
Ii
2.31
2.01
1.96
2.38
2.23 .2.i9
2.15 ' 2.10
2.45
2.02,
2.06
2.34
2.19
2.15
2.11
1.92
IE
2.41
2..27
1.97
IS
2.16
2.11
2.07,
2.03
1.98
1.93
1.88
2.38
2.31
2.23
.
2(
1.95 . 1.90
1.99
2.12
2.08
1'.84
2.04
2.35 ' 2.28 . 2.20
2.2S, ,2.18
2.32
2.10
iM 2.01 1.96 1.92 1.87 1.81 21
2.07 ' 2.03
2.23
2.15
1.94
1.84
1.78
:z:
1.98
1.89
2.30
2,
2.13
1.91
1.81
1.76
2.20
2.OS
2.01
1.96
1.86
2.27
2.18
2.11
U4 1.89 1.84 1.79 1.73
2l
2.03
1.98
2.25
1.92
2.09
i.87
1.82
1.77
2.16
1.96
1.71
2!
2.01
2.24
2.07
:u
2.15
1.99
1.90
1.85
1.80
1.75
'1.69
.1.95
2.22
To
2.13
1.97
1.93
1.84
2.20
2.06
1.88
1.79
h73
1.67
1.82
2.12
1.77
1.71
21
2.04
t96
1.91
1.87
1.65
2.19
2.03
1.90
1.81
1.75
~
2.18
2.10
1.94
1.85
1.70 '1.64
J(
1.79
1.74
2.01
2.09
1.93
1.89
1.84
1.68
1.62
2.16
4(
2.00
1.92
1.84
1.74
1.69
!.S8
1.51
1.64
1.79
2.08
IS),
1.92
1.84
1.75
1.70
1.59
1.99
1.47
1.39
61
1.65
12(
1:61'
1.75
1.66
ISO
1.83
1.55
1.43
1.25
1.91
1.35
1.75
1.67
1.57
1.52
1.46
1.39
1.32
1.83'
1.22
1.00
00
From -rabIes of Percentage PoinlS of the Inverted Beta IF) Distribution,~ BWIIlerrikn, Vol. 33 (194
pp. 73-88, by M8lIinc !l«mington and catherine M. Thompson. Reproduced by pcnnission of Prof_
E. S. Pearson.,
.
~J ~
: lib' ~~ ~~
~,.\?t:
~~ fit>+­
1_ ~
~
rt'r
o)lt
m
m
61
~ ~
IID ~
~ H
~ llIIl
iI~
.-..
.....
'-'
•
DIll
~
I­
~
~
~
~ J
TABLE 7
Percenlage POinl3o/the F DislribuJkm;a. = .01 _
~
" (d.r.i
10
OF.
'
Ill.
.1
2
3
,4
·5
6
7
8
9
10
11
12
13
14
. 15
16
17
18
19
20
·21
.22
23
.24
25
2627
28
29
30
40
60
120
00
_. 2
3
4
12
IS
20
24
30
40
60 .
120
<0
'l(d
6106
6157
6209
6235
6261
6287
6313
6339
6366
1
99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.50 '. 2
27.23 ·27.05 26.87 . 26.69 26.60 26.50 26.41, 26.32 ' 26.22 26.13
3
14,55 14.3.7 14.20 14.02 13.93 13.84 ' 13.75
13.65
13.56" 13.46'
4
10.05
9.89 '9.72
9,55
9.38
9.29
9.20
9.11
9.02
9·'F
~
7.87
7.72
7.56
7.40
7.-31
7.23
7.14
7.06
697
6.88
E
6.62­
6.47
6.31
6.16
6.07
5,99
591 " 5.82
5.74
5.6S
i
5.81
'5,52
5.67
5.36
5.28
4.95 . 4.86
5.20
5.12
5.03
~
5.26
5.11
4.96
4.81
4.73
4,57
4.65
<;
4.40
4.48·
4.31
4.85
4.71
4.56
4.41
4.33
4.25
4.17
4.08
4.00
3.91
IC
4.54
4.40
4.25
4.10
'3.86
4.02
3.94
3.78
3.69
3.60
11 4.30
:
3.70
.
4.16
4.01
3:86
3.78
3.62
3.54 . 3.45
1:; 3.36
4.10
3.96
3.82
3.66
3.59
3.51'
3.43
3.34
3.15. . 3.17
13 3.94
3.80.
3.66
3,51
3.43
3.35
3.27
3.18
3.09
3.00
14 3,80
3.67
3.52
3.37
3.29
3.21
3.13
3.05
2.96
2.87
1~
3,55
3.69
.3.41
3.U
3.18
f{
3.10
3.02
2.93
2.84
2.75
3.59
3.16
3.08 . 3.00
3.'\6 . 3.31
2.922.83
275
2.65
Ii
3.08 ...... '··3.00
3.51
3.37
3.23
2.922.84
2.75
2.66
2.57
U
3.43
3.30
3.15
2.84
3.00- " . 2.92­
2.76
2.67
2S8
2.49
IS
3.09 . 2.94
3.37
3.23
2(
2.86
2.78
2.69
2.61
2.52
2.42
3.31
3.0J' 2.88 . 2.80
3.17
2.72
2.64
2.55
246
2.36
21
2.83 . 275
3.26
3.12
2.98
267
2.SS
2.50
2.40
2.31
2:i
3.21
3.07
2.78' . 2.70', 2.62
2.93
2S4
245
235
2.26
2.:'
3.17
3.03
274 ' 2.66
289
25S
2.49 ·240
231
2.21
2A
'3.13
2.99
285.
