Statistics for Management and Economics, Sixth Edition

Formulas

Numerical Descriptive Techniques

Population mean

µ = N

xN

ii∑

= 1

Sample mean

n

xx

n

ii∑

== 1

Range

Largest observation - Smallest observation

Population variance

2σ = N

xN

ii∑

=

−1

2)( µ

Sample variance

2s = 1

)(1

2

−

−∑=

n

xxn

ii

Population standard deviation

σ = 2σ

Sample standard deviation

s = 2s

Population covariance

COV(X,Y) =N

yxN

iyixi∑

=

−−1

))(( µµ

Sample covariance

cov(x,y) =1

))((1

−

−−∑=

n

yyxxn

iii

Population coefficient of correlation

yx

YXCOVσσ

ρ),(

=

Sample coefficient of correlation

yx ssyx

r),cov(

=

Least Squares: Slope coefficient

21

),cov(

xsyx

b =

Least Squares: y-Intercept

xbyb 10 −=

Probability Conditional probability

P(A|B) = P(A and B)/P(B)

Complement rule

P( CA ) = 1 – P(A)

Multiplication rule

P(A and B) = P(A|B)P(B)

Addition rule

P(A or B) = P(A) + P(B) - P(A and B)

Random Variables and Discrete Probability Distributions

Expected value (mean)

E(X) = ∑=µx_all

)x(xp

Variance

V(x) = ∑ µ−=σx_all

22 )x(p)x(

Standard deviation

2σσ =

Covariance

COV(X, Y) =∑ −− ),())(( yxpyx yx µµ

Coefficient of Correlation

yx

YXCOVσσ

ρ),(

=

Laws of expected value

1. E(c) = c

2. E(X + c) = E(X) + c

3. E(cX) = cE(X)

Laws of variance

1.V(c) = 0

2. V(X + c) = V(X)

3. V(cX) = 2c V(X)

Laws of expected value and variance of the sum of two variables

1. E(X + Y) = E(X) + E(Y)

2. V(X + Y) = V(X) + V(Y) + 2COV(X, Y)

Laws of expected value and variance for the sum of more than two independent variables

1. ∑∑==

=k

ii

k

ii XEXE

11

)()(

2. ∑∑==

=k

ii

k

ii XVXV

11

)()(

Mean and variance of a portfolio of two stocks

E(Rp) = w1E(R1) + w2E(R2)

V(Rp) = 21w V(R1) + 2

2w V(R2) + 2 1w 2w COV(R1, R2)

