KEY FORMULAS Jaggia and Kelly
� Business Statistics Communicating with Numbers
Chapter 3: Numerical Descriptive Measures
Section
Topic
Formula
3.1
Sample Mean
x =
3.2
Percentile Location
3.1
3.3
3.3
3.4
3.4
3.4
3.4
3.4
3.4
3.4
3.4
3.5
3.7
3.7
3.7
3.7
3.7
Population Mean
∑ xi
n
∑ xi
µ=
N
( n + 1)
L=
p
p
100
(1 + R1 )(1 + R2 ) ⋅⋅⋅ (1 + Rn ) − 1
Geometric Mean Return
G=
R
n
G=
g
n
Range
Range = Maximum Value – Minimum Value
Average Growth Rate
Sample MAD
Population MAD
Sample Variance
Sample Standard Deviation
Population Variance
Gg
(1 + g1 )(1 + g 2 ) ⋅⋅⋅ (1 + g n ) − 1 or=
∑ ( xi − x )
s2 =
σ =
2
∑ ( xi − µ )
s2 =
Sample CV =
x=
2
σ
or=
2
N
Sample Variance for Grouped Data
The Weighted Mean
∑x
2
i
−
n −1
nx 2
n −1
s = s2
Sharpe Ratio =
Population Variance for Grouped
Data
N
=
s2
or
n −1
Sharpe Ratio
Population Mean for Grouped Data
∑ xi − µ
2
σ = σ2
Sample Mean for Grouped Data
n
Population MAD =
Population Standard Deviation
Coefficient of Variation
∑ xi − x
Sample MAD =
∑x
sI
∑ ( mi − x ) fi
=
or s 2
n −1
∑ mi fi
N
∑ ( mi − m ) fi
or σ 2
=
σ =
N
2
x=
∑w x
i i
N
xI − R f
∑ mi fi
n
2
− µ2
s
σ
; Population CV =
x
µ
2
m=
2
i
∑m
2
i fi
n −1
∑m
−
2
i fi
N
nx 2
n −1
− m2
n −1
xn
−1
x1
KEY FORMULAS Jaggia and Kelly
3.8
3.8
3.8
3.8
� Business Statistics Communicating with Numbers
sxy =
Sample Covariance
Population Covariance
σ xy =
Sample Correlation Coefficient
Population Correlation Coefficient
rxy =
ρ xy =
∑ ( xi − x )( yi − y )
n −1
(
∑ ( xi − µ x ) yi − µ y
)
N
sxy
sx s y
σ xy
σ xσ y
Chapter 4: Introduction to Probability
Section
Topic
Formula
4.2
Complement Rule
P Ac = 1 − P ( A )
4.2
Addition Rule for Mutually
Exclusive Events
4.2
4.2
4.2
4.2
4.4
( )
Addition Rule
P ( A ∪ B=
) P ( A) + P ( B ) − P ( A ∩ B )
Conditional Probability
P ( A | B) =
Multiplication Rule
Multiplicative Rule for
Independence Event
Total Probability Rule
P ( A ∪ B=
) P ( A) + P ( B )
P ( A ∩ B)
P ( B)
P ( A ∩ B) =
P ( A | B) P ( B)
P ( A ∩ B) =
P ( A) P ( B )
(
P ( A )= P ( A ∩ B ) + P A ∩ B c
(
)
) ( )
or P ( A ) P ( A | B ) P ( B ) + P A | B c P B c
=
P ( B | A) =
P ( A ∩ B)
(
P ( A ∩ B ) + P A ∩ Bc
)
4.4
Bayes’ Theorem
4.4
P (=
A ) P ( A ∩ B1 ) + P ( A ∩ B2 ) + ⋅⋅⋅ + P ( A ∩ Bn )
Extensions of the Total
Probability Rule
=
P ( A ) P ( A | B1 ) P ( B1 ) + P ( A | B2 ) P ( B2 ) + ⋅⋅⋅ + P ( A | Bn ) P ( Bn )
4.4
Extensions of Bayes’
Theorem
4.5
Factorial Formula
or P ( B | A ) =
P ( Bi | A ) =
P ( Bi | A ) =
P ( A | B) P ( B)
(
) ( )
P ( A | B ) P ( B ) + P A | Bc P Bc
P ( A ∩ Bi )
P ( A ∩ B1 ) + P ( A ∩ B2 ) + ⋅⋅⋅ + P ( A ∩ Bn )
P ( A | Bi ) P ( Bi )
P ( A | B1 ) P ( B1 ) + P ( A | B2 ) P ( B2 ) + ⋅⋅⋅ + P ( A | Bn ) P ( Bn )
n ! =n × ( n − 1) × ( n − 2 ) × ( n − 3) ×⋅⋅⋅× 1
KEY FORMULAS Jaggia and Kelly
4.5
4.5
� Business Statistics Communicating with Numbers
Combination Formula
Permutation Formula
Cx
n=
n Px
=
n
=

 x
n!
( n − x )! x !
n!
( n − x )!
Chapter 5: Discrete Probability Distributions
Section
Topic
Formula
5.2
Expected Value of a Discrete
Random Variable
E(X ) =
µ=
∑ xi P ( X =
xi )
5.2
Standard Deviation of a Discrete
Random Variable
5.2
5.3
Variance of a Discrete Random
Variable
Portfolio Expected Return
Var ( X ) =
σ2 =
∑ ( xi − µ ) P ( X =
xi ) =
∑ xi2 P ( X =
xi ) − µ 2
2
SD ( X =
) σ=
σ2
( )
=
E Rp
wA E ( RA ) + wB E ( RB )
( )
Var ( R p ) = wA2 σ A2 + wB2 σ B2 + 2 wA wB r ABσ Aσ B
Var R p = wA2 σ A2 + wB2 σ B2 + 2 wA wBσ AB or
5.3
Portfolio Variance
5.4
Binomial Distribution
5.4
Variance of a Binomial Random
Variable
2
Var ( X=
npq
) σ=
Poisson Distribution
P ( X= x=
)
Variance of a Poisson Random
Variable
2
Var ( X=
µ
) σ=
5.4
5.4
5.5
5.5
5.5
5.5
Expected Value of a Binomial
Random Variable
𝑛
𝑛!
𝑃(𝑋 = 𝑥) = � � 𝑝 𝑥 𝑞 𝑛−𝑥 =
𝑝 𝑥 (1 − 𝑝)𝑛−𝑥
𝑥!(𝑛−𝑥)!
𝑥
E(X=
) µ= np
Standard Deviation of a Binomial
Random Variable
SD ( X =
) σ=
Expected Value of a Poisson
Random Variable
E(X ) = µ
Standard Deviation of a Poisson
Random Variable
5.6
Hypergeometric Distribution
5.6
Expected Value of a
Hypergeometric Random Variable
SD ( X =
) σ=
npq
e− µ µ x
x!
µ
 S  N − S 
 

