formula sheet and statistical tables

8/17/2019 Formula Sheet and Statistical Tables

1/11

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


2/11

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=

−−

−

=

∑∑

∑

=

−

=

−

=

−−


3/11

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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