On the strategic equivalence of multiple-choice test

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Nov 17, 2005 - and lowers test reliability in measuring students' knowledge.2 This is the case .... of 1. M−1 points for each incorrect answer. If a student does not know the ...... The original instructions were given in Spanish and Basque:what ...
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B$XIFO.DMJL.B$X"iQPRB$AGlAGIDu€[XIHPR@ DKSgijD†HnŽAGIDrLA>IS"DXhAoMXISkS"DKFO.DMJLB$X"iQPRB$AGkAGIDuS"DiZO.DKD†Hn"@$ATkHn AGIDbB$A.DcŠ\`bGIDbC"O.HJvMv"B$@$B$AT

i

zg@ DMYhB$X"i

N −1

B$A.Dc

OB$iZGhAsHJOonMB$@]\|åuDXIHJA.D ̎DA

` {i}

Hnn]MB$@$B$X"i†AGIDbB$A.DcÎB L

1−qi

Lo>"XMXILP|DO.DKS:\|`bGIDO.DWMO.D†AP|H

s(r, w, o)

\”uL.L>"c

D}AGIDRLA>IS"DXpA}MXILP|DO.LtHJX"@$TNB$A.Dc

C HZL.LB$v"@ D†DYjDXhA.L

AGIDWLFKHJO.DˆHJv"AMB$XIDKSŠn„O.HJc

r

OB$iZGpA.Lz

S"DXIHJA.DWAGIDr@ HJAA.DOTkB$XISg>IFKDKSŠvpTwMXILP}DOB$X"iHJX"@$TlB$A.Dc

i

z

w

â

LGIDrceMTkijDA{AGIDWMXILP}DO PRO.HJX"ijLrMXIS

o

HJcQB L.L.B HJXIL\

` {i} = [qi ◦ s(1, 0, N − 1) ⊕ (1 − qi ) ◦ s(0, 1, N − 1)] . »uHJA.DNAGMA†AGIDyL.FKHJOB$X"iwO>"@ D v">"AbAGIDrC"O.HJvMv"B$@$B$AT ç

qi

M› DKFA.LrAGIDyYZM@$>IDKL†Hn|AGIDyL.FKHJO.DZz

B L}O>"@ DE]B$XIS"DC8DXIS"DXhA\|Ì0DA

U (` {i})

ø ÔZÚÛá)Õ0áœáœà7ë ç7Ó¥ÚÛÝ]Öoæ7ݪÒá)ÖZÝӒ҉òŠäâÛà^æŠÒç^Õáœá cߞÕ0ÚÛâU҉ò7Õë áÚ©Ö ’ÔZÚÛäŸÔ „

Á

o

s(1, 0, N − 1)

MXIS

s(0, 1, N − 1)

S"DXIHJA.DrAGIDr>"AB$@$B$ATlS"DOB$YjDKSlnO.HJcæAG"B L

u0 = 0

à7ÖªÓ¥Ú©âWÓ¥Ô7Òç^Õáœá)áœä‰Ý0Ù¥Ò2ÚÛá’Ù¥ÒÕ0äŸÔZÒfæsî

@ HJAA.DOTjz

U (` {i}) = qi u(s(1, 0, N − 1)) + (1 − qi )u(s(0, 1, N − 1)).

•]nAGIDbLA>IS"DXhAtMXILP|DO.LB$A.Dc PRGIDXŠAGIDrAP|HQB$A.Dc

2)

MXIS

s(0, 2, N − 2)

AGID†n„HJ@$@ HPRB$X"i

L

i

MXIS

j

zJ@ DMYhB$X"i

LsMO.D†FKHJOO.DKFA@$TkMXILP}DO.DKS:z

N −2

B$A.Dc

L>"XMXILP|DO.DKS:zhLGID|ijDA.L

s(1, 1, N − 2)

s(2, 0, N −

PRGIDXŠLGIDrijDA.LRHJXIDrB$A.Dc,PRO.HJX"i

PRGIDXVLGIDyijDA.L†v HJAG‡PRO.HJX"iI\¿è¬`bGIDQCMKTjH›:LWMXIS‡C"O.HJvMv"B$@$B$AB DKLNMO.DQiZB$YjDX‡B$X

AMv"@ D

«[FKHJO.DWMXISqé!O.HJvMv"B$@$B$AT «[FKHJO.D

é!O.HJvMv"B$@$B$AT

s(2, 0, N − 2)

q i qj

s(0, 2, N − 2)