x  n − x 

P ( X= x=
)
N
 
n
S
E(X=
) µ= n  
N
KEY FORMULAS Jaggia and Kelly
5.6
5.6
� Business Statistics Communicating with Numbers
S  N − n 
 S 
2
Var ( X=
n   1 −  
) σ=
N   N − 1 
 N 
Variance of a Hypergeometric
Random Variable
Standard Deviation of a
Hypergeometric Random Variable
S  N − n 
 S 
n   1 −  

N
N
 
  N −1 
SD ( X =
) σ=
Chapter 6: Continuous Probability Distributions
Section
6.1
Topic
Expected Value of a Uniform
Distribution
6.2
Normal Distribution
Standard Deviation of a Uniform
Distribution
6.3
Standard Transformation of the
Normal Random Variable
6.4
Exponential Distribution
6.3
6.4
6.4
6.4
6.4
 1
for a ≤ x ≤ b, and

f ( x) = b − a
0
for x < a or x > b

Continuous Uniform Distribution
6.1
6.1
Formula
SD ( X =
) σ=
Mean of an Exponential
Distribution
E(X ) =
Standard Deviation of an
Exponential Distribution
1
λ
SD=
(X )
( X ) E=
1
λ
P ( X ≤ x ) =−
1 e−λ x
 ln ( y ) − µ 2
(
)
exp  −
2

yσ 2p
2σ


1
f ( y)
=
Mean of the Lognormal Distribution
6.4
Mean of the Normal Variable X =
ln(y)
Standard Deviation of the
Lognormal Distribution




f ( x ) = λ e−λ x
6.4
6.4
12
 ( x − µ )2
exp  −

2σ 2
σ 2p

x−µ
z=
σ
x= µ + Z σ
Lognormal Distribution
( b − a )2
1
f ( x)
=
Inverse Transformation of the
Normal Random Variable
Cumulative Exponential
Distribution Function
a+b
2
E(X=
) µ=
 2µ + σ 2 

2


µY = exp 
σY =
( exp (σ ) − 1) exp ( 2µ + σ )
2




2
2 
+
µ
σ
γ
γ 


µ = ln 
µγ2
2




KEY FORMULAS Jaggia and Kelly
6.4
� Business Statistics Communicating with Numbers
Standard Deviation of the Normal
Variable X = ln(y)
 σ γ2
ln 1 + 2
 µγ

=
σ




Chapter 7: Sampling and Sampling Distributions
Section
Topic
Formula
7.2
Expected Value of the Sample Mean
E X =µ
7.2
Standard Transformation of the
Sample Mean
7.2
7.3
7.3
7.3
7.4
7.4
7.5
7.5
7.5
7.5
( )
σ
n
( )
Standard Deviation of the Sample
Mean
SD X =
Expected Value of the Sample
Proportion
E P =p
Standard Deviation of the Sample
Proportion
Standard Transformation of the
Sample Proportion
z=
x −µ
σ
n
( )
p (1 − p )
( )
SD P =
n
p− p
z=
p (1 − p )
n
( )
Finite Population Correction Factor
for the Sample Mean
SD X =
Upper Control Limit for the x chart
µ +3
Finite Population Correction Factor
for the Sample Proportion
Lower Control Limit for the x chart
Upper Control Limit for the p
chart
Lower Control Limit for the p
chart
σ  N −n 

n 
p (1 − p )  N − n 


n
 N −1 
( )
SD P =
µ −3
p+3
p −3

N − 1 
σ
n
σ
n
p (1 − p )
n
p (1 − p )
n
KEY FORMULAS Jaggia and Kelly
� Business Statistics Communicating with Numbers
Chapter 8: Estimation
Section
8.2
8.3
8.4
8.5
8.5
Topic
Confidence Interval for µ when
σ is Known
Confidence Interval for µ when
σ is Unknown
Confidence Interval for p
Required Sample Size when
Estimating the Population Mean
Formula
σ
x ± zα
2
x ± tα
2,df
p ± zα
2
n
s
p (1 − p )
n
 zα 2σˆ
n = 
 D
Required Sample Size when
Estimating the Population =
n
Proportion
; df = n − 1
n
 zα 2

 D



2
2

 pˆ (1 − pˆ )

Chapter 9: Hypothesis Testing
Section
Topic
9.2
Test Statistic for µ when σ is
Known
9.3
9.4
Test Statistic for µ when σ is
Unknown
Test Statistic for p
Formula
z=
x − µ0
σ
tdf =
z=
n
x − µ0
s
n
; df = n − 1
p − p0
p0 (1 − p0 ) n
Chapter 10: Statistical Inference Concerning Two Populations
Section
10.1
10.1
Topic
Formula
Confidence Interval for µ1 − µ2 if σ 12
( x1 − x2 ) ± zα 2
and σ 22 are known
Confidence Interval for µ1 − µ2 if σ 12
and σ 22 are unknown but assumed
equal
( x1 − x2 ) ± tα 2,df
s 2p =
σ12
n1
+
σ 22
n2
1 1 
s 2p  +  ;
 n1 n2 
(n1 − 1) s12 + (n2 − 1) s22
; df = n1 + n2 − 2
n1 + n2 − 2
KEY FORMULAS Jaggia and Kelly
10.1
10.1
10.1
� Business Statistics Communicating with Numbers
Confidence Interval
for µ1 − µ2 if σ 12
and σ 22 are unknown and cannot be
assumed equal
Test Statistic for µ1 − µ2 if σ 12 and
σ 22 are known
Test Statistic for µ1 − µ2 if σ 12 and
σ 22 are unknown but assumed
equal
df =
z=
10.1
10.2
10.2
Test Statistic
σ 22
(s
2
1
Confidence Interval for µ D
Test Statistic for µ D
10.3
Confidence Interval for p1 − p2
10.3
Test Statistic for Testing p1 − p2 if d 0
is zero
σ12
n1
s 2p
s 2p =
10.3
Test Statistic for Testing p1 − p2 if d 0
is not zero
df =
σ 22
n2
1 1 
 + 
 n1 n2 
( x1 − x2 ) − d0
 s12 s22
 +
 n1 n2



