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Formula Sheet for Final Exam

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Formula Sheet for Final Exam
Mean
!
!!! π‘₯!
π‘₯=
𝑛
=
π‘₯! + π‘₯! + π‘₯! + β‹― + π‘₯!
𝑛
Median:Ifnisodd,middlevalueofsequence;ifniseven,averageof2middlevalues
Range=Max–Min
Variance
s2 =
n
(
∑ xi − x
i =1
2
)
n −1
Standarddeviation
s = s2
IQR=Q3-Q1Outlier:1.5*IQRrule
P(A)=
!"#$%& !" !
!"#$%& !" !"!#$
c
P(A )=1–P(A) P(AorB)=P(A)+P(B)–P(AandB)
P(A|B)=
P(AandB)
P(B)
P(AandB)=P(A|B)*P(B)=P(B|A)*P(A)
c
c
P(B)=P(B|A)*P(A)+P(B|A )*P(A )
MutuallyExclusiveEvents:P(AandB)=0 Independentevents:P(A|B)=P(A)P(B|A)=P(B)P(AandB)=P(A)*P(B)
Combination 𝐢 𝑛, π‘Ÿ =
!!
!! !!! !
𝑃𝑉 ! = 𝑃(π‘‘π‘–π‘ π‘’π‘Žπ‘ π‘’|𝑑𝑒𝑠𝑑 ! ) =
π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘‘π‘Ÿπ‘’π‘’ π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’π‘ 
π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘‘π‘Ÿπ‘’π‘’ π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’π‘  + π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘“π‘Žπ‘™π‘ π‘’ π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’π‘ 
𝑃𝑉 ! = 𝑃 π‘›π‘œ π‘‘π‘–π‘ π‘’π‘Žπ‘ π‘’ 𝑑𝑒𝑠𝑑 ! =
π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘‘π‘Ÿπ‘’π‘’ π‘›π‘’π‘”π‘Žπ‘‘π‘–π‘£π‘’π‘ 
π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘‘π‘Ÿπ‘’π‘’ π‘›π‘’π‘”π‘Žπ‘‘π‘–π‘£π‘’π‘  + π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘“π‘Žπ‘™π‘ π‘’ π‘›π‘’π‘”π‘Žπ‘‘π‘–π‘£π‘’π‘ 
𝑆𝑒𝑛𝑠𝑖𝑑𝑖𝑣𝑖𝑑𝑦 = 𝑃(𝑑𝑒𝑠𝑑 ! |π‘‘π‘–π‘ π‘’π‘Žπ‘ π‘’) =
π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘‘π‘Ÿπ‘’π‘’ π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’π‘ 
π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘‘π‘Ÿπ‘’π‘’ π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’π‘  + π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘“π‘Žπ‘™π‘ π‘’ π‘›π‘’π‘”π‘Žπ‘‘π‘–π‘£π‘’π‘ 
𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑖𝑑𝑦 = 𝑃 𝑑𝑒𝑠𝑑 ! π‘›π‘œ π‘‘π‘–π‘ π‘’π‘Žπ‘ π‘’ =
π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘‘π‘Ÿπ‘’π‘’ π‘›π‘’π‘”π‘Žπ‘‘π‘–π‘£π‘’π‘ 
π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘‘π‘Ÿπ‘’π‘’ π‘›π‘’π‘”π‘Žπ‘‘π‘–π‘£π‘’π‘  + π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘“π‘Žπ‘™π‘ π‘’ π‘π‘œπ‘ π‘–π‘‘π‘–π‘£π‘’π‘ 
1
π‘ƒπ‘Ÿπ‘’π‘£π‘Žπ‘™π‘’π‘›π‘π‘’ =
π‘π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘π‘’π‘œπ‘π‘™π‘’ π‘€π‘–π‘‘β„Ž π‘‘π‘–π‘ π‘’π‘Žπ‘ π‘’ π‘π‘’π‘Ÿπ‘Ÿπ‘’π‘›π‘‘π‘™π‘¦
π‘‡π‘œπ‘‘π‘Žπ‘™ π‘›π‘’π‘šπ‘π‘’π‘Ÿ π‘œπ‘“ π‘π‘’π‘œπ‘π‘™π‘’ 𝑖𝑛 π‘‘β„Žπ‘’ 𝑠𝑑𝑒𝑑𝑦 π‘π‘œπ‘π‘’π‘™π‘Žπ‘‘π‘–π‘œπ‘›
Discreteprobabilitydistribution:µ=ΣxP(x) σ =Σ(x-µ)2P(x)
βŽ›n⎞
n!
Binomialprobability: p( x) = ⎜ ⎟ p x q n − x =
p x (1 − p)n − x
x ! (n − x)!
⎝ x⎠
2
Forbinomialrandomvariable:µ = np and σ = npq
e− µ µ k
Poissonprobability:eisapproximately2.71828
P( X = k ) =
, k = 0,1,2,...
k!
ForPoissonrandomvariable:E(X)=Var(X)=µ
Themeanandstandarderrorofthesamplingdistributionofasamplemeanare:
µx = μ
and
σx=
σ
n
Formulaforaz-scorewithmeanμandsdσ:
X–µ
Z=
σ
The(1–α)*100%confidenceintervalforpopulationmeanisπ‘₯ ± 𝑧!/!
!
!