(1 − qj )(1 − qi )

s(1, 1, N − 2)

PRGIDO.D

qi

MXIS

qj

qi (1 − qj ) + (1 − qi )qj

MO.D}AGIDbC"O.HJvMv"B$@$B$AB DKLƒHn3ijDAAB$X"iˆAGID}OB$iZGpA|MXILP}DO!A.HuB$A.Dc

`bG"B L{@ HJAA.DOTlceMTkv8DrPROB$AA.DX?MJL

L

i

MXIS

j

O.DKLC DKFAB$YjD@$Tj\

` {i, j} = qi ◦ [qj ◦ s(2, 0, N − 2) ⊕ (1 − qj ) ◦ s(1, 1, N − 2)] ⊕ (1 − qi ) ◦ [qj ◦ s(1, 1, N − 2) ⊕ (1 − qj ) ◦ s(0, 2, N − 2)] . `bGIDˆ>"AB$@$B$ATwS"DOB$YjDKSkn„O.HJc¶AG"B LR@ HJAA.DOTkB L

U (` {i, j}) = qi qj u(s(2, 0, N − 2)) + (qi (1 − qj ) + (1 − qi )qj )u(s(1, 1, N − 2)) + (1 − qi )(1 − qj )u(s(0, 2, N − 2)). «hB$cQB$@~MO@$Tjz"B$XŠMXŠD"McêPRB$AG

L(N )

N

B$A.Dc

L{MXhTlL>"vILDAbHn"XISw@ HJAA.DOB DKLRB$XqMXŠDIMcæPRB$AG

@ HJAA.DOTjz[S"DXIHJA.DKSvpT PRB$AGqAP|HeB$A.Dc

` {0}

LbB L{M@ L.HQMyFKHJcQC8HJ>"XISk@ HJAA.DOTj\ƒÌ0DA

N

B$A.Dc

zpPRG"B F.GkFKHJOO.DKLC HJXIS"LƒA.HdHJcQB$AAB$X"iNM@$@B$A.Dc

LRB$XIF@$>ISgB$X"iMyS"DijDXIDOMA.D

L\!¢IHJO|DIMcQC"@ DZzpB$XkMXeDIMc

LzgAGIDˆL.DA{HntM@$@ŽFKHJcQC HJ>"XISw@ HJAA.DOB DKL{B L

L(N ) = {` {0} , ` {1} , ` {2} , ` {1, 2}} , AGMAB LzJAGID}S"DijDXIDOMA.DR@ HJAA.DOTdPRG"B F.G

FKHJOO.DKLC HJXIS"LIS"DXpAQFGIHhHZL.DKLNAGIDlv DKLAdFKHJcQC HJ>"XIS¥@ HJAA.DOTVB$X

L\

L(N ) ë

¢IHJOceM@$@$TjztLGIDceMgB$cQB$‘DKLyAGID

DgC DKFA.DKSw>"AB$@$B$ATwHYjDOoAGIDˆL.DA{HntM@$@0C8HZLLB$v"@ D†FKHJcQC8HJ>"XISŠ@ HJAA.DOB DKL

max U (`)

’£

` ∈ L(N ) `bG"B LeC"O.HJv"@ DcìL.>"cQceMOB$‘DKLHJ>"OMJLL>"cQC"AB HJXILHJXaLA>IS"DXpA.L|v DGMKY[B HJO\ãŸbDOXMO.S"H㒠( Z Zí î

Ñ=ÚíÓ¥ÔZÝ0à7ӒâÛÝ]áœá)Ý0ß,ã0ÒÖZҘٟÕ0âÛÚíÓOÞ Ò2Õ0áœáœà7ë Ò»Ó¥Ô^Õ~Ó4Õ0âÛâHÚíÓ¥Òë á ފÚÛÒâ©æÓ¥Ô7Òá¥Õ0ë ÒçZÕfÞQÝ sáî o