(s
2
1
;
/ n1 + s22 / n2 )
2
( s12 / n1 ) 2 /(n1 − 1) + ( s22 / n2 ) 2 /(n2 − 1)
2,df
d − d0
sD
n ; df = n − 1
sD
n
; df = n − 1
( p1 − p2 ) ± zα 2
z=
;
(n1 − 1) s12 + (n2 − 1) s22
; df = n1 + n2 − 2
n1 + n2 − 2
d ± tα
tdf =
+
( x1 − x2 ) − d0
tdf =
z=
2
( s12 / n1 ) 2 /(n1 − 1) + ( s22 / n2 ) 2 /(n2 − 1)
and
are unknown and cannot be
assumed equal
/ n1 + s22 / n2 )
( x1 − x2 ) − d0
tdf =
for µ1 − µ2 if σ 12
s12 s22
+
;
n1 n2
( x1 − x2 ) ± tα 2,df
p1 (1 − p1 )
n1
+
p2 (1 − p2 )
n2
x1 + x2 n1 p1 + n2 p2
=
; p =
n
n1 + n2
1 + n2
1
1 
p (1 − p )  + 
 n1 n2 
p1 − p2
( p1 − p2 ) − d0
p1 (1 − p1 ) p2 (1 − p2 )
+
n1
n2
Chapter 11: Statistical Inference Concerning Variance
Section
Topic
11.1
Confidence Interval for σ
Formula
2
 ( n − 1) s 2 ( n − 1) s 2 
 2
 where df = n − 1
,
 χα /2, df χ12−α /2, df 
KEY FORMULAS Jaggia and Kelly
11.1
� Business Statistics Communicating with Numbers
( n − 1) s 2
χ df2 =
Test Statistic for σ 2
for σ 12
11.2
Confidence Interval
11.2
Test Statistic for σ 12 / σ 22
/ σ 22
s 02
𝑠2
where df = n − 1
𝑠12
2 𝐹𝛼/2,(𝑑𝑓2 ,𝑑𝑓1 )
2 𝐹𝛼/2,(𝑑𝑓1 ,𝑑𝑓2 ) 𝑠2
1
�𝑠12
,
df=
n2 − 1
2
n1 − 1 and
�, df=
1
n1 − 1 and df=
n2 − 1
F( df , df ) = s12 / s 22 where df=
1
2
1
2
Chapter 12: Chi-Square Tests
Section
Topic
Formula
12.1
Goodness-of-Fit Test Statistic
χ df2 =
12.2
12.2
12.3
12.3
Expected Frequencies for a Test of
Independence
Test Statistic for a Test of
Independence
Goodness-of-Fit Test Statistic for
Normality
Jarque-Bera Test Statistic for
Normality
eij =
χ df2
∑
( oi − ei )2
ei
( Row i total )( Column
Sample Size
=
∑∑
i
χ df2 =
( oij − eij )
∑
Topic
( oi − ei )2
ei
( n / 6 )  S 2 + K 2 / 4
2
JB
= χ=
2
Formula
ni
c
13.1
13.1
13.1
13.1
13.1
Grand Mean
∑∑ x
ij
x=
=i 1 =j 1
nT
Sum of Squares due to Treatments
SSTR
Mean Square for Treatments MSTR
Error Sum of Squares SSE
Mean Square Error MSE
MSTR =
SSTR
c −1
c
=
SSE
∑ ( n − 1)s
i
i =1
MSE =
2
eij
j
Chapter 13: Analysis of Variance
Section
j total )
SSE
nT − c
2
i
KEY FORMULAS Jaggia and Kelly
13.1
13.2
13.2
13.2
� Business Statistics Communicating with Numbers
MSTR
; df1= c − 1 and df=
nT − c
F( df , df ) =
2
1
2
MSE
Test Statistic for One-way ANOVA
1 1 
MSE  + 
 ni n j 


Fisher’s Confidence Interval for
µi − µ j
( xi − x j ) ± tα /2,n −c
Tukey’s Confidence Interval for
µi − µ j for balanced data
( xi − x j ) ± qα ,(c,n −c )
MSE
n
Tukey’s Confidence Interval for
µi − µ j for unbalanced data
( xi − x j ) ± qα ,(c,n −c )
MSE  1 1 
 + 
2  ni n j 
( n=
n=
nj
i
T
)
T
( ni ≠ n j )
T
Column Means: F( df , df ) =
1
13.3
2
MSA
; df1= c − 1 and
MSE
df 2 = nT − c − r + 1
Test Statistics for Two-way ANOVA
with No Interaction
Row Means: F( df , df ) =
MSB
; df1= r − 1 and
MSE
Interaction: F( df , df ) =
MSAB
; df1 =(r − 1)(c − 1) and
MSE
1
2
df 2 = nT − c − r + 1
1
2
=
df 2 rc( w − 1)
13.4
Column Means: F( df , df ) =
Test Statistics for Two-way ANOVA
with Interaction
1
Row Means: F( df , df ) =
2
=
df 2 rc( w − 1)
Chapter 14: Regression Analysis
Section
Topic
Formula
14.1
Sample Covariance
s xy =
Sample Correlation Coefficient
rxy =
∑ ( x − x )( y − y )
i
i
n −1
s xy
sx s y
rxy rxy n − 2
=
sr
1 − rxy2
14.1
Test Statistic for ρ xy
t=
df
14.2
Simple Linear Regression Model
y =β 0 + β1 x + ε
14.2
Sample Regression Equation for
the Simple Linear Regression
Model
MSA
; df1= c − 1 and
MSE
=
df 2 rc( w − 1)
1
14.1
2
=
ŷ b0 + b1 x
MSB
; df1= r − 1 and
MSE
KEY FORMULAS Jaggia and Kelly
14.2
14.2
14.2
14.3
14.3
14.4
14.4
14.4
14.4
14.4
14.4
� Business Statistics Communicating with Numbers
Slope Estimate b1 for the Simple
Linear Regression Model
Intercept Estimate b0 for the
Simple Linear Regression Model
b1 =
∑ ( x − x )( y − y )
∑( x − x )
i
i
2
i
b0= y − b1 x
sy
s
Relationship between b1 and
rxy
=
and rxy b1 x
b1 r=
xy
sx
sy
Multiple Linear Regression Model
Sample Regression Equation for
the Multiple Linear Regression
Model
=
y β 0 + β1 x1 + β 2 x2 + ⋅⋅⋅ + β k xk + ε
ˆ b0 + b1 x1 + b2 x2 + ⋅⋅⋅ + bk xk
y=
SSE
=
n − k −1
Standard Error of the Estimate=
=
s
s2 =
MSE
e
Total Sum of Squares
Sum of Squares due to Regression
Sum of Squares due to Error
Coefficient of Determinant R 2
Adjusted R 2
e
SST
=
∑(y
=
SSR
∑ ( yˆ
SSE
=
∑(y
R2 =
∑
ei2
=
n − k −1
∑ ( y − yˆ )
i
n − k −1
− y )2
i
i
i
− y )2
− yˆ i ) 2
SSR
SSE
= 1−
SST
SST
(
)
 n −1 
Adjusted R 2 =1 − 1 − R 2 