= 21w 2

1σ + 22w 2

2σ + 2 1w 2w ρ 1σ 2σ

Mean and variance of a portfolio of k stocks

E(Rp) = ∑=

k

iii REw

1

)(

V(Rp) = ∑ ∑∑= +==

+k

i

k

ijjiji

k

iii RRCOVwww

1 11

22 ),(2σ

Binomial probability

P(X = x) = )!xn(!x

!n−

xnx )p1(p −−

np=µ

)1(2 pnp −=σ

)1( pnp −=σ

Poisson probability

P(X = x) = !x

e xµµ−

Continuous Probability Distributions Expected value of the sample mean

=)( XE µµ =x

Variance of the sample mean

=)( XVnx

22 σ

σ =

Standard error of the sample mean

n

xσ

σ =

Standardizing the sample mean

n

XZ

/σ

µ−=

Expected value of the sample proportion

=)ˆ(PE pp =ˆµ

Variance of the sample proportion

n

ppPV p

)1()ˆ( 2

ˆ−

== σ

Standard error of the sample proportion

n

ppp

)1(ˆ

−=σ

Standardizing the sample proportion

npp

pPZ

)1(

ˆ

−

−=

Expected value of the difference between two means

=− )( 21 XXE 2121µµµ −=− xx

Variance of the difference between two means

2

22

1

212

21 21)(

nnXXV xx

σσσ +==− −

Standard error of the difference between two means

2

22

1

21

21 nnxxσσ

σ +=−

Standardizing the difference between two sample means

2

22

1

21

2121 )()(

nn

XXZ

σσ

µµ

+

−−−=

Introduction to Estimation Confidence interval estimator of µ

n

zxσ

α 2/±

Sample size to estimate µ

2

2/

=

W

zn

σα

Introduction to Hypothesis Testing

Test statistic for µ

n

xz

/σ

µ−=

Inference about One Population

Test statistic for µ

ns

xt

/

µ−=

Interval estimator of µ

n

stx 2/α±

Test statistic for 2σ

2

22 )1(

σχ

sn −=

Interval Estimator of 2σ

LCL = 2

2/

2)1(

αχ

sn −

UCL = 2

2/1

2)1(

αχ −

− sn

Test statistic for p

npp

ppz

/)1(

ˆ

−

−=

Interval estimator of p

nppzp /)ˆ1(ˆˆ 2/ −± α

Sample size to estimate p

2

2/ )ˆ1(ˆ

−=

W

ppzn α

Inference about Two Populations Equal-variances t-test of 21 µµ −

+

−−−=

21

2

2121

11

)()(

nns

xxt

p

µµ 221 −+= nnν

Equal-variances interval estimator of 21 µµ −

+±−

21

22/21

11)(

nnstxx pα 221 −+= nnν

Unequal-variances t-test of 21 µµ −

+

−−−=

2

22

1

21

2121 )()(

ns

ns

xxt

µµ

−+

−

+=

1)/(

1)/(

)//(

2

22

22

1

21

21

22

221

21

nns

nns

nsnsν

Unequal-variances interval estimator of 21 µµ −

2

22

1

21

2/21 )(ns

ns

txx +±− α

−+

−

+=

1)/(

1)/(

)//(

2

22

22

1

21

21

22

221

21

nns

nns

nsnsν

t-Test of Dµ

DD

DD

ns

xt

/

µ−= 1−= Dnν

t-Estimator of Dµ

D

DD

n

stx 2/α± 1−= Dnν

F-test of 22

21 / σσ

F = 22

21

ss

111 −= nν and 122 −= nν

F-Estimator of 22

21 / σσ

LCL =

21 ,,2/22

21 1

νναFss

UCL = 12,,2/2

2

21

νναFs

s

z-Test and estimator of 21 pp −

Case 1:

+−

−=

21

21

11)ˆ1(ˆ

)ˆˆ(

nnpp

ppz

Case 2:

2

22

1

11

2121

)ˆ1(ˆ)ˆ1(ˆ

)()ˆˆ(

npp

npp

ppppz

−+

−

−−−=

z-Interval estimator of 21 pp −

2

22

1

112/21

)ˆ1(ˆ)ˆ1(ˆ)ˆˆ(

npp

npp

zpp−

+−

±− α

Analysis of Variance One-Way Analysis of variance

SST = ∑=

−k

jjj xxn

1

2)(

SSE = ∑∑==

−jn

ijij

k

j

xx1

2

1

)(

MST = 1−k

SST

MSE = kn

SSE−

F = MSEMST

Two-way analysis of Variance (randomized block design of experiment)

