Download Laws of expected value

Statistics for management and Economics, Seventh Edition
Formulas
Numerical Descriptive techniques
Population mean
N
x
i
i 1
 =
N
Sample mean
n
x
x
i
i 1
n
Range
Largest observation - Smallest observation
Population variance
N
( x   )
2
i
2 =
i 1
N
Sample variance
n
s2 =
 (x
i 1
i
 x)2
n 1
Population standard deviation
 =
2
Sample standard deviation
s=
s2
Population covariance
N
 xy 
( x
  x )( y i   y )
i
i 1
N
Sample covariance
n
( x
i
 x )( y i  y )
i 1
s xy 
n 1
Population coefficient of correlation
 xy

 x y
Sample coefficient of correlation
r
s xy
sx sy
Slope coefficient
b1 
cov( x , y )
s x2
y-intercept
b0  y  b1 x
Probability
Conditional probability
P(A|B) = P(A and B)/P(B)
Complement rule
C
P( A ) = 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)
Bayes’ Law Formula
P(A i | B) 
P( A i ) P( B | A i )
P( A 1 ) P( B | A 1 )  P( A 2 ) P( B | A 2 )  . . .  P( A k ) P( B | A k )
Random Variables and Discrete Probability Distributions
Expected value (mean)
E(X) =  
 xP( x )
all x
Variance
V(x) =  2 
( x   ) P( x )
2
all x
Standard deviation
  2
Covariance
COV(X, Y)
= ( x   x )( y   y )P( x , y )
Coefficient of Correlation

COV ( X , Y )
 x y
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)
2
3. V(cX) = c 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 variables
1. E (
2. V (
k
k
i 1
i 1
k
k
i 1
i 1
 X i )   E( X i )
 X i )  V ( X i ) if the variables are independent
Mean and variance of a portfolio of two stocks
E(Rp) = w1E(R1) + w2E(R2)
V(Rp) = w12 V(R1) + w22 V(R2) + 2 w1 w2 COV(R1, R2)
= w12 12 + w22  22 + 2 w1 w2  1  2
Mean and variance of a portfolio of k stocks
k
E(Rp) =
 w E(R )
i
i
i 1
k
V(Rp) =

k
k
  w w COV (R , R )
wi2  i2  2
i 1
i
j
i 1 j i 1
Binomial probability
P(X = x) =
n!
p x ( 1  p ) n x
x! ( n  x )!
  np
 2  np( 1  p )
  np( 1  p )
Poisson probability
P(X = x) =
e   x
x!
Continuous Probability Distributions
Standard normal random variable
i
j
Z
X 

Exponential distribution
    1/ 
P(X  x)  e x
P(X  x)  1  e x
P(x1  X  x 2 )  P(X  x 2 )  P(X  x1 )  e x1  e x 2
F distribution
F1 A, 1 , 2 =
1
FA, 2 , 1
Sampling Distributions
Expected value of the sample mean
E( X )   x  
Variance of the sample mean
V ( X )   2x 
2
n
Standard error of the sample mean
x 

n
Standardizing the sample mean
Z
X 
/ n
Expected value of the sample proportion
E (Pˆ )   pˆ  p
Variance of the sample proportion
p (1  p )
V ( Pˆ )   2pˆ 
n
Standard error of the sample proportion
 pˆ 
p (1  p )
n
Standardizing the sample proportion
Z
Pˆ  p
p (1  p ) n
Expected value of the difference between two means
E ( X 1  X 2 )   x1 x2  1   2
Variance of the difference between two means
V ( X 1  X 2 )   x21  x2 
 12
n1