 n − k −1 
Chapter 15: Inference with Regression Models
Section
Topic
Formula
15.1
Test Statistics for the Test of
Individual Significance
tdf =
b j − b j0
sb j
; df = n − k − 1
Confidence Interval for β j
b j ± tα
15.1
SSR / k
MSR
;
F( df , df ) =
Test Statistic for the Test of Joint=
1
2
SSE / ( n − k − 1) MSE
Significance
df1 = k and df 2 = n − k − 1
15.2
Test Statistic for the Test of Linear
Restrictions
F( df , df ) =
1
2
15.1
15.3
Confidence Interval for the
Expected Value of y
2, df sb j
; df = n − k − 1
( SSER − SSEU ) / df1 ;
SSEU / df 2
df1 = number of linear restrictions and df 2 = n − k − 1
yˆ 0 ± tα
2, df
( )
se yˆ 0 ; df = n − k − 1
i
2
KEY FORMULAS Jaggia and Kelly
15.3
15.4
� Business Statistics Communicating with Numbers
Prediction Interval for an Individual
Value of y
Residuals for the Regression Model
yˆ 0 ± tα
( se ( yˆ ))
0
2, df
2
+ se2 ; df = n − k − 1
e=
yi − yˆi
i
Chapter 16: Regression Models for Nonlinear Relationships
Section
Topic
Formula
16.1
Quadratic Regression Model
y =β 0 + β1 x + β 2 x 2 + ε
16.1
Cubic Regression Model
y =β 0 + β1 x + β 2 x 2 + β3 x3 + ε
Log-log Regression Model
ln ( y ) =
β 0 + β1 ln ( x ) + ε
Logarithmic Model
y=
β 0 + β1 ln ( x ) + ε
16.1
16.1
16.2
16.2
16.2
16.2
16.2
16.2
16.2
Sample Regression Equation for the
Quadratic Regression Model
Sample Regression Equation for the
Cubic Regression Model
ŷ =b0 + b1 x + b2 x 2
ŷ =b0 + b1 x + b2 x 2 + b3 x3
(
Predictions with the Log-log
Regression Model
yˆ = exp b0 + b1 ln ( x ) + se2 / 2
Predictions with the Logarithmic
Model
yˆ b0 + b1 ln ( x )
=
Exponential Model
Predictions with the Exponential
Model
Coefficient of determination R 2
ln ( y ) =β 0 + β1 x + ε
(
=
yˆ exp b0 + b1 x + se2 / 2
( )
R 2 = ryyˆ
2
Chapter 17: Regressions Models with Dummy Variables
Section
Topic
17.3
Logit Model
Formula
exp ( b0 + b1 x )
Pˆ =
1 + exp ( b0 + b1 x )
)
)
KEY FORMULAS Jaggia and Kelly
� Business Statistics Communicating with Numbers
Chapter 18: Time Series and Forecasting
Section
Topic
Formula
18.1
Residuals
e=
yt − yˆt
t
18.1
Mean Square Error (MSE)
( y − yˆ ) ∑ e
∑
=
2
=
MSE
t
∑
2
t
t
n
n
∑
18.2
| yt − yˆt |
| et |
Mean Absolute Deviation (MAD)=
MAD =
n
n
Sum of the m most recent observations
Moving Average =
m-period Moving Average
m
18.3
Linear Trend Model
yt = β 0 + β1t + ε t
Exponential Trend Model
ln ( yt ) = β 0 + β1t + ε t
18.1
18.2
18.3
18.3
18.3
18.3
18.3
18.4
18.4
Exponential Smoothing
At = α yt + (1 − α ) At −1
Predictions with the Linear Trend
Model
yˆ=
b0 + b1t
t
Predictions with the Exponential
Trend Model
=
yˆt exp(b0 + b1t + se2 / 2)
Polynomial Trend Model
β 0 + β1t + β 2t 2 + β3t 3 + ⋅⋅⋅ + β q t q + ε t
y=
t
Linear Trend Model with Seasonal
Dummy Variables
y = β 0 + β1d1 + β 2 d 2 + β3 d3 + β 4 t + ε
Predictions with the Polynomial
Trend Model
yˆ=
b0 + b1t + b2 t 2 + b3t 3 + ⋅⋅⋅ + bq t q
t
Exponential Trend Model with
Seasonal Dummy Variables
ln ( y ) = β 0 + β1d1 + β 2 d 2 + β3 d3 + β 4 t + ε
Chapter 19: Returns, Index Numbers, and Inflation
Section
Topic
Formula
19.1
Investment Return
Rt =
19.1
19.1
Adjusted Close Prices
Fisher Equation
Rt =
Pt − Pt −1 + I t
Pt −1
Pt* − Pt*−1
Pt*−1
1+ R
1+ r =
1+ i
KEY FORMULAS Jaggia and Kelly
19.2
19.2
19.2
19.2
19.3
19.3
� Business Statistics Communicating with Numbers
Simple Price Index
Unweighted Aggregate Price Index
in Period t
Laspeyres Price Index
Paasche Price Index
Real Value
Inflation Rate
pt
× 100
p0
∑ P ×100
∑P
∑ p q ×100
∑p q
∑ p q ×100
∑p q
it
i0
it i 0
i0 i0
it in
i 0 in
𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑉𝑉𝑉𝑉𝑉
× 100
𝑃𝑃𝑃𝑃𝑃 𝐼𝐼𝐼𝐼𝐼
CPI t − CPI t −1
it =
CPI t −1
𝑅𝑅𝑅𝑅 𝑉𝑉𝑉𝑉𝑉 =
Chapter 20: Nonparametric Tests
Section
Topic
Formula
z=
20.1
20.2
20.3
20.4
20.4
20.5
20.6
T − µT
; where
σT
Wilcoxon Signed-Rank Test Statistic
for n ≥ 10
n(n + 1)
n(n + 1)(2n + 1)
=
µT =
and σ T
4
24
W − µW
(n1 + n2 + 1) × min(n1 , n2 )
; where mW =
and
z=
σW
2
Wilcoxon Rank-Sum Test Statistic
for n1 ≥ 10 and n2 ≥ 10
n n (n + n + 1)
σW = 1 2 1 2
12
 12
H 
Kruskal-Wallis Test Statistic =
 n ( n + 1)