Â

Ó

“

MXIS

“

Ÿ}>"OijHZL{’£Z¤Z¤Á

MO.DbAGIDbHJX"@$TdHJAGIDOƒMAA.DcQC"A.LtAGMA!P|Db€[XIHP¨Hn8MA!c H[S"D@$B$X"iˆLA>IS"DXpA.Lv DGMKY[B HJO!B$X “ cd>"@$AB$C"@ DEF.GIHJB FKDbA.DKLA\IS"DXpA.L"OijHZLo’£Z¤Z¤Á >IL.DKLC"O.HZL.C8DKFAAGIDKHJOTNA.HsC HZLA>"@~MA.D{Ms>"AB$@$B$AT “ n„>"XIFAB HJXqAGMArMJL.LB$iZXILsSgBœ› DO.DXhAsYZM@$>IDKLoA.H @ HZL.DKLsMXISqipMB$XIL\R™sHP|DYjDOz3XIDB$AGIDOuHn!AGIDKL.DNM>"AGIHJO.L A.DKL.A.DKSwAGIDB$OsMJL.L>"cQC"AB HJXILoDcQC"B$OB FM@$@$Tj\ `ŽHmiZB$YjD?LC DKFBœ¹FqFKHJXpA.DXhAA.HVAG"B LeipMcdv"@$B$X"ifB$XhA.DOC"O.DAMAB HJX¬HnrB$A.Dc LFKHJOB$X"i

O>"@ D†AGMA{C DXM@$B$‘DKL{nHJORPRO.HJX"ijLz

MN@ HJAA.DOT

\tïoXIS"DOoAG"B LoL.FKHJOB$X"i

O>"@ DdMXILP|DOB$X"ilHJX"@$TlB$A.Dc

i

B L

â

•XqAG"B LuFMJL.DZz3PRB$AGqC"O.HJvMv"B$@$B$AT

1 − qi

S2

Lzb@ DAe>ILeFKHJXIL.B S"DOeAGID

L.GID†B$XIF>"O.LRAGID†C DXM@$AT

`bGIDRO.DP}MO.S



 qi ◦ 1 ⊕ (1 − qi ) ◦ −

qi

AGIDdLA>IS"DXpArijDA.LrM

1 M −1



.

L.FKHJO.DdHnƒHJXIDWC HJB$XpArMXISqPRB$AGxC"O.HJvMv"B$@$B$AT

1 M −1 .

n„HJOƒHJcQB LLB HJXILtL.FKHJOB$X"iˆO>"@ D{GMJL|MrLB$cQB$@~MOtB$XhA.DOC"O.DAMAB HJX0\ƒïsXIS"DO

GMJL}MˆL>"O.D{CMKTjH›

S3

AGID{LA>IS"DXhA

\ N Bœn0LGIDsS"HhDKLƒXIHJA}MXILP}DO}MXpTQB$A.DcŠzhAGMAƒB LzjBœn0LGID{FGIHhHZL.DKLƒAGID{@ HJAA.DOT ` {0} M …ãGIDXkAGIDsL.A>IS"DXpAoMXILP}DO.LbB$A.Dc LGIDs@ HZL.L.DKL}AGIDrL>"O.DsO.DP}MO.SlnHJORHJcQB$AAB$X"i 1 \!•]nŽLGIDuijDA.L}AGID i M OB$iZGhAoMXILP|DOsLGIDuipMB$XIL{MyL.FKHJO.DrHn

i

1

B LsMy@ HJAA.DOT



MXISlBœn"@ DKLNMO.Dkøá„ÝùZáÜßJÞú.ùJûû_ümÜ.ýÙhސþùJû ÜØáˆBœnbAGIDTmM@$PbMKTgLWB$XISg>IFKDeAGID v8DGMYhB HJOoB$X?MNOMAB HJXM@5DIMcæAM€jDOd’MXqDgC8DKFA.DKSŠ>"AB$@$B$ATkceM[B$cQB$‘DO •XwAG"B LRCMC DO{P|DWF.GIDKF.€kPRGIDAGIDO