ConfidenceLevel
zα/2
.90
1.645
.95
1.96
.99
2.575
Whenσisunknown,the(1–α)*100%confidenceintervalforpopulationmeanis
𝑠
π‘₯ ± 𝑑!,!"!(!!!) !
𝑛
Ztest:
One-SidedTest
Two-SidedTest
H0:µ=µ0
H0:µ=µ0
H1:µ<µ0
H1:µ≠µ0
(orH0:µ=µ0
TestStatistic:
H1:µ>µ0)
TestStatistic:
Rejectionregion:
|z|>z (z<-z orz>z )
Rejectionregion:
wherez ischosensothat
z<–z P(Z>z )=α/2
(orz>z whenH1:µ>µ0)
P-value:
p=2P(Z>|z|)=2P(Z<-|z|)
wherez ischosensothat
P(Z>z )=α
P-value:
p=P(Z<z)(orP(Z>z)whenH1:µ>µ0)
α/2
α/2
α
α/2
α
α
α
2
α/2
α/2
Z-testrejectionregion
ttest:
One-TailedTest
H0:µ=µ0
H1:µ<µ0
(orH0:µ=µ0
H1:µ>µ0)
TestStatistic:
Two-SidedTest
H0:µ=µ0
H1:µ≠µ0
TestStatistic:
Rejectionregion:
|t|>t ,df=n-1(t<-t ,df=n-1,ort>t
Wheret ,df=n-1ischosensothat
P(T>t ,df=n-1)=α/2
P-value:
p=2P(T>|t|)=2P(T<-|t|)
Rejectionregion:
t<–t ,df=n-1
(ort>t whenH1:µ>µ0)
Wheret ,df=n-1ischosensothat
P(T>t ,df=n-1)=α
P-value:
p=P(T<t)
(orP(T>t)whenH1:µ>µ0)
Proportiontest:
One-SidedTest
H0:p≥p0
H1:p<p0
(orH0:p≤p0
H1:p>p0)
TestStatistic:
α/2
α/2
α
α/2
α
α
α
Two-SidedTest
H0:p=p0
H1:p≠p0
TestStatistic:
Rejectionregion:
|z|>z (z<-z orz>z )
wherez ischosensothat
P(Z>z )=α/2
P-value:
p=2P(Z>|z|)=2P(Z<-|z|)
Rejectionregion:
z<–z (orz>z whenH1:µ>µ0)
wherez ischosensothat
P(Z>z )=α
P-value:
p=P(Z<z)
(orP(Z>z)whenH1:µ>µ0)
α/2
α
α/2
α
α/2
α
α
3
α/2
α/2
α/2
α/2
,df=n-1)
Chi-squaretest:
H0:ThereisnoassociationbetweenAandB
H1:ThereisassociationbetweenAandB
TestStatistic:
E = Expected =
,where
Rejectionregion:
2
2
χ >χ 1-α(df),df=(r-1)(c-1)
P-value:
2
p=P(χ >teststatistic)
Twosamplesparametrictests:
t-testforindependentsamples:
H0:μ1=μ2vs.H1:μ1≠μ2orμ1>μ2orμ1<μ2
TestStatistic:
Unequalvariancefortwopopulations:
( x1 − x2 ) − ( µ1 − µ2 ) t=
2
2
s1 s2
+
n1 n2
wheredf=approximateddfbySPSSorthesmaller
ofn1-1orn2-1.
Equalvariancefortwopopulations:
(Row Total)(Column Total)
Overall Total
t-testforpairedsamples:
H0:Δ=0vs.H1:Δ≠0orΔ>0orΔ<0,
whereΔ=μAfter-μBefore
TestStatistic:
t=
wheredf=n-1
x1 − x2
1
1
sp
+
n1 n2 where𝑑𝑓 = 𝑛! + 𝑛! − 2
t=
Proportiontestforindependentsamples:
H0:p1=p2vs.H1:p1≠p2orp1>p2orp1>p2
TestStatistic:
( pˆ 1 − pˆ 2 ) − 0
z=
pˆ 1 (1 − pˆ 1 ) pˆ 2 (1 − pˆ 2 )
+
n1
n2
(1 − α ) *100% confidence interval for ( pˆ 1 − pˆ 2 ) is
( pˆ 1 − pˆ 2 ) ± za / 2
pˆ 1 (1 − pˆ 1 ) pˆ 2 (1 − pˆ 2 )
+
n1
n2
4
d x After − xBefore
=
sd
sd n ANOVATable:
ANOVAFTest:
H0:μ1=μ2=μ3=···vs.H1:μ1≠μ2≠μ3≠···
LinearRegression:
y=a+b*x,a=Intercept,b=Slopeoftheregressionline.
RegressionANOVAFTest:
H0:β =0vs.H1:β ≠0
CorrelationTest:
H0:ρ=0vs.H1:ρ≠0
Teststatistic:
½
⎑ n−2 ⎀
t =r⎒
βŽ₯ ≈ t( n − 2 )
⎣1 − r 2 ⎦
RejectionRegion:|t|>t
α/2
,df=n-2(t<-tα/2,df=n-2,ort>tα/2,df=n-2)
RelativeRiskandOddsRatio:
Diseased Healthy
Exposed
a
b
NotExposed
c
d
𝑅𝑅 =
π‘Ž/(π‘Ž + 𝑏)
π‘Ž/𝑏
, 𝑂𝑅 =
𝑐/(𝑐 + 𝑑)
𝑐/𝑑
5
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