Spearman Rank Correlation
Coefficient
Spearman Rank Correlation
Coefficient Test Statistic for n ≥ 10
rS =
∑
i =1
Ri2
ni
∑d
1−
n ( n − 1)
6

 − 3 ( n + 1)


2
i
2
=
z rs n − 1
Test Statistic for the Sign Test for
n ≥ 10
z=
Test Statistic for the WaldWolfowitz Runs Test for n1 ≥ 10 and
z=
n2 ≥ 10
k
p − 0.5
0.5 / n
R − µR
σR =
σR
µR
; where=
2n1n2 (2n1n2 − n)
n 2 (n − 1)
2n1n2
+ 1 and
n
Final PDF to printer
APPENDIX B
Tables
TABLE 1 Standard Normal Curve Areas
Entries in this table provide cumulative probabilities, that is, the area
under the curve to the left of −z. For example, P(Z ≤ −1.52) = 0.0643.
P (Z ≤ –z)
–z
0
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
−3.9
−3.8
−3.7
−3.6
−3.5
−3.4
−3.3
−3.2
−3.1
−3.0
0.0000
0.0001
0.0001
0.0002
0.0002
0.0003
0.0005
0.0007
0.0010
0.0013
0.0000
0.0001
0.0001
0.0002
0.0002
0.0003
0.0005
0.0007
0.0009
0.0013
0.0000
0.0001
0.0001
0.0001
0.0002
0.0003
0.0005
0.0006
0.0009
0.0013
0.0000
0.0001
0.0001
0.0001
0.0002
0.0003
0.0004
0.0006
0.0009
0.0012
0.0000
0.0001
0.0001
0.0001
0.0002
0.0003
0.0004
0.0006
0.0008
0.0012
0.0000
0.0001
0.0001
0.0001
0.0002
0.0003
0.0004
0.0006
0.0008
0.0011
0.0000
0.0001
0.0001
0.0001
0.0002
0.0003
0.0004
0.0006
0.0008
0.0011
0.0000
0.0001
0.0001
0.0001
0.0002
0.0003
0.0004
0.0005
0.0008
0.0011
0.0000
0.0001
0.0001
0.0001
0.0002
0.0003
0.0004
0.0005
0.0007
0.0010
0.0000
0.0001
0.0001
0.0001
0.0002
0.0002
0.0003
0.0005
0.0007
0.0010
−2.9
−2.8
−2.7
−2.6
−2.5
−2.4
−2.3
−2.2
−2.1
−2.0
0.0019
0.0026
0.0035
0.0047
0.0062
0.0082
0.0107
0.0139
0.0179
0.0228
0.0018
0.0025
0.0034
0.0045
0.0060
0.0080
0.0104
0.0136
0.0174
0.0222
0.0018
0.0024
0.0033
0.0044
0.0059
0.0078
0.0102
0.0132
0.0170
0.0217
0.0017
0.0023
0.0032
0.0043
0.0057
0.0075
0.0099
0.0129
0.0166
0.0212
0.0016
0.0023
0.0031
0.0041
0.0055
0.0073
0.0096
0.0125
0.0162
0.0207
0.0016
0.0022
0.0030
0.0040
0.0054
0.0071
0.0094
0.0122
0.0158
0.0202
0.0015
0.0021
0.0029
0.0039
0.0052
0.0069
0.0091
0.0119
0.0154
0.0197
0.0015
0.0021
0.0028
0.0038
0.0051
0.0068
0.0089
0.0116
0.0150
0.0192
0.0014
0.0020
0.0027
0.0037
0.0049
0.0066
0.0087
0.0113
0.0146
0.0188
0.0014
0.0019
0.0026
0.0036
0.0048
0.0064
0.0084
0.0110
0.0143
0.0183
−1.9
−1.8
−1.7
−1.6
−1.5
−1.4
−1.3
−1.2
−1.1
−1.0
0.0287
0.0359
0.0446
0.0548
0.0668
0.0808
0.0968
0.1151
0.1357
0.1587
0.0281
0.0351
0.0436
0.0537
0.0655
0.0793
0.0951
0.1131
0.1335
0.1562
0.0274
0.0344
0.0427
0.0526
0.0643
0.0778
0.0934
0.1112
0.1314
0.1539
0.0268
0.0336
0.0418
0.0516
0.0630
0.0764
0.0918
0.1093
0.1292
0.1515
0.0262
0.0329
0.0409
0.0505
0.0618
0.0749
0.0901
0.1075
0.1271
0.1492
0.0256
0.0322
0.0401
0.0495
0.0606
0.0735
0.0885
0.1056
0.1251
0.1469
0.0250
0.0314
0.0392
0.0485
0.0594
0.0721
0.0869
0.1038
0.1230
0.1446
0.0244
0.0307
0.0384
0.0475
0.0582
0.0708
0.0853
0.1020
0.1210
0.1423
0.0239
0.0301
0.0375
0.0465
0.0571
0.0694
0.0838
0.1003
0.1190
0.1401
0.0233
0.0294
0.0367
0.0455
0.0559
0.0681
0.0823
0.0985
0.1170
0.1379
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
−0.0
0.1841
0.2119
0.2420
0.2743
0.3085
0.3446
0.3821
0.4207
0.4602
0.5000
0.1814
0.2090
0.2389
0.2709
0.3050
0.3409
0.3783
0.4168
0.4562
0.4960
0.1788
0.2061
0.2358
0.2676
0.3015
0.3372
0.3745
0.4129
0.4522
0.4920
0.1762
0.2033
0.2327
0.2643
0.2981
0.3336
0.3707
0.4090
0.4483
0.4880
0.1736
0.2005
0.2296
0.2611
0.2946
0.3300
0.3669
0.4052
0.4443
0.4840
0.1711
0.1977
0.2266
0.2578
0.2912
0.3264
0.3632
0.4013
0.4404
0.4801
0.1685
0.1949
0.2236
0.2546
0.2877
0.3228
0.3594
0.3974
0.4364
0.4761
0.1660
0.1922
0.2206
0.2514
0.2843
0.3192
0.3557
0.3936
0.4325
0.4721
0.1635
0.1894
0.2177
0.2483
0.2810
0.3156
0.3520
0.3897
0.4286
0.4681
0.1611
0.1867
0.2148
0.2451
0.2776
0.3121
0.3483
0.3859
0.4247
0.4641
Source: Probabilities calculated with Excel.
727
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727
09/11/20 11:24 PM
Final PDF to printer
TABLE 1 (Continued)
Entries in this table provide cumulative probabilities, that is, the area
under the curve to the left of z. For example, P(Z ≤ 1.52) = 0.9357.
P(Z ≤ z)
0
z
z