S2

MXIS

S3

“

LMc

D

\

MO.DˆDK‹h>"B$YJM@ DXhAoHJO{XIHJA{>"XIS"DOoSgBœ›:DO.DXpAoOB L€C"O.DnDO‰E

DXIFKDKLsMXISŠDKLAMv"@$B LGŠYZMOB HJ>ILRO.DKL>"@$A.LsHJXwAGIDˆLAOMA.DiZB FˆDK‹h>"B$YZM@ DXIFKDˆHnLFKHJOB$X"i

O>"@ DKL\



ÿ:*

0ÇI2"O.DZzC DXM@$B$‘KB$X"ilPRO.HJX"ijLuMXISqO.DP}MO.SgB$X"iwHJcQB$A.Lz

MJLoS"D¹IXIDKSwB$XŠAGIDrC"O.DY[B HJ>ILoL.DKFAB HJX0zMO.DˆXIHJARB$XŠijDXIDOM@ŽLAOMA.DiZB FM@$@$TŠDK‹h>"B$YJM@ DXhA\

  

ö tö ô;õôö3ó

÷

S2

MXIS

S3

MO.D†XIHJAsLAOMA.DiZB FM@$@$TwDK‹p>"B$YZM@ DXpARnHJO{OB L€kMKYjDO.L.DdD"McæAM€jDO.L\

`5HeLGIHP½AG"B LzIB$A{B LsL>gÀeFB DXpArA.Hy¹IXISxMX?DIMcQC"@ DWPRGIDO.DNMQOMAB HJXM@5DIMc,AM€jDOuP|HJ>"@ S?v DGMKYjD SgBœ›:DO.DXpA@$Tq>"XIS"DO AB$YjDKLz

M = 2,

S2

AGMXq>"XIS"DO

S3 .



HJXILB S"DO†MXUD"Mc©PRB$AGxHJXIDdB$A.DcŠz

MkLA>IS"DXhA†PRB$AG‡YJM@$>MAB HJX

ijDAAB$X"iqAGIDyOB$iZGpAdMXILP}DO\y”oXILP}DOB$X"iŠAG"B LˆDIMcB LˆMl@ HJAA.DOT MJL

S2

\"XIS"DO

PRG"B$@ DRHJcQB$AAB$X"iWThB D@ S"LƒMrL.>"O.D|CMTjH›ŠHn 1 \5ïsXIS"DO √ 2 >"AB$@$B$ATen„O.HJc¶MXILP|DOB$X"ieB Lb@ HP|DOoAGMXwAGMAbnO.HJcæHJcQB$AAB$X"i Bœn

l3 = [qi ◦ 1 + (1 − qi ) ◦ 0]

™uHP}DYjDOz >"XIS"DO

AP}HlM@$A.DOXME

√ z:PRB$AG MXISUC"O.HJvMv"B$@$B$AT Hn a+s a ≥ N, qi

u(s) =

PRG"B$@ DbHJcQB$AAB$X"irT[B D@ S"LMsL>"O.D|CMKTjH›kHn‘DO.Ho>"XIS"DO

N = 1,

MeOMAB HJXM@DIMc–AM€jDOs¹IXIS"L†M

qi

S3

FMXyv8DƒPROB$AA.DX

AGIDRD[C DKFA.DKS

S2 , √ √ a + 1 + (1 − qi ) a − 1 < a.

G"B$iZGIDOrDgC DKFA.DKSq>"AB$@$B$ATŠn„O.HJcÍMXILP}DOB$X"ikBœn

S3 q √ √ •AyB LyDMJLTVA.HxYjDOBœn„TmAGMAynHJO MXIS v HJAG qi a + 1 + (1 − qi ) a > a + 12 . a = 1 qi = 0.6 Ã

B$XIDK‹h>M@$B$AB DKL}GIHJ@ S:\!¢MJFB$X"iyAG"B L|DIMcŠz[AGIDsL.A>IS"DXpA}P|HJ>"@ SkHJcQB$A|>"XIS"DO `bG"B LoLGIHP{LoAGMA{AGID†AP}H