270
245
262.. 254
236
227
2.17
~
3.09
2.96
2.81
2.66
2S8
·2.50
2.42
2.33
2.23
2.13
2f
3.06
2.93 .278
263
2.55
2.47
2.38
229
2.20
2.10
2i
3.03'
275
290
260
252
2.44
235'
2.26
217 .2.06
~
3.00
2.S7
2.13
257 . 249
233 . 2.23'
241
2.14
2.03
2S
298
2.84
270
255
2.47
239
230
221 . 211
2.01. 3(
2S0
2.66
2.52
2.37
2.29
220
2.11
4(
2.02
1.92
1.80
263
2.50
2.35
2.20
2.12
1.94'
2.03
6(
1.84
1.73
1.60
2.47
234
219
2.03
195
1.86
1.76
1.66
1.53
1.38 12(
2.18
232
204
1.88
1.70'
1.79
1.59
1.47
1.32
1.00 00
6OS6
" (dr.)
v. (dr.)
$
TABLE 7 (Continlled)
S
6
7
8
Ii
"
4052
4999,5 5403
5615
5764
SS59
5928 '5982­
6022
98.50
99.00
99.17
99.25 , 99.30
99.33
99.39'
99·36 . 99.37
34.12
29.4(;
30.82
28.71
28.24
27.91
27.67
27.49
27.35
. 21.20
18.00
16.69
15.98
15,52
15.21
1498
14.80
14.66
16.26
11.39 . 10.97
13.27
12.06
10.67
10.46
10.29
10.16
.l3.7S
10.92
9.78
9.15
8,47
8.75
8.26
• 8~10
7.98
12.25
955
8.45
7.85.
7.46
7.19
6.99 . ~g.. .··6·77
11.26
8.65
7.59
7.01
6.63
(j.37
6.18
.6.03
5.91
10.56
6.06 . 5.80
8.02
6.99
6.42
5.61
. S.!,7
• 5.35
10.04
.• 6,55
7.56
5.99
5.64
5.39
5.20
5.06
4.94
9.65
7.21
6.22
5.67
5.32
4.89
5.C17
4.74
4.63
9.33
693
595
5.41
5.06'
'4.82
.4.64
·.4.50
4.39
·9.07
6.10": . 5.74.
5.21
4:86:
4.62
4.44
4.30
4.19
8.86
6.51
4.69 .
5.56
5.04
4.46
4.28
4.14
4.03 ..
8.68
6.36·
5.42
4.89
4.56
4.32
4.14
4.00
3.89
8,53
6.23
5.29
4.77
4.4<J
4.20
4.03
3.89
3.78
8.40
6.11
4;):oi . 4.10
5.18
4.67
3.93
3.79
3.68
8.29 . 6.01
5.09
4,58
4.25
4.01
3.84
3.71
3.60
8.18
5.93
5.01
4.50
4.17
3.94
3.77
3.63
3.52
8.l0
5.85.
4.43 . 4.10
4.94
3.87
,3.70
l56
3.46
8.02
5.78
4.87
4.37
4.0:$
3.81
3,51
3.64
3.40
7.95
4.82,
5.72
4.31
3.99
3.76
3.59
" 3.45
.3.35
7.88
5.66
4.76
4.26
3.94
3.71
3.54
3.30
3.41
5.61 . " 4.72
7.82
4.22
3,90
3.67
3:50
3,36
3.26
7.77
4.68 . 4.1S
557
3.85
3.63
3.46
3.32
3.22
7.72
4.14 . 3.82
5.53
4.64
l59
3.42
3.29
3.18
7.68
5.49
4.60
4.11
3,56
3.78
3.39
3.26
l15
7.64
.
4.57
·5.45.
4.07
3.75
3,53
3.36
3.23
.~..!J
7.60
4,54
5.42
3.73
4,04
3.50
. 3.09
3.33
3.20
7.56
4,51
5.39
4.02
3.70
3.47
3.30
3.17
3.07
7.31·
5.1S
4.31
3.83
3.51
3.29
3.12
299
2.89
'7.08
4.98
4.13.
. 2.72
3.65
3:34
3.12
295
2.82
6.85
4.79
3.95
3.48
3.17
2.66'
296­
2.79
256
6.63
4.61
3.78
3.32
3.02
280
2.51
264
2.41
§1Il
I~r ~.
~*
~:tti
1>4f
rt
m>t
iill
I
~
m
lit
~
m
m
If
:t!£
illI
~
*;m H
Il!JJI
'\lJ;m
1.>\
~
nu)
....
,.-,
From "Tab!... or Percentage Poinll or !he Invened Beta (F) Distributioll," Biometrika, Vol. 33 (19<
pp. 73-88, by Maxine M~ngton and Catherine M. Thompson. Reproduced by permission or.Profes
E. So PearSol'l.
'-'
~
nuR
*
~ '" ~