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9987
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
0.9990
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
1.0000
0.9991
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
1.0000
0.9991
0.9994
0.9995
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.9991
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.9992
0.9994
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.9992
0.9995
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.9993
0.9995
0.9996
0.9997
0.9998
0.9999
0.9999
0.9999
1.0000
0.9993
0.9995
0.9997
0.9998
0.9998
0.9999
0.9999
0.9999
1.0000
Source: Probabilities calculated with Excel.
728
B U S I N E S S S TAT I S T I C S
jag16309_appB_727-738
728
Appendix B
TABLES
09/11/20 11:24 PM
Final PDF to printer
TABLE 2 Student’s t Distribution
Entries in this table provide the values of tα,df that correspond to a given
upper-tail area α and a specified number of degrees of freedom df. For
­example, for α = 0.05 and df = 10, P(T10 ≥ 1.812) = 0.05.
Area in Upper
Tail, α
0
t α ,df
tdf
α
df
0.20
0.10
0.05
1
2
3
4
5
6
7
8
9
10
1.376
1.061
0.978
0.941
0.920
0.906
0.896
0.889
0.883
0.879
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
0.025
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
0.01
31.821
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
11
12
13
14
15
16
17
18
19
20
0.876
0.873
0.870
0.868
0.866
0.865
0.863
0.862
0.861
0.860
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
21
22
23
24
25
26
27
28
29
30
0.859
0.858
0.858
0.857
0.856
0.856
0.855
0.855
0.854
0.854
1.323
1.321
1.319
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.518
2.508
2.500
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.831
2.819
2.807
2.797
2.787
2.779
2.771
2.763
2.756
2.750
Appendix B
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729
TABLES
0.005
B U S I N E S S S TAT I S T I C S
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TABLE 2 (Continued)
α
df
0.20
0.10
0.05
0.025
0.01
0.005
31
32
33
34
35
36
37
38
39
40
0.853
0.853
0.853
0.852
0.852
0.852
0.851
0.851
0.851
0.851
1.309
1.309
1.308
1.307
1.306
1.306
1.305
1.304
1.304
1.303
1.696
1.694
1.692
1.691
1.690
1.688
1.687
1.686
1.685
1.684
2.040
2.037
2.035
2.032
2.030
2.028
2.026
2.024
2.023
2.021
2.453
2.449
2.445
2.441
2.438
2.434
2.431
2.429
2.426
2.423
2.744
2.738
2.733
2.728
2.724
2.719
2.715
2.712
2.708
2.704
41
42
43
44
45
46
47
48
49
50
0.850
0.850
0.850
0.850
0.850
0.850
0.849
0.849
0.849
0.849
1.303
1.302
1.302
1.301
1.301
1.300
1.300
1.299
1.299
1.299
1.683
1.682
1.681
1.680
1.679
1.679
1.678
1.677
1.677
1.676
2.020
2.018
2.017
2.015
2.014
2.013
2.012
2.011
2.010
2.009
2.421
2.418
2.416
2.414
2.412
2.410
2.408
2.407
2.405
2.403
2.701
2.698
2.695
2.692
2.690
2.687
2.685
2.682
2.680
2.678
51
52
53
54
55
56
57
58
59
60
0.849
0.849
0.848
0.848
0.848
0.848
0.848
0.848
0.848
0.848
1.298
1.298
1.298
1.297
1.297
1.297
1.297
1.296
1.296
1.296
1.675
1.675
1.674
1.674
1.673
1.673
1.672
1.672
1.671
1.671
2.008
2.007
2.006
2.005
2.004
2.003
2.002
2.002
2.001
2.000
2.402
2.400
2.399
2.397
2.396
2.395
2.394
2.392
2.391
2.390
2.676
2.674
2.672
2.670
2.668
2.667
2.665
2.663
2.662
2.660
80
100
150
200
500
1000
∞
0.846
0.845
0.844
0.843
0.842
0.842
0.842
1.292
1.290
1.287
1.286
1.283
1.282
1.282
1.664
1.660
1.655
1.653
1.648
1.646
1.645
1.990
1.984
1.976
1.972
1.965
1.962
1.960
2.374
2.364
2.351
2.345
2.334
2.330
2.326
2.639
2.626
2.609
2.601
2.586
2.581
2.576
Source: t values calculated with Excel.
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TABLES
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TABLE 3 χ2 (Chi-Square) Distribution
2
Entries in this table provide the values of ​χ​  α ​​
,df​ that correspond to a given upper-tail
area α and a specified number of degrees of freedom df. For example, for α = 0.05
2
and df = 10, P( ​χ​ 10​​ ​  ≥ 18.307) = 0.05.
Area in Upper
Tail, α
χα2,df
χdf2
α
df
0.995
0.990
0.975
0.950
0.900
0.100
0.050
0.025
0.010
0.005
1
2
3
4
5
6
7
8
9
10
0.000
0.010
0.072
