2016 mathematical methods written examination 2€¦ · mathematical methods written examination 2...
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MATHEMATICAL METHODSWritten examination 2
Thursday 3 November 2016 Reading time: 11.45 am to 12.00 noon (15 minutes) Writing time: 12.00 noon to 2.00 pm (2 hours)
QUESTION AND ANSWER BOOK
Structure of bookSection Number of
questionsNumber of questions
to be answeredNumber of
marks
A 20 20 20B 4 4 60
Total 80
• Studentsarepermittedtobringintotheexaminationroom:pens,pencils,highlighters,erasers,sharpeners,rulers,aprotractor,setsquares,aidsforcurvesketching,oneboundreference,oneapprovedtechnology(calculatororsoftware)and,ifdesired,onescientificcalculator.CalculatormemoryDOESNOTneedtobecleared.Forapprovedcomputer-basedCAS,fullfunctionalitymaybeused.
• StudentsareNOTpermittedtobringintotheexaminationroom:blanksheetsofpaperand/orcorrectionfluid/tape.
Materials supplied• Questionandanswerbookof26pages.• Formulasheet.• Answersheetformultiple-choicequestions.
Instructions• Writeyourstudent numberinthespaceprovidedaboveonthispage.• Checkthatyournameandstudent numberasprintedonyouranswersheetformultiple-choice
questionsarecorrect,andsignyournameinthespaceprovidedtoverifythis.• Unlessotherwiseindicated,thediagramsinthisbookarenot drawntoscale.• AllwrittenresponsesmustbeinEnglish.
At the end of the examination• Placetheanswersheetformultiple-choicequestionsinsidethefrontcoverofthisbook.• Youmaykeeptheformulasheet.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
©VICTORIANCURRICULUMANDASSESSMENTAUTHORITY2016
SUPERVISOR TO ATTACH PROCESSING LABEL HEREVictorian Certificate of Education 2016
STUDENT NUMBER
Letter
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2016MATHMETHEXAM2 2
SECTION A – continued
Question 1Thelinearfunction f:D → R, f(x)=5–x hasrange[–4,5).ThedomainDisA. (0,9]B. (0,1]C. [5,–4)D. [–9,0)E. [1,9)
Question 2
Letf:R → R, f x x( ) cos= −
1 2
2π .
TheperiodandrangeofthisfunctionarerespectivelyA. 4and[–2,2]B. 4and[–1,3]C. 1and[–1,3]D. 4πand[–1,3]E. 4πand[–2,2]
SECTION A – Multiple-choice questions
Instructions for Section AAnswerallquestionsinpencilontheanswersheetprovidedformultiple-choicequestions.Choosetheresponsethatiscorrect forthequestion.Acorrectanswerscores1;anincorrectanswerscores0.Markswillnotbedeductedforincorrectanswers.Nomarkswillbegivenifmorethanoneansweriscompletedforanyquestion.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
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3 2016MATHMETHEXAM2
SECTION A – continuedTURN OVER
Question 3Partofthegraphy=f(x)ofthepolynomialfunction f isshownbelow.
13
10027
,
–3
−12 1
O
(–2, –9)
y
x
f ′(x)<0for
A. x∈ − ∪ ∞
( , ) ,2 0 1
3
B. x∈ −
9 100
27,
C. x∈ −∞ − ∪ ∞
( , ) ,2 1
3
D. x∈ −
2 1
3,
E. x∈ −∞ − ∪ ∞( )( , ] ,2 1
Question 4
Theaveragerateofchangeofthefunction f withrule f x x x( ) = − +3 2 12 ,betweenx=0andx=3,is
A. 8
B. 25
C. 539
D. 253
E. 139
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2016MATHMETHEXAM2 4
SECTION A – continued
Question 5
Whichoneofthefollowingistheinversefunctionofg :[3,∞)→ R, g x x( ) = −2 6 ?
A. g–1:[3,∞)→ R, g x x− =+1
2 62
( )
B. g–1:[0,∞)→ R, g–1(x)=(2x −6)2
C. g–1:[0,∞)→ R, g x x− = +1
26( )
D. g–1:[0,∞)→ R, g x x− =+1
2 62
( )
E. g–1:R → R, g x x− =+1
2 62
( )
Question 6Considerthegraphofthefunctiondefinedby f :[0,2π]→ R, f(x)=sin(2x).
