Post on 06-Jul-2018
8/17/2019 Formula Sheet and Statistical Tables
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Formula Sheet
Mean of a uniform distribution
2
ba += µ
Hypothesis Tests for the Mean, t distribution
n
s
xt
µ −=
−
Standard deviation of a uniform
distribution
( )
12
2ab −
=σ
Hypothesis Tests for Proportions
n
PP
PP z s
)1( −
−=
Uniform probability distribution
ab xP
−=
1)( ,
if b xa ≤≤ and 0 elsewhere
Test statistics for difference between two
large sample means
( )
2
2
2
1
2
1
2121
ns
ns
x x
z
+
−−
−
=
−−
µ µ
The Standardized Normal Distribution
σ
µ −= x
z
The confidence interval for 21 µ µ − ,large
sample
+±
−
−−
2
2
2
1
2
121
n
s
n
s z x x
z-value for Sampling Distribution of the
Sample Mean
n
s
x z
µ −=
−
Pooled variance
2)1()1(
21
2
22
2
112
−+−+−=
nnsnsns p
Confident Interval for z-Distribution
±
−
n z x σ α
2
Two sample test of means-small samples
( )
+
−−
−
=
−−
21
2
2121
11
nns
x x
t
p
µ µ
Confident Interval for t-Distribution
±
−
n
st x
df ,2
α
The confidence interval for 21 µ µ − , small
samples
+±
−
−+
−−
21
2
2,2
21
11
21 nnst x x p
nnα
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Confident Interval for proportion
( )
−±
n
PP zP sss
1
2
α
Expected Opportunity Loss
( )∑=
=
N
i
iij P L j EOL1
)(
Confident Interval for proportion
( 21 PP − )
( )
−
+
−
±−
212
2211
21
11
n
p p
n
p p
zPP
S S S S
S S α
df of two population means with unequal
variances
11 2
2
2
22
1
2
1
12
2
2
22
1
12
−
+−
+
=
n
n
S
n
n
S
n
S
n
S
v
Sample size for estimating population
mean2
=
e
zn
σ
Z test approximation to the Wilcoxon
Signed Rank Test:
24
)12)(1(
4
)1(
++
+−
=
nnn
nnw
z
Sample size for proportion2
)1(
−=
e
zPPn
Z test approximation to the Wilcoxon Rank-
Sum Test:
1
11 1
w
w
S
W z
µ −=
Hypothesis Tests for the Mean,
z distribution
n
x z
σ
µ −=
−
Kruskal-Wallis H test statistics
)1(3)1(
12
1
2
+−
+= ∑
=
nn
R
nn H
c
j j
j
Confident Interval for D µ
±
−
−
n
st D D
n 1,2
α
Spearman Rank Correlation Coefficient
)1(
6
12
1
2
−−=
∑=
nn
d
r
n
i
i
s
Pooled proportion
21
21
nn
X X P
+
+=
−
Pearson Product Moment CorrelationCoefficient
y xn
i
i
n
i
i
n
i
ii
S S
y xCov
yn y xn x
y xn y x
r ),(
1
22
1
22
1=
−−
−
=
∑∑
∑
=
−
=
−
=
−−
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Two sample test of proportions
( )
+
−
−−
−
=−−
21
21
111
21
nnPP
p p p p
zS S
Expected Monetary Value
( )∑=
=
N
i
iijP X j EMV 1
)(
Tukey-Kramer Critical Range
Critical Range =
+
−
ji
k nk nn
MSW Q 112
,,α ,
where ji ≠
Coefficient of Variation
%100)(
)( ×= j EMV
s jCV j
where [ ]∑=
−=
N
i
i X P X s1
2 )()( µ
Chi-square test statistics
( )∑
−=
e
eo
f
f f 2
2 χ
Return-to-Risk Ratio
js
j EMV j RTRR
)()( =
Expected frequency
ncolumnrow f e ∑ ∑×=
Standard error of regression model
1−−=
k n
SSE SE
F-test for the entire regression model in
multiple regression:
MSE
MSRF =
R- square of regression
SST
SSR R =
2, where SSR is sum square of
regression and SST is total sum square.
Testing for the slope in multiple
regression:
−=
∧
∧
i
ii
SE
t
β
β β
Adjusted R-square of regression
−−
−−−=
−
1
1)1(1
22
k n
n R R
Confidence interval for the population
slope, i β
±
∧
−−
∧
ik n
i SE t β β α 1,
2
Paired t test
n
s
Dt
D
D µ −=
−
, where:n
D
D
n
i
i∑=
−
=1
1
1
2
−
−
=
∑=
−
n
D D
s
n
i
D
The Partial F-Test Statistic
( ) ( )
Full
duced Fullduced Full
MSE
k k SSRSSRF ReRe
/ −−=
The F test statistic for population Variances
2
2
2
1
S
S F =
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Statistical Tables
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