 22
n2
Standard error of the difference between two means
 x1  x2 
12  22

n1 n 2
Standardizing the difference between two sample means
Z
( X 1  X 2 )  (1   2 )
12  22

n1 n 2
Introduction to Estimation
Confidence interval estimator of 
x  z / 2

n
Sample size to estimate 

z
n    / 2 
W


2
Introduction to Hypothesis Testing
Test statistic for 
z
x
/ n
Inference about One Population
Test statistic for 
t
x
s/ n
Confidence interval estimator of 
x  t / 2
s
n
Test statistic for  2
2 
( n  1 )s 2
2
Confidence interval Estimator of  2
LCL =
UCL =
( n  1 )s 2
 2 / 2
( n  1 )s 2
 12 / 2
Test statistic for p
z
p̂  p
p( 1  p ) / n
Confidence interval estimator of p
p̂  z / 2 p̂( 1  p̂ ) / n
Sample size to estimate p
z
p̂( 1  p̂ ) 
n /2


W


2
Confidence interval estimator of the total of a large finite population

s 
N  x  t / 2

n

Confidence interval estimator of the total number of successes in a large finite population

p̂( 1  p̂ 
N  p̂  z / 2

n 

Confidence interval estimator of  when the population is small
x  t / 2
s
n
N n
N 1
Confidence interval estimator of the total in a small population

s
N  x  t / 2

n

N  n 
N  1 
Confidence interval estimator of p when the population is small

p  z / 2
p̂( 1  p̂ )
n
N n
N 1
Confidence interval estimator of the total number of successes in a small population

N  p̂  z / 2


p̂( 1  p̂ ) N  n 
n
N  1 
Inference About Two Populations
Equal-variances t-test of 1   2
t
( x1  x 2 )  ( 1   2 )

s 2p 
1
1 


n
n
2 
 1
  n1  n2  2
Equal-variances interval estimator of 1   2
 1
1
( x1  x 2 )  t / 2 s 2p  
 n1 n 2
Unequal-variances t-test of 1   2



  n1  n2  2
t
( x1  x 2 )  (  1   2 )

 s12 s 22 



 n1 n 2 


( s12 / n1  s22 / n2 )2
( s12 / n1 )2 ( s22 / n2 )2

n1  1
n2  1
Unequal-variances interval estimator of 1   2
( x1  x 2 )  t / 2
s12 s 22

n1 n 2

( s12 / n1  s22 / n2 )2
( s12 / n1 )2 ( s22 / n2 )2

n1  1
n2  1
t-Test of  D
t
xD   D
  nD  1
sD / nD
t-Estimator of  D
x D  t / 2
sD
  nD  1
nD
F-test of 12 / 22
F=
s12
 1  n1  1 and  2  n2  1
s22
F-Estimator of 12 / 22
 s2
LCL =  12
s
 2

1

 F / 2 , ,

1 2
 s2
UCL =  12
s
 2

 F / 2 , ,
2 1


z-Test and estimator of p1  p 2
Case 1: z 
Case 2:
z
( p̂1  p̂ 2 )
 1
1 

p̂( 1  p̂ ) 
n
n
2 
 1
( p̂1  p̂2 )  ( p1  p2 )
p̂1( 1  p̂1 ) p̂2 ( 1  p̂2 )

n1
n2
z-estimator of p1  p 2
p̂1( 1  p̂1 ) p̂2 ( 1  p̂2 )

n1
n2
( p̂1  p̂2 )  z / 2
Analysis of Variance
One-way analysis of variance
k
n ( x
SST =
j
j
 x )2
j 1
k
nj
 ( x
SSE =
j 1
i 1
MST =
SST
k 1
MSE =
SSE
nk
F=
 x j )2
ij
MST
MSE
Two-way analysis of variance (randomized block design of experiment)
k
SS(Total) =
b
 ( x
j 1
ij
 x )2
i 1
k
SST =
b( x [ T ]
j
 x )2
i 1
b
SSB =
 k( x [ B ]  x )
2
i
i 1
k
SSE =
b
 ( x
ij
j 1
i 1
MST =
SST
k 1
MSB =
SSB
b 1
 x [ T ] j  x [ B ]i  x )2
MSE =
SSE
n  k  b 1
F=
MST
MSE
F=
MSB
MSE
Two-factor experiment
a
SS(Total) =
b
r
 ( x
ijk
 x )2
i 1 j 1 k 1
a
SS(A) = rb
( x [ A ]  x )
2
i
i 1
b
SS(B) = ra
( x [ B ]
j
 x )2
j 1
a
SS(AB) = r
b
( x [ AB ]
ij
 x [ A ]i  x [ B ] j  x )2
i 1 j 1
a
SSE =
b
r
 ( x
ijk
 x [ AB ]ij )2
i 1 j 1 k 1
F=
MS(A)
MSE
F=
MS(B)
MSE
F=
MS(AB)
MSE
Least Significant Difference Comparison Method
 1
1 
LSD = t / 2 MSE 
 ni n j 