Thesquareofthelengthofthelinesegmentjoiningthepointsonthegraphforwhich x = π4and x = 3
4π is
A. π 2 164+
B. π + 4
C. 4
D. 3 164
2π π+
E. 1016
2π
Question 7Thenumberofpets,X,ownedbyeachstudentinalargeschoolisarandomvariablewiththefollowingdiscreteprobabilitydistribution.
x 0 1 2 3
Pr(X=x) 0.5 0.25 0.2 0.05
Iftwostudentsareselectedatrandom,theprobabilitythattheyownthesamenumberofpetsisA. 0.3B. 0.305C. 0.355D. 0.405E. 0.8
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5 2016MATHMETHEXAM2
SECTION A – continuedTURN OVER
Question 8TheUVindex,y,forasummerdayinMelbourneisillustratedinthegraphbelow,wheretisthenumberofhoursafter6am.
O
2
4
6
8
10
2 4 6 8 10 12 14
y
t
Thegraphismostlikelytobethegraphof
A. y t= +
5 5
7cos π
B. y t= −
5 5
7cos π
C. y t= +
5 5
14cos π
D. y t= −
5 5
14cos π
E. y t= +
5 5
14sin π
Question 9
Giventhat d xedx
kx ekx
kx( )= +( )1 ,then kxxe dx isequalto
A. xekx
ckx
++
1
B. kxk
e ckx+
+
1
C. 1 kxe dxk ∫
D. 1kxe e dx ckx kx−( ) +∫
E. 12kxe e ckx kx−( ) +
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2016MATHMETHEXAM2 6
SECTION A – continued
Question 10Forthecurvey=x2–5,thetangenttothecurvewillbeparalleltothelineconnectingthepositivex-interceptandthey-interceptwhenxisequalto
A. 5
B. 5
C. –5
D. 52
E. 15
Question 11Thefunction f hastheproperty f (x)–f (y)=(y–x)f (xy)forallnon-zerorealnumbersxandy.Whichoneofthefollowingisapossibleruleforthefunction?
A. f (x)=x2
B. f (x)=x2 + x4
C. f (x)=xloge(x)
D. f xx
( ) = 1
E. f xx
( ) = 12
Question 12Thegraphofafunction f isobtainedfromthegraphofthefunctiongwithrule g x x( ) = −2 5 bya
reflectioninthex-axisfollowedbyadilationfromthey-axisbyafactorof12.
Whichoneofthefollowingistheruleforthefunctionf ?
A. f x x( ) = −5 4
B. f x x( ) = − − 5
C. f x x( ) = + 5
D. f x x( ) = − −4 5
E. f x x( ) = − −4 10
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7 2016MATHMETHEXAM2
SECTION A – continuedTURN OVER
Question 13Considerthegraphsofthefunctions f andgshownbelow.
y = f (x)
y = g (x)
a b c d
y
x
Theareaoftheshadedregioncouldberepresentedby
A. f x g x dxa
d( ) ( )−( )∫
B. f x g x dxd
( ) ( )−( )∫0C. f x g x dx f x g x dx
b
b
c( ) ( ) ( ) ( )−( ) + −( )∫ ∫0
D. f x dx f x g x dx f x dxa
a
c
b
d( ) ( ) ( ) ( )
0∫ ∫ ∫+ −( ) +
E. f x dx g x dxd
a
c( ) ( )
0∫ ∫−
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2016MATHMETHEXAM2 8
SECTION A – continued
Question 14Arectangleisformedbyusingpartofthecoordinateaxesandapoint(u,v),whereu>0ontheparabolay=4–x2.
(u, v)
y
x
5
4
3
2
1
54321O
Whichoneofthefollowingisthemaximumareaoftherectangle?A. 4
B. 2 33
C. 8 3 43−
D. 83
E. 16 39
Question 15Aboxcontainssixredmarblesandfourbluemarbles.Twomarblesaredrawnfromthebox,withoutreplacement.Theprobabilitythattheyarethesamecolouris
A. 12
B. 2845
C. 715
D. 35
E. 13
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9 2016MATHMETHEXAM2
SECTION A – continuedTURN OVER
Question 16Therandomvariable,X,hasanormaldistributionwithmean12andstandarddeviation0.25Iftherandomvariable,Z,hasthestandardnormaldistribution,thentheprobabilitythatXisgreaterthan12.5isequaltoA. Pr(Z<–4)B. Pr(Z<–1.5)C. Pr(Z<1)D. Pr(Z≥1.5)E. Pr(Z >2)
Question 17Insideacontainerthereareonemillioncolouredbuildingblocks.Itisknownthat20%oftheblocksarered.Asampleof16blocksistakenfromthecontainer.Forsamplesof16blocks,P̂ istherandomvariableofthedistributionofsampleproportionsofredblocks.(Donotuseanormalapproximation.)