SS(Total) = ∑∑==

−b

iij

k

j

xx1

2

1

)(

SST =∑=

−k

ij xTxb

1

2)][(

SSB =∑=

−b

ii xBxk

1

2)][(

SSE = ∑∑==

+−−b

iijij

k

j

xBxTxx1

2

1

)][][(

MST = 1−k

SST

MSB = 1−b

SSB

MSE = 1+−− bkn

SSE

F = MSEMST

F= MSEMSB

Two-factor experiment

SS(Total) = ∑ ∑ ∑= = =

−a

i

b

j

r

kijk xx

1 1 1

2)(

SS(A) = ∑=

−a

ii xAxrb

1

2)][(

SS(B) = ∑=

−b

jj xBxra

1

2)][(

SS(AB) = ∑ ∑= =

+−−a

i

b

jjiij xBxAxABxr

1 1

2)][][][(

SSE = ∑ ∑ ∑= = =

−a

i

b

j

r

kijijk ABxx

1 1 1

2)][(

F = MSE

AMS )(

F = MSE

BMS )(

F = MSE

ABMS )(

Least Significant Difference Comparison Method

LSD =

+

ji nnMSEt

112/α

Tukey’s multiple comparison method

gn

MSEkq ),( νω α=

Chi-Squared Tests Test statistic for all procedures

∑=

−=

k

i i

ii

e

ef

1

22 )(

χ

Nonparametric Statistical Techniques Wilcoxon rank sum test statistic

1TT =

E(T) = 2

)1( 211 ++ nnn

12

)1( 2121 ++=

nnnnTσ

T

TETz

σ

)(−=

Sign test statistic

x = number of positive differences

n

nxz

5.

5.−=

Wilcoxon signed rank sum test statistic

+= TT

E(T) = 4

)1( +nn

24

)12)(1( ++=

nnnTσ

T

TETz

σ

)(−=

Kruskal-Wallis Test

)1(3)1(

12

1

2

+−

+= ∑

=

nn

T

nnH

k

j j

j

Friedman Test

)1(3)1)((

12

1

2+−

+= ∑

=

kbTkkb

Fk

jjr

Simple Linear Regression Sample slope

21

),cov(

xsyx

b =

Sample y-intercept

xbyb 10 −=

Sum of squares for error

SSE = ∑=

−n

iii yy

1

2)ˆ(

Standard error of estimate

2−

=nSSE

sε

Test statistic for the slope

1

11

bsb

tβ−

=

Standard error of 1b

2)1(1

x

bsn

ss

−= ε

Coefficient of determination

22

22 )],[cov(

yx ssyx

R = ∑ −

−=2)(

1yy

SSE

i

Prediction interval

2

2

2,2/)1(

)(11ˆ

x

gn

sn

xx

nsty

−

−++± − εα

Confidence interval estimator of the expected value of y

2

2

2,2/)1(

)(1ˆ

x

gn

sn

xx

nsty

−

−+± − εα

Sample coefficient of correlation

yxss

yxr

),cov(=

Test statistic for testing ρ = 0

21

2

r

nrt

−

−=

Sample Spearman rank correlation coefficient

ba

S ss

bar

),cov(=

Test statistic for testing Sρ = 0 when n > 30

11/1

0−=

−

−= nr

n

rz S

S

Multiple Regression Standard Error of Estimate

1−−

=kn

SSEsε

Test statistic for iβ

ib

ii

sb

tβ−

=

Coefficient of Determination

22

22 )],[cov(

yx ssyx

R = ∑ −

−=2)(

1yy

SSE

i

Adjusted Coefficient of Determination

Adjusted ∑ −−

−−−=

)1/()()1/(

12

2

nyyknSSE

Ri

Mean Square for Error

MSE = SSE/k

Mean Square for Regression

MSR = SSR/(n-k-1)

F-statistic

F = MSR/MSE

Durbin-Watson statistic

∑

∑

=

=

−−

=n

ii

n

iii

e

ee

d

1

2

2

21 )(

Time Series Analysis and Forecasting Exponential smoothing

1)1( −−+= ttt SwwyS

Statistical Process Control Centerline and control limits for x chart using S

Centerline = x

Lower control limit = n

Sx 3−

Upper control limit = n

Sx 3+

Centerline and control limits for the p chart

Centerline = p

Lower control limit = n

ppp

)1(3

−−

Upper control limit = n

ppp

)1(3

−+

Decision Analysis Expected Value of perfect Information

EVPI = EPPI - EMV*

Expected Value of Sample Information

EVSI = EMV' - EMV*