0.207
0.412
0.676
0.989
1.344
1.735
2.156
0.000
0.020
0.115
0.297
0.554
0.872
1.239
1.646
2.088
2.558
0.001
0.051
0.216
0.484
0.831
1.237
1.690
2.180
2.700
3.247
0.004
0.103
0.352
0.711
1.145
1.635
2.167
2.733
3.325
3.940
0.016
0.211
0.584
1.064
1.610
2.204
2.833
3.490
4.168
4.865
2.706
4.605
6.251
7.779
9.236
10.645
12.017
13.362
14.684
15.987
3.841
5.991
7.815
9.488
11.070
12.592
14.067
15.507
16.919
18.307
5.024
7.378
9.348
11.143
12.833
14.449
16.013
17.535
19.023
20.483
6.635
9.210
11.345
13.277
15.086
16.812
18.475
20.090
21.666
23.209
7.879
10.597
12.838
14.860
16.750
18.548
20.278
21.955
23.589
25.188
11
12
13
14
15
16
17
18
19
20
2.603
3.074
3.565
4.075
4.601
5.142
5.697
6.265
6.844
7.434
3.053
3.571
4.107
4.660
5.229
5.812
6.408
7.015
7.633
8.260
3.816
4.404
5.009
5.629
6.262
6.908
7.564
8.231
8.907
9.591
4.575
5.226
5.892
6.571
7.261
7.962
8.672
9.390
10.117
10.851
5.578
6.304
7.042
7.790
8.547
9.312
10.085
10.865
11.651
12.443
17.275
18.549
19.812
21.064
22.307
23.542
24.769
25.989
27.204
28.412
19.675
21.026
22.362
23.685
24.996
26.296
27.587
28.869
30.144
31.410
21.920
23.337
24.736
26.119
27.488
28.845
30.191
31.526
32.852
34.170
24.725
26.217
27.688
29.141
30.578
32.000
33.409
34.805
36.191
37.566
26.757
28.300
29.819
31.319
32.801
34.267
35.718
37.156
38.582
39.997
21
22
23
24
25
26
27
28
29
30
8.034
8.643
9.260
9.886
10.520
11.160
11.808
12.461
13.121
13.787
8.897
9.542
10.196
10.856
11.524
12.198
12.879
13.565
14.256
14.953
10.283
10.982
11.689
12.401
13.120
13.844
14.573
15.308
16.047
16.791
11.591
12.338
13.091
13.848
14.611
15.379
16.151
16.928
17.708
18.493
13.240
14.041
14.848
15.659
16.473
17.292
18.114
18.939
19.768
20.599
29.615
30.813
32.007
33.196
34.382
35.563
36.741
37.916
39.087
40.256
32.671
33.924
35.172
36.415
37.652
38.885
40.113
41.337
42.557
43.773
35.479
36.781
38.076
39.364
40.646
41.923
43.195
44.461
45.722
46.979
38.932
40.289
41.638
42.980
44.314
45.642
46.963
48.278
49.588
50.892
41.401
42.796
44.181
45.559
46.928
48.290
49.645
50.993
52.336
53.672
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TABLE 3 (Continued)
α
df
0.995
0.990
0.975
0.950
0.900
0.100
0.050
0.025
0.010
0.005
31
32
33
34
35
36
37
38
39
40
14.458
15.134
15.815
16.501
17.192
17.887
18.586
19.289
19.996
20.707
15.655
16.362
17.074
17.789
18.509
19.233
19.960
20.691
21.426
22.164
17.539
18.291
19.047
19.806
20.569
21.336
22.106
22.878
23.654
24.433
19.281
20.072
20.867
21.664
22.465
23.269
24.075
24.884
25.695
26.509
21.434
22.271
23.110
23.952
24.797
25.643
26.492
27.343
28.196
29.051
41.422
42.585
43.745
44.903
46.059
47.212
48.363
49.513
50.660
51.805
44.985
46.194
47.400
48.602
49.802
50.998
52.192
53.384
54.572
55.758
48.232
49.480
50.725
51.966
53.203
54.437
55.668
56.896
58.120
59.342
52.191
53.486
54.776
56.061
57.342
58.619
59.893
61.162
62.428
63.691
55.003
56.328
57.648
58.964
60.275
61.581
62.883
64.181
65.476
66.766
41
42
43
44
45
46
47
48
49
50
21.421
22.138
22.859
23.584
24.311
25.041
25.775
26.511
27.249
27.991
22.906
23.650
24.398
25.148
25.901
26.657
27.416
28.177
28.941
29.707
25.215
25.999
26.785
27.575
28.366
29.160
29.956
30.755
31.555
32.357
27.326
28.144
28.965
29.787
30.612
31.439
32.268
33.098
33.930
34.764
29.907
30.765
31.625
32.487
33.350
34.215
35.081
35.949
36.818
37.689
52.949
54.090
55.230
56.369
57.505
58.641
59.774
60.907
62.038
63.167
56.942
58.124
59.304
60.481
61.656
62.830
64.001
65.171
66.339
67.505
60.561
61.777
62.990
64.201
65.410
66.617
67.821
69.023
70.222
71.420
64.950
66.206
67.459
68.710
69.957
71.201
72.443
73.683
74.919
76.154
68.053
69.336
70.616
71.893
73.166
74.437
75.704
76.969
78.231
79.490
55
60
31.735
35.534
33.570
37.485
36.398
40.482
38.958
43.188
42.060
46.459
68.796
74.397
73.311
79.082
77.380
83.298
82.292
88.379
85.749
91.952
65
70
75
80
85
90
95
100
39.383
43.275
47.206
51.172
55.170
59.196
63.250
67.328
41.444
45.442
49.475
53.540
57.634
61.754
65.898
70.065
44.603
48.758
52.942
57.153
61.389
65.647
69.925
74.222
47.450
51.739
56.054
60.391
64.749
69.126
73.520
77.929