Tukey’s multiple comparison method
  q ( k , )
Chi-Squared Tests
MSE
ng
Test statistic for all procedures
(f i  e i ) 2
ei
k

2 
i 1
Simple Linear Regression
Sample slope
b1 
s xy
s x2
Sample y-intercept
b0  y  b1 x
Sum of squares for error
n
SSE =
( y
i
 ŷ i ) 2
i 1
Standard error of estimate
s 
SSE
n2
Test statistic for the slope
t
b1  1
s b1
Standard error of
s b1 
b1
s
( n  1 )s x2
Coefficient of determination
R2 
2
s xy
s x2 s 2y
 1
SSE
( y
i
 y )2
Prediction interval
ŷ  t / 2 ,n 2 s 1 
2
1 ( xg  x )

n ( n  1 )s x2
Confidence interval estimator of the expected value of y
ŷ  t / 2 ,n 2 s
2
1 ( xg  x )

n ( n  1 )s x2
Sample coefficient of correlation
r
s xy
sx sy
Test statistic for testing  = 0
tr
n2
1 r 2
Multiple Regression
Standard Error of Estimate
s 
SSE
n  k 1
Test statistic for  i
t
bi   i
s bi
Coefficient of Determination
R2 
2
s xy
s x2 s 2y
 1

SSE
( yi  y )2
Adjusted Coefficient of Determination
Adjusted R 2  1 
SSE /( n  k  1 )
( y  y )
2
i
Mean Square for Error
MSE = SSE/k
Mean Square for Regression
MSR = SSR/(n-k-1)
/( n  1 )
F-statistic
F = MSR/MSE
Durbin-Watson statistic
n
d
( e
i
 ei 1 ) 2
i2
n
e
2
i
i 1
Time Series Analysis and Forecasting
Exponential smoothing
S t  wyt  (1  w)S t 1
Nonparametric Statistical techniques
Wilcoxon rank sum test statistic
T  T1
E(T) =
n1 ( n1  n 2  1 )
2
T 
n1 n 2 ( n1  n 2  1 )
12
z
T  E( T )
T
Sign test statistic
x = number of positive differences
z
x  .5n
.5 n
Wilcoxon signed rank sum test statistic
T T
E(T) =
n( n  1)
4
T 
z
n( n  1 )( 2n  1 )
24
T  E( T )
T
Kruskal-Wallis Test
 12
H 
 n( n  1 )

T j2 
  3( n  1 )
n
j 1 j 

k

Friedman Test

12
Fr  
 b( k )( k  1 )


T j2   3b( k  1 )

j 1

k

Spearman rank correlation coefficient
rS 
s ab
s a sb
Spearman Test statistic for n > 30
z  rS n  1
Statistical Process Control
Centerline and control limits for x chart using S
Centerline = x
Lower control limit = x  3
S
n
Upper control limit = x  3
S
n
Centerline and control limits for the p chart
Centerline = p
Lower control limit = p  3
p( 1  p )
n
Upper control limit = p  3
p( 1  p )
n
Decision Analysis
Expected Value of perfect Information
EVPI = EPPI - EMV*
Expected Value of Sample Information
EVSI = EMV' - EMV*