Probability
Basic
Equations
PCAIB )=DCAnB ) / PCB)
Independence
* Neither #
PCAVBJ
1-
↳ DCA / B) =DCA )
PIBIA )=DlB )
Hinde pendent
PIANBt-PIAHDCBJDCAorBJ-PCBJ-DCAJ-DCANBJPCAIB.sc
)=PCA^Bnc ) / NBN )
PCAORBORC )=pCA ) -111
+
Tree
ex :
Mutually
Diagram
¥
>
PIAORB)
Rz
( never
•¥B÷¥Bz
q¥→Rz
Exclusive
-
-
PCAJTPCB)
independent)
-
p( A) ( DLB ) )
( 1- PCAI )( 1- PCB ) ) / PCC ))
EEX.FM/--x1lxj+PlX=XJCXz)t
SDCX]=F×ÉEF+
.
.
.
.
.
.
.
.
.
.
.
Vara ]=SD[✗32
←
manipulating
random variables
* Bz
5-2
SDLCX ]=cSD[X ]
varf.CI/J--C2VarCx]
E[cX]=cE[x ]
parameter
✗ NIEM ,SDHD
-
vtenormalcdfor
normalpdf
expected valve
interpretation
lfwe
overt
-
over again
average
,
E[ X-Y]=E[X ]
-
EEY ]
E[Xt6]=E[X] -16
sD[Xt6]=SD[X ]
sD[X±Y]=ÑÉD[YT
-
randomly
select
E[XtY]=E[X]tE[Y ]
the
Var[aXtbY ]
=jÉ+ÑY]2
-
the selected
would be about
in
-
E[X , -1×2-1
.
.
.
.
]=NlE[X ] )
.tn#lSDsxDllnthelonqrun,the
expected
is
average
going -10 been
-
sD[X , -1kt
.
.
.
Binomial Random Variable / fixed
oftr.ca/s,XisnUMberofsvccesses1mvststatevaviableincontext,ptsq
exiPCX-4JBinompdfln.pt )
1444117 Binomcdtcn ,p,7)
*
-
#
aswell
Binomcdfcn ,p,z,
X-D /
Kp )
callbinomonlygoestole.tt#
M=np
e-
Conditions
↳ fixed # Oftrials
↳
Independent trials
↳ only Iovtcomet success / fail
:
in
↳
model binomial random variable w/ normal model
success failure condition
-
potsvccessissametoreach
-1 " "
-
A.
PHON -9=10
More
Probability
Geometric Random Variable lfixe.cl#otsoicesseslthetirstI,covntstrialD
probability
x~Glp )
M=kp
that he/she first
misses or makes
geometcdtlpixl
✗
→
number of trials
sD=rq,p
conditions
PIK-nt.pl/-pjn-1
↳ ✗ istriallvntillstsvices ,
Plant 1- Dlxen )
↳
trials
Independent
↳ only Iovtcomet success / fail
__
:
↳
potsvccessissametoreach
trial
Central
Limit Theorem
27a) conditions met
•
•
these 801am represent
27b)
asampleforthistypeofcar
N-pl2.cl 0.0447 )
,
normalcdf
80carswovldbelessthanlo%01-allcarsotth.is type
-0.0126
wears
The
(3,3-1,2-910.0447)
probability
large enough
thatyisbetween3-3.lgmlmiisabout.co
emission forcarsare mutually independent
0.0126
•
should be
ng=My=2 -99M / Mi
05=71--0.tt#Y-mi--o.o447mq1m ;
N~PH.cl , 0.0447)
27c)
INVNOVM / 0.950,1 , / eft)
-4.6449
z=Y¥
1.6449=0%41-7
4=2.9736
Thereisonlyatolochanlethatthe
fleet 's
meancolevelisgreaterthan
2.97369mm;
1) conditions met
-
SRS
mentioned
25finchesshovldbes-10%01-entirehovsetinchpop.nl
-
-25<30
*
assuming
lnotbigenovghsamplesizeproceedwithcavtion )
lheshapeottheorq.pop.is/-air1ysymmetrictheCLTapp1ies*Mx---
-
Ng
1.5min
?
pix
a>
1.71
.FI#-- 0.18min
F- Nlliimin , 0.18min )
=O -1333
" 7min
=
1.5min
Thereisao -1333
probability that the song duration
will be
greater
than 1.7min however weshovldnitpvttoomvchfaithinthis
because Ztisnotalargeenovghsampleandweare assuming that
theorq.pop.is symmetric
3)
.
conditions met
-
-
srhsmentionedil
36hotdoqsa-10%0f-wholepop.othotdoythatmanvfaltormaket.ir
enough sampler
Cltapplies
-36>301 big
i.
If
,gg18!4g
ni=18g
05-1%6=16-9
?
PIIH8-4JI-N1189.t.gs
-0.0082
-
Themanvfactorsclaimisnotwrongasitisshownthatthe
probability forthetatcontenttobe18.4qorgreater.is 0.0082
Which is very unlikely and provides evidence thatthemeantat
content's around
18g making
the manufacturer 's claim correct
.