3Pr16
P ≥ ̂ isclosestto
A. 0.6482B. 0.8593C. 0.7543D. 0.6542E. 0.3211
Question 18Thecontinuousrandomvariable,X,hasaprobabilitydensityfunctiongivenby
f xx x
( )cos
=
≤ ≤
14 2
3 5
0
elsewhere
π π
ThevalueofasuchthatPr( )X a< =+3 2
4is
A. 196π
B. 143π
C. 103π
D. 296π
E. 173π
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2016MATHMETHEXAM2 10
END OF SECTION A
Question 19ConsiderthediscreteprobabilitydistributionwithrandomvariableXshowninthetablebelow.
x –1 0 b 2b 4
Pr(X=x) a b b 2b 0.2
ThesmallestandlargestpossiblevaluesofE(X)arerespectivelyA. –0.8and1B. –0.8and1.6C. 0and2.4D. 0.2125and1E. 0and1
Question 20ConsiderthetransformationT,definedas
T R R Txy
xy
: ,2 2 1 00 3
05
→
=
−
+
ThetransformationTmapsthegraphofy=f(x)ontothegraphofy=g(x).
If f x dx( ) =∫ 50
3,then g x dx( )
−∫ 3
0isequalto
A. 0B. 15C. 20D. 25E. 30
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11 2016MATHMETHEXAM2
SECTION B – Question 1–continuedTURN OVER
Question 1 (11marks)
Let f :[0,8π]→ R, f xx( ) cos=
+2
2π .
a. Findtheperiodandrangeof f. 2marks
b. Statetheruleforthederivativefunction f ′. 1mark
c. Findtheequationofthetangenttothegraphof f at x=π. 1mark
SECTION B
Instructions for Section BAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.
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2016MATHMETHEXAM2 12
SECTION B – continued
d. Findtheequationsofthetangentstothegraphof f :[0,8π]→ R, f x x( ) cos=
+2
2π that
haveagradientof1. 2marks
e. Theruleof f ′canbeobtainedfromtheruleof f underatransformationT,suchthat
T R R Txy a
xy b
: ,2 2 1 00
→
=
+
−
π
Findthevalueofaandthevalueofb. 3marks
f. Findthevaluesofx,0≤x≤8π,suchthat f (x)=2f ′(x)+π. 2marks
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13 2016MATHMETHEXAM2
SECTION B – Question 2–continuedTURN OVER
Question 2 (12marks)
Considerthefunction f (x)=– 13(x+2)(x–1)2.
a. i. Giventhatg′(x)=f (x)andg (0)=1,showthat g xx x x( ) .= − + − +
4 2
12 223
1 1mark
ii. Findthevaluesofxforwhichthegraphofy=g (x)hasastationarypoint. 1mark
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2016MATHMETHEXAM2 14
SECTION B – Question 2–continued
Thediagrambelowshowspartofthegraphofy=g (x),thetangenttothegraphatx=2andastraightlinedrawnperpendiculartothetangenttothegraphatx=2.Theequationofthetangentat
thepointAwithcoordinates(2,g (2))is yx
= −3 43.
Thetangentcutsthey-axisatB.Thelineperpendiculartothetangentcutsthey-axisatC.
B
A
C
O
y
x
b. i. FindthecoordinatesofB. 1mark
ii. FindtheequationofthelinethatpassesthroughAandCand,hence,findthecoordinatesofC. 2marks
iii. FindtheareaoftriangleABC. 2marks
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15 2016MATHMETHEXAM2
SECTION B – continuedTURN OVER
c. ThetangentatDisparalleltothetangentatA.ItintersectsthelinepassingthroughAandC at E.
D
A
CE
O
y
x
i. FindthecoordinatesofD. 2marks
ii. FindthelengthofAE. 3marks
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2016 MATHMETH EXAM 2 16
SECTION B – Question 3 – continued
Question 3 (16 marks)A school has a class set of 22 new laptops kept in a recharging trolley. Provided each laptop is correctly plugged into the trolley after use, its battery recharges.On a particular day, a class of 22 students uses the laptops. All laptop batteries are fully charged at the start of the lesson. Each student uses and returns exactly one laptop. The probability that a student does not correctly plug their laptop into the trolley at the end of the lesson is 10%. The correctness of any student’s plugging-in is independent of any other student’s correctness.
a. Determine the probability that at least one of the laptops is not correctly plugged into the trolley at the end of the lesson. Give your answer correct to four decimal places. 2 marks
b. A teacher observes that at least one of the returned laptops is not correctly plugged into the trolley.