L.FKHJOB$X"i

DˆLAOMA.DiZB FM@$@$TwDK‹h>"B$YJM@ DXhA\

ö tö ô;õôö3ó

`5HqLGIHP

÷R¢IHJOsMNOB L.€XID>"AOM@ŽDIMcQB$XIDKDZz

AG"B Lz5B$AWB LWXIDKFKDKL.LMOTUA.HŠC"O.HYjD

DwS"DKFB LB HJX¦>"XIS"DO

FGIHhHZLDo@ HJAA.DOT AGMAoB L

DB$AGIDO

nHJO{M@$@

MXIS

S3

MO.DˆLAOMA.DiZB FM@$@$TwDK‹p>"B$YZM@ DXpA\

HnsAGIDwL.FKHJOB$X"i‡O>"@ DKL\fïsXIS"DO

S2

z|M?OB L€fXID>"AOM@oLA>IS"DXpAQP}HJ>"@ S

X

qi` s`2i ≥

zgPRGIDO.D

s`2i

0

X

’ ¡

0

qi` s`2i

MO.DrAGID†L.FKHJO.DKLR>"XIS"DO

S2

MJL.L.H[FB~MA.DKSUC"O.HJvMv"B$@$B$AB DKLdHn}AGIHZL.DQHJ>"A.FKHJc

X

qi`

i

`0 ∈ L(N ).



ïuLB$X"iq’ (

M −1 ` N + s2i M M “

HntM@$@:C8HZLLB$v"@ DrHJ>"A.FKHJc DKLzŽLHAGMA

i

nHJOoM@$@

`0 ∈ L(N ).

P

LDKDc

≥

qi` s`3i ≥

X

qi` s`3i ,

i

0

S2

MXIS

S3

S3 .

` i qi

D†B$Xw@ HJAA.DOT

=1



\

MXIS

`

HJXISgB$AB HJXa’ ¡

B L “

 M − 1 `0 N + s2i , M M

0 qi`

0

i

•XqP|HJO.S"Lz"AGIDWLA>IS"DXpAsP|HJ>"@ SqFGIHhHZLDˆ@ HJAA.DOT

MXISqHJX"@$TBœn!LGIDrP|HJ>"@ S?F.GIH[HZL.D†B$AR>"XIS"DO «[FKHJOB$X"iO>"@ DKL



X

P}DˆFMXŠPROB$A.D

X



“

i

DK‹h>"B$YZM@ DXpA{A.H

nHJOoM@$@

AGID

PRGIDXIDYjDObB$A.L}D[C DKFA.DKSeCMTjH›qB LbMA|@ DMJLARMJL|G"B$iZGkMJL|AGMA}Hn5MXhTeHJAGIDO}@ HJAA.DOTjz

`

`0 ∈ L(N )

MO.DyAGID

S2

AGMAyMwOB L€UXID>"AOM@|L.A>IS"DXpANP|HJ>"@ S—M@$P}MKTgLdceM€jD

i

S3 .

O>"@ DKLuMO.DrXIHJA{B$XwijDXIDOM@5L.AOMA.DiZB FM@$@$TkDK‹h>"B$YZM@ DXpA\

   

qi`

MXISlMXILP}DO|>"XIS"DO

>"O}L.DKFKHJXISlO.DKL>"@$AbLAMA.DKL}AGMA|n„HJORMˆCMOAB F>"@~MObAThC DsHn0OB L.€MAAB$A>IS"DZz[AGIDsAP|HyL.FKHJOB$X"iNO>"@ DKL

v DKFKHJc

LMc

S2

`

B$XŠAGIDWL.DA

L(N )

>"XIS"DO

`bG"B LoFKHJcQC"@ DA.DKL{AGID†C"O.H[HnHn!DK‹h>"B$YJM@ DXIFKDZ\

MO.DWDK‹h>"B$YZM@ DXpA{nHJOoOB L.€lXID>"AOM@LA>IS"DXhA.L\{™uHP|DYjDOz DgC8DOB$c

A.HˆB$XISgB FMA.DsAGMAƒLA>IS"DXhA.L}S"HˆXIHJA}M@$P}MT[L|v DGMKYjDsB S"DXpAB FM@$@$T



Bœn

S2

DXhA.L

>"XIS"DO|AGID{AP}HWL.FKHJOB$X"idO>"@ DKL\



>"ORC HJB$XpAoB LbAGMAROB L€C"O.DnDO.DXIFKDKLRceMTlGMYjD†v DKDXkSgB L.cQB L.L.DKSŠMJL HJAGIDORCILTgF.GIHJ@ HJiZB FM@ŽnMJFA.HJO.L