F) conditions met
srsmentionedv-36shovldbec.IO/ootallprodvcedo.tboltsV
-
-36>3011941 enough samples
#
0.49in
o.g.no
-51in
Plo -49<1%0.51 )
-0.9976
1-0.9976=0.0024
-
iii. 0.5in
0i=0¥=o -0033in
I
-
N / 0.5in 0.0033in )
,
The probability the process
onanygivendayiso -0024
.
will be shutdown
confidence Interval
2) Themarginoterrormeansthatthe
the true
medical researcher believes with some % confidence -1ha -1
,
,
proportion otchildrenexposedtolead-basedpaintiswithin34ooth.is estimate
lollcanconclvdethatweare 95% confident that the true
Nebraska Boardofparoleis between 0.5610 and 0.6252
.
proportion otdecisionsmadebythe
.
Bblweare 95% confident that the true proportion of
auto accidents that involve teenagers is between
But
conditions met
sprsmentionedv
0-1268 and 0.1860
-582910dm -1140%01-911 accident, ✓
ni > long > 10582194-821--91>-10 ✓ 13495%10 nfidenle means that within these range
58214945821--491>-10
Otvalves Weare 91-40 certain / twill contain the
true pop proportion
pimp
-
.
-
,F¥s
F- Nlp
,
.
0.01511
%É¥i%
.
Bdlltcontradiltsthis statement because the
politician assaying -1ha -10.201 theallidentsinvolve
-
-
ateenaqerbvtbasedonovrc-1-ithevalveshovldbelessthano.NO
95%
-1268,018601 and
thatthetrvepop proportion
.
95%11=9%-82
'
-
'
-
"
zany )
.
0.1860
Weare
sure
lies within 0.1268$
'
1582+-11.91-9963986110.011-1 ,
(0.1268/0.1860)
* Remember 'HsP-hat*
Confidence Levetssamplesize affects confidence interval
↳ bigger samples
ME .=Z*hF¥ )
.ee/smallermarginoferrorl
↳ half
confidence interval
M.EE#P-rn )
confidence level The probability thatitthetestwasrepeatedoversover again
-
obtained would be the
,
the results
same
Vponrepeatedsamplingvsinqthesarnesamplesiesmethod
otthec.l.pro dviedwillcaptvrethetrve
-
_#%
Hypothesis Test
one proportion
one tail
two tail
2- test
1
proportions)
TWO
Proportion
Making
Decisions
2-
Interval
using test statistics
TWO
Proportion
2- test
12 groups independent
* It
iii. out
two
proportions
sample , they
are
single
independent *
come from
not
Critical Value
Difference between 2 proportion
2-
test $2
proportion
z
-
interval
a
.
Type
Type
-1-31 # errors
lfnvllhypothesisistrvebvtwerejectit-ype2-l
tnvllhypothes.is
wrong
/
→
but we don't
is
power
→
The
Probability
reject
probability otcorrectlyrejectingthetalsetlo
getting Type -1#
of
Type
→
>
-
p
Xlsamexllevelof
power-up
9 sample size $9
↳ have more
into onpop.io
there is abetter
estimate ottrve
pop .la power )
significance
will
↳ Willa
.
significance )J
tPlTypeIJ
PITYPEI )
tnchanceot
rejecting How
to PITYPEII)
-
SENKSD
SEM
SDM SEM
.
sD→
-10 measure
spread
of measurements
Interpretation onaveraqe
-
SEM> measure
Margin
,
-
uncertainty
Willditterbysfrommean
around the estimate ofthemean
of error
M.E.az#fFFqhorz*fn2waysolevelo1-
confidence
MET it
-
out
,nh
or
isao%C2-a.in
2=1.65
Clf
lonovtpvttablesE-F.nl
Raise Power
9×9
,
samples 're ,lvsD
Point Estimate
,
change
Ha
↳ center / middle of CI
conditions for Inference procedures
T
-
Distribution
1- sample -1
-
interval
sometimes might
¥83Bn
have to calculate
mean
→
no extreme outliers
or use NPPIONCAKI
it approx normal
it meets the condition
.
I
sample
••µl
-
t test
,
SD
yourself
2
sample
t
-
test
0
0
2 sample t interval
-
lxi-i.t #fEFE.lIiIzl~tlx-,SElXi-Xi
))
Matched Pairs
-1
-
test
Matched Pairs
-1 interval
-
mean differences
Chi
Chi
-
square
Square Goodness
Of Fit
/ Do
-
reflect -1 / consistent )
Chi
Square test
of
Homogeneity
expected
row
total
✗
column total
total
Chi
square
test of
Residuals for Chi square
-
Independence
Testotln dependence
2cateqor.ua/variab1eareassoiiatedW1oneanother
↳
not causation
↳ observational units are collected
categorical variables are observed
atrandom $2
lvnliketestofhomoqeneitywheresrsistromh
↳
stat-test-XZ-test-smutrix-edit-enterrowacolvmntorns.us
groups separately)
→
gobackslclickcalc
←
ltyoohavemvltipleqvestionsforonegrovp
↳
"
association
Testot
↳
"
"
or
dependence / independence
Homogeneity
distribution "
Residual forx '
↳
"
"
same /
procedure
c=l0bfɥ
homogenous
"
"
Linear
Regression
-1 Test
-
actual
# Halve
lope coefficient te stat
g-intercept
Rvalve tells
-
us
about
strength , directional form of the scatter plot
linear
Regression
T interval
-
ltlow
{
-
-
-
-
Regression
*
Reg
Have
↳ Igo
to vars , stats,
eqn Req) Replace
,
Ly ( Lz Lst Obs Exp
2nd y= , 1st plot keep 4$14
-
-
is the association )
Linear
Plot Residuals
put data to list
Lin
strong
-
✗
w/ Li