Giventhis,findtheprobabilitythatfewerthanfivelaptopsarenot correctly plugged in. Give your answer correct to four decimal places. 2 marks
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17 2016MATHMETHEXAM2
SECTION B – Question 3–continuedTURN OVER
Thetimeforwhichalaptopwillworkwithoutrecharging(thebatterylife)isnormallydistributed,withameanofthreehoursand10minutesandstandarddeviationofsixminutes.Supposethatthelaptopsremainoutoftherechargingtrolleyforthreehours.
c. Foranyonelaptop,findtheprobabilitythatitwillstopworkingbytheendofthesethreehours.Giveyouranswercorrecttofourdecimalplaces. 2marks
Asupplieroflaptopsdecidestotakeasampleof100newlaptopsfromanumberofdifferentschools.Forsamplesofsize100fromthepopulationoflaptopswithameanbatterylifeofthreehoursand10minutesandstandarddeviationofsixminutes, P̂ istherandomvariableofthedistributionofsampleproportionsoflaptopswithabatterylifeoflessthanthreehours.
d. FindtheprobabilitythatPr(P̂ ≥0.06|P̂ ≥0.05).Giveyouranswercorrecttothreedecimalplaces.Donotuseanormalapproximation. 3marks
Itisknownthatwhenlaptopshavebeenusedregularlyinaschoolforsixmonths,theirbatterylifeisstillnormallydistributedbutthemeanbatterylifedropstothreehours.Itisalsoknownthatonly12%ofsuchlaptopsworkformorethanthreehoursand10minutes.
e. Findthestandarddeviationforthenormaldistributionthatappliestothebatterylifeoflaptopsthathavebeenusedregularlyinaschoolforsixmonths,correcttofourdecimalplaces. 2marks
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2016 MATHMETH EXAM 2 18
SECTION B – Question 3 – continued
The laptop supplier collects a sample of 100 laptops that have been used for six months from a number of different schools and tests their battery life. The laptop supplier wishes to estimate the proportion of such laptops with a battery life of less than three hours.
f. Suppose the supplier tests the battery life of the laptops one at a time.
Findtheprobabilitythatthefirstlaptopfoundtohaveabatterylifeoflessthanthreehoursisthe third one. 1 mark
Thelaptopsupplierfindsthat,inaparticularsampleof100laptops,sixofthemhaveabatterylifeof less than three hours.
g. Determinethe95%confidenceintervalforthesupplier’sestimateoftheproportionofinterest.Give values correct to two decimal places. 1 mark
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19 2016MATHMETHEXAM2
SECTION B – continuedTURN OVER
h. Thesupplieralsoprovideslaptopstobusinesses.Theprobabilitydensityfunctionforbatterylife,x(inminutes),ofalaptopaftersixmonthsofuseinabusinessis
f xx e
x
x
( )( )
=−
≤ ≤
−
210400
0
0 210
21020
elsewhere
i. Findthemeanbatterylife,inminutes,ofalaptopwithsixmonthsofbusinessuse,correcttotwodecimalplaces. 1mark
ii. Findthemedianbatterylife,inminutes,ofalaptopwithsixmonthsofbusinessuse,correcttotwodecimalplaces. 2marks
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2016MATHMETHEXAM2 20
SECTION B – Question 4–continued
Question 4 (21marks)
a. Express 2 12
xx++
intheform a bx
++ 2
,whereaandbarenon-zerointegers. 2marks
b. Let f :R\{–2}→ R, f x xx
( ) = ++
2 12.
i. Findtheruleanddomainof f –1,theinversefunctionof f. 2marks
ii. Partofthegraphsof f andy=xareshowninthediagrambelow.
y = x
y = f (x)
2
1
–1
–2
1O 2–2 –1
y
x
Findtheareaoftheshadedregion. 1mark
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21 2016MATHMETHEXAM2
SECTION B – Question 4–continuedTURN OVER
iii. Partofthegraphsof f and f –1areshowninthediagrambelow.
y = f –1 (x)
y = f (x)
2
1
–1
–2
1 2–2 –1
y
xO
Findtheareaoftheshadedregion. 1mark
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2016MATHMETHEXAM2 22
SECTION B – Question 4–continued
c. Partofthegraphof f isshowninthediagrambelow.