B$XwAGIDWD[C DOB$c •X

S3∗

DXpA.LsS"DKLB$iZXIDKSwA.H

DYZM@$>MA.D

MXIS

S3

\

AG"B L|CMC DOtP|DsC"O.HJC8HZLDoM†XIHJOceM@$B$‘MAB HJXkHn:AGIDoL.FKHJOB$X"idO>"@ DKL

\ƒ`bGIDKL.DrXIHJOceM@$B$‘DKSŠO>"@ DKLuMO.DˆMJLbn„HJ@$@ HP{L

S2∗

=

S3∗ PRGIDO.D

p =

B LWAGIDC DXM@$ATj\ 1 M −1

XIDipMAB$YjDqYJM@$>IDKL\¨»sHJAB FKDŠAGMA ‹h>IDKLAB HJXILR>"XMXIL.P|DO.DKS:\

    ö tö ô;õôö3ó

¢HJO

S2

S2∗

MXIS

S3∗

â

S2∗

MXIS

S3∗

S2∗

S2∗

MXIS



r − pw +p N

1 = N MXIS

S3∗

 r+



S2

MXIS

S3 ,

S"DXIHJA.DKS

MXIS

1 , 1+p

 p o , 1+p

@$B DB$X¥AGIDe>"X"B$ANB$XpA.DOYZM@]z!PRGIDO.DMJL

S3∗

S2∗

S2

ceMTmAM€jD

MJLLB$iZXaMUC HZLB$AB$YjDŠL.FKHJO.DkA.HVD"McQB$XIDKDKL@ DMKY[B$X"ifM@$@

MO.DˆLAOMA.DiZB FM@$@$TwDK‹p>"B$YZM@ DXpA{n„HJOsM@$@:AThC DKL{HnOB L€lC"O.Dn„DO.DXIFKDKL\

A.Hˆv8DRLAOMA.DiZB FM@$@$T

DK‹h>"B$YZM@ DXpAƒAGIDoLA>IS"DXpAƒL.GIHJ>"@ SeF.GIH[HZL.DRAGIDoLMc

A.HUMXILP}DOd>"XIS"DOyAGIDAP|HUL.FKHJOB$X"ixO>"@ DKLz"AB HJX¥A.H?C"O.HJv"@ Dc Ä

’£ “

DRL.DAƒHn B$A.Dc

B LNAGIDLMc

DZ\?¢IHJO

L

AGMAbP}D†LB$cQC"@$TeGMYjDrA.HNL.GIHP­AGMAbAGID†L.DA AP|H

L(N )

B LbB S"DXpAB FM@:>"XIS"DO

S2∗

MXIS

S3∗

\5…mD†S"HNAG"B L}B$X

L.A.DCIL\!¢"cWv DOuHn!B$A.Dc

LsB$XqAGIDdD"Mc¶B LuDK‹h>M@0A.H

OB$iZGhA.LoC"@$>ILsPRO.HJX"ijL

C"@$>ILoHJcQB$A.L{P|DˆGMKYjD†AGMA

S2∗

=



r − pw +p N



1 = 1+p



r − p(N − r − o) +p N



1 1+p

p r + 1+p o (1 + p)r + po = = S3∗ , = N (1 + p) N

LHeCMTjH› LˆMO.DQM@ L.HlB S"DXpAB FM@!>"XIS"DO†v HJAGUL.FKHJOB$X"ikO>"@ DKL\ˆ`bG"B LrFKHJcQC"@ DA.DKL†AGIDNC"O.HhHn}Hn|DK‹p>"B$YZME @ DXIFKDZ\ `bGIDƒXIHJOceM@$B$‘DKSyL.FKHJOB$X"isO>"@ DKL

S2∗

MXIS

S3∗

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