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Nutl hypotthests
Aemate fhpethest^ pt^ 06.
a) Evaluat
ejectono^ denoBer o the^ obabiliyhut
-P(thart
POX-)-bC, 15, o) (jeton
tthct p 06
Samplr
ypothens
of commHtng
PCOsxSS
AoP=O.
null hy^ ptthes's^ ohe^ n^
it's tue)
Tutortal no- 4
tohen H 's^ toue
the numbex of college^ araduaes^
in the 's belueen
S) 06 Ct-06)
type-T or, whteh
o and^5 or^
betoeen 19 and^15
or (3<^ Xsls^ )
O0338+ O2|
Csorn Stattbteat^ table )
2bC I5,^ 06)^ t^ bl2;^
5, o-s)
Þ
O
b (*;15, 06)
foa aHrattve p 05
denoes he poobab'ity
nbn- ectba of
The pnbabity
fos p- 0,
cbG,n,p))
of toyittrg^ a^ type-^ I^ ror,^ tohtth^ th the nul^ hypathestr^ ben^ is^ -btse.
of commha
12
rof oood Orough
bl6,15,^ os^ )
+ BC^ (o;^ (5,0:5)
Y-
of hon- ertton
he null hypothest
B
B; 15, o5)
6
ond pr t,
n the^ sonmple^ be-teeen^
6 and^ t
O 0'
b(#;(5,^ 0s)^ +^ b(8;^ 5,^ 0-5)^4
bC9, (5,o-5)
4 bCM;^ I5,^ 0-5)^ t^ b(12;(5,^
ttho num bey
nocluafes inhesample
6(1;5, 02)
2 bCa;^ 15,^ 07)
o Fox p- 0'5,
The pobabtlty^ of^ committog^ ype-I
of colleg
00034
when i^ s^ fal^ )
of co'lee^ qyaduate^ ,^ ohen
Alothe pobabiltey
qe ehtnely^ hiah.^ So^ th^ ts not
Cfiom satstal fabte)
t betoteen^6 Rle^
rtohen p^ 4)
h(a:l5, 0-4) Cfom stattttal^ table^ )
IS O06,^ wh^ rch^ t
of cemmitn9 ftypeI good poteduar
436 Variance Decomposition. We (^) start (^) with (^) the (^) following (^) equation: If (^) we (^) square both (^) sides ve (^) obtain
The (^) last (^) term (^) on the (^) nght-hand side is
We therefore obtain
which equates to
Ni-= (; - )-(Þ - ).
The Relation between R and r.
sOReidual
= bSxy
i=
= Syy +bs-26s.y
= Syy - bSr = Syy
SORegression Syy- SOResidual
(Sy)? Sxx (S.)?
i=
Sxx
i=
Appendix C: (^) Technical (^) Appendix
i=
i=
i=
Appendix C Technical Appendix
We thercfore obtain R (^) SORegression Syy
(S? S Sy TheLeast Squares Estimators areUnbiased.
E(Ò) =E(X'X)'X'y) Giventhat Xin the model is assumed to be fixed (i.e., non-stochastic and not following any distribution), we obtain
E(Ô) =(Xx)x'E().
Since E(e)^0 it^ follows^ that^ Ey^ XØ^ and^ therefore E(Ø) =(X'X)X'xø-8.
How to^ Obtain^ the^ Variance^ of^ the^ Least^ Squares^ Estimator.^ With^ the^ same
arguments as^ above^ (i.e^ X^ is^ fixed^ and^ non-stochastic)^ and^ applying^ the^ rule Var(bX)) =Var(X)^ from^ the^ scalar^ case^ to^ matrices^ we^ obtain:
Var() =Var(N'N)'X'y)^ =(X'X)^ x'Var(y)^ X(N'x)-=o'(N
437
Maximum Likelihood^ Estimation^ in^ the^ Linear^ Model.^ The^ lincar^ model follows anormal distribution: y XØ+e^ N(XO,^ o').
ihood function^ is^ given^ by
Therefore, the^ likelihood^ function^ of^ yalso^ follows^ a^
nomal distribution:
A1}Pe) of cenHol charl I) Aribue dala:^ ohen^ dat a^ is^ in
- Numericu 0ata
hen data^ ig^ in^ the^ Form^ o^ an ttibte ot^ count^ form,^ hen^
we wi USe^ con^ Ho)^ CharH^ Ike
P chart
U chart
when dat^ a^ is^ in^
the
a Continous^ +ype then^ we^
oi we ConHo) charH ike
X- bar chart
R boar charL
bar chart
Form al
x bur chart
UcL 3061S
*= 3. o
LCL 3-615S
Q12) Gven
No
mean: 95
We
S
To chenle
The rneon
toheheY proess
neove
LCL 36
ten e, the
Upors (onto tnit
o catculafe
35- 3x
UCE u t 36
contol
3o
o
39025
liraits
tus tn conhol
does nof
(ouerc (onrol tihnit CLCL)
(Uct)
the thi one have eans
af thid^ subqroup^ s^ belouw^ LCL^
ohrkk
fo be conto!
) find aetocovelrton fureion RO) EO)
c) (^) Nandom eleonogh prores is sttorory. Because,
Given
at) £x))^ t^ tndeperdent^ of^4 O Rt,to) ROo, t,-) R(T)
3 RO)^ :F))^ < Rordom hene t
\anables Sone fred
tcle qaph S
tutll be
set of posible
State
Sippeose
t (6,2.... m
sate of we say thart^ susm^ is^ In^ sterte Systern of
a Stguenoe^ ot^ ondem^ Yashbles
Cfinite secord momenf) proess
T
ie P( Xht|=
of Sone Systensusern
atsfy all aböve^ rondfon is widesense,
Ratnn tocoy
Values of thesheg^ ndom^ anah^ les
to iteoee^ Xn^ as^ being
Mos kov Chain is^ a if each me system ts
tn next stat.
hof aining
tocoy
ts hetofal
Xhci, Xh-
Cdefniton
pobability.
in-t,
n' & therfore
fox cohich
0sjs N
at timt^ n'^ ifi
y uen ce cf onom
in slafey and
he sysm
Xo io
of tonsètion ProbabiliHg
Psobabi fty of vaining tornoroo t
pobabily oy P
o hen it was hot
anihg
twhen wasn of aining foday
)
fio
Poo
tomosrow
tomorgowohenit ias
ainirq oday)
too nstston matnx
PLrainng
eP (not uhing
fo PCX: pXo- O)
P
P
ormorCrainihg