P(c, d)y = f (x)
12
1
–1
–2
2
x
y
1 2–2 –1−
12
O
ThepointP(c,d)isonthegraphof f.
Findtheexactvaluesofcanddsuchthatthedistanceofthispointtotheoriginisaminimum,andfindthisminimumdistance. 3marks
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23 2016MATHMETHEXAM2
SECTION B – Question 4–continuedTURN OVER
Letg :(–k,∞)→ R, g x kxx k
( ) = ++
1,wherek>1.
d. Showthatx1<x2impliesthatg (x1)<g (x2),wherex1 ∈(–k,∞)andx2 ∈(–k,∞). 2marks
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2016MATHMETHEXAM2 24
SECTION B – Question 4–continued
e. i. LetXbethepointofintersectionofthegraphsofy=g (x)andy=–x.
FindthecoordinatesofXintermsofk. 2marks
ii. FindthevalueofkforwhichthecoordinatesofXare −
12
12
, . 2marks
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25 2016MATHMETHEXAM2
SECTION B – Question 4–continuedTURN OVER
iii. LetZ(–1,–1),Y(1,1)andX betheverticesofthetriangleXYZ.Lets(k)bethesquareoftheareaoftriangleXYZ.
y
x
Y
X
1
–1
1–1 O
Z
y = g (x)
y = x
Findthevaluesofksuchthats(k)≥1. 2marks
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2016MATHMETHEXAM2 26
END OF QUESTION AND ANSWER BOOK
f. Thegraphofgandtheliney=xenclosearegionoftheplane.Theregionisshownshadedinthediagrambelow.
y
x
y = x
y = g (x)
1–1
1
–1O
LetA(k)betheruleofthefunctionAthatgivestheareaofthisenclosedregion.ThedomainofAis(1,∞).
i. GivetheruleforA(k). 2marks
ii. Showthat0<A(k)<2forallk>1. 2marks
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MATHEMATICAL METHODS
Written examination 2
FORMULA SHEET
Instructions
This formula sheet is provided for your reference.A question and answer book is provided with this formula sheet.
Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
Victorian Certificate of Education 2016
© VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2016
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MATHMETH EXAM 2
Mathematical Methods formulas
Mensuration
area of a trapezium 12a b h+( ) volume of a pyramid 1
3Ah
curved surface area of a cylinder 2π rh volume of a sphere
43
3π r
volume of a cylinder π r 2h area of a triangle12bc Asin ( )
volume of a cone13
2π r h
Calculus
ddx
x nxn n( ) = −1 x dxn
x c nn n=+
+ ≠ −+∫ 11
11 ,
ddx
ax b an ax bn n( )+( ) = +( ) −1 ( )( )
( ) ,ax b dxa n
ax b c nn n+ =+
+ + ≠ −+∫ 11
11
ddxe aeax ax( ) = e dx a e cax ax= +∫ 1
ddx
x xelog ( )( ) = 1 1 0x dx x c xe= + >∫ log ( ) ,
ddx
ax a axsin ( ) cos( )( ) = sin ( ) cos( )ax dx a ax c= − +∫ 1
ddx
ax a axcos( )( ) −= sin ( ) cos( ) sin ( )ax dx a ax c= +∫ 1
ddx
ax aax
a axtan ( )( )
( ) ==cos
sec ( )22
product ruleddxuv u dv
dxv dudx
( ) = + quotient ruleddx
uv
v dudx
u dvdx
v
=
−
2
chain ruledydx
dydududx
=
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3 MATHMETH EXAM
END OF FORMULA SHEET
Probability
Pr(A) = 1 – Pr(A′) Pr(A ∪ B) = Pr(A) + Pr(B) – Pr(A ∩ B)
Pr(A|B) = Pr
PrA BB∩( )( )
mean µ = E(X) variance var(X) = σ 2 = E((X – µ)2) = E(X 2) – µ2
Probability distribution Mean Variance
discrete Pr(X = x) = p(x) µ = ∑ x p(x) σ 2 = ∑ (x – µ)2 p(x)
continuous Pr( ) ( )a X b f x dxa
b< < = ∫ µ =
−∞
∞
∫ x f x dx( ) σ µ2 2= −−∞
∞
∫ ( ) ( )x f x dx
Sample proportions
P Xn
=̂ mean E(P̂ ) = p
standard deviation
sd P p pn
(ˆ ) ( )=
−1 approximate confidence interval
,p zp p
np z
p pn
−−( )
+−( )
1 1ˆ ˆ ˆˆˆ ˆ