50.883
55.329
59.795
64.278
68.777
73.291
77.818
82.358
79.973
85.527
91.061
96.578
102.079
107.565
113.038
118.498
84.821
90.531
96.217
101.879
107.522
113.145
118.752
124.342
89.177
95.023
100.839
106.629
112.393
118.136
123.858
129.561
94.422
100.425
106.393
112.329
118.236
124.116
129.973
135.807
98.105
104.215
110.286
116.321
122.325
128.299
134.247
140.169
Source: χ2 values calculated with Excel.
732
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TABLES
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jag16309_appB_727-738
733
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
1
2
3
4
Appendix B
TABLES
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
5
6
7
0.10
0.05
0.025
0.01
α
df2
3.59
5.59
8.07
12.25
3.78
5.99
8.81
13.75
4.06
6.61
10.01
16.26
4.54
7.71
12.22
21.20
5.54
10.13
17.44
34.12
8.53
18.51
38.51
98.50
39.86
161.45
647.79
4052.18
1
3.26
4.74
6.54
9.55
3.46
5.14
7.26
10.92
3.78
5.79
8.43
13.27
4.32
6.94
10.65
18.00
5.46
9.55
16.04
30.82
9.00
19.00
39.00
99.00
49.50
199.50
799.50
4999.50
2
3.07
4.35
5.89
8.45
3.29
4.76
6.60
9.78
3.62
5.41
7.76
12.06
4.19
6.59
9.98
16.69
5.39
9.28
15.44
29.46
9.16
19.16
39.17
99.17
53.59
215.71
864.16
5403.35
3
2.96
4.12
5.52
7.85
3.18
4.53
6.23
9.15
3.52
5.19
7.39
11.39
4.11
6.39
9.60
15.98
5.34
9.12
15.10
28.71
9.24
19.25
39.25
99.25
55.83
224.58
899.58
5624.58
4
2.88
3.97
5.29
7.46
3.11
4.39
5.99
8.75
3.45
5.05
7.15
10.97
4.05
6.26
9.36
15.52
5.31
9.01
14.88
28.24
9.29
19.30
39.30
99.30
57.24
230.16
921.85
5763.65
5
2.83
3.87
5.12
7.19
3.05
4.28
5.82
8.47
3.40
4.95
6.98
10.67
4.01
6.16
9.20
15.21
5.28
8.94
14.73
27.91
9.33
19.33
39.33
99.33
58.2
233.99
937.11
5858.99
6
2.78
3.79
4.99
6.99
3.01
4.21
5.70
8.26
3.37
4.88
6.85
10.46
3.98
6.09
9.07
14.98
5.27
8.89
14.62
27.67
9.35
19.35
39.36
99.36
58.91
236.77
948.22
5928.36
7
2.75
3.73
4.90
6.84
2.98
4.15
5.60
8.10
3.34
4.82
6.76
10.29
3.95
6.04
8.98
14.80
5.25
8.85
14.54
27.49
9.37
19.37
39.37
99.37
59.44
238.88
956.66
5981.07
df1
8
2.72
3.68
4.82
6.72
2.96
4.10
5.52
7.98
3.32
4.77
6.68
10.16
3.94
6.00
8.90
14.66
5.24
8.81
14.47
27.35
9.38
19.38
39.39
99.39
59.86
240.54
963.28
6022.47
9
10
2.70
3.64
4.76
6.62
2.94
4.06
5.46
7.87
3.30
4.74
6.62
10.05
3.92
5.96
8.84
14.55
5.23
8.79
14.42
27.23
9.39
19.40
39.40
99.40
60.19
241.88
968.63
6055.85
Entries in this table provide the values of Fα,(df1,df2) that correspond to a given upper-tail area α and a specified number of degrees
of freedom in the numerator df1 and degrees of freedom in the denominator df2. For example, for α = 0.05, df1 = 8, and df2 = 6,
P(F(8,6) ≥ 4.15) = 0.05.
TABLE 4 F Distribution
15
2.63
3.51
4.57
6.31
2.87
3.94
5.27
7.56
3.24
4.62
6.43
9.72
3.87
5.86
8.66
14.20
5.20
8.70
14.25
26.87
9.42
19.43
39.43
99.43
61.22
245.95
984.87
6157.28
25
2.57
3.40
4.40
6.06
2.81
3.83
5.11
7.30
3.19
4.52
6.27
9.45
3.83
5.77
8.50
13.91
5.17
8.63
14.12
26.58
9.45
19.46
39.46
99.46
62.05
249.26
998.08
6239.83
50
2.52
3.32
4.28
5.86
2.77
3.75
4.98
7.09
3.15
4.44
6.14
9.24
3.80
5.70
8.38
13.69
5.15
8.58
14.01
26.35
9.47
19.48
39.48
99.48
62.69
251.77
1008.12
6302.52
2.50
3.27
4.21
5.75
2.75
3.71
4.92
6.99
3.13
4.41
6.08
9.13
3.78
5.66
8.32
13.58
5.14
8.55
13.96
26.24
9.48
19.49
39.49
99.49
63.01
253.04
1013.17
6334.11
100
Fα,(df1,df2)
2.48
3.24
4.16
5.67
2.73
3.68
4.86
6.90
3.11
4.37
6.03
9.04
3.76
5.64
8.27
13.49
5.14
8.53
13.91
26.15
9.49
19.49
39.50
99.50
63.26
254.06
1017.24
6359.50
500
Area in Upper
Tail, α
Final PDF to printer
B U S I N E S S S TAT I S T I C S
733
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734
α
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
df2
8
9
10
11
12
13
14
15
16
1
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Appendix B
3.05
4.49
6.12
8.53
3.07
4.54
6.20
8.68
3.10
4.60
6.30
8.86
3.14
4.67
6.41
9.07
3.18
4.75
6.55
9.33
3.23
4.84
6.72
9.65
3.29
4.96
6.94
10.04
3.36
5.12
7.21
10.56
3.46
5.32
7.57
11.26
TABLE 4 (Continued)
2.67
3.63
4.69
6.23
2.70
3.68
4.77
6.36
2.73
3.74
4.86
6.51
2.76
3.81
4.97
6.70
2.81
3.89
5.10
6.93
2.86
3.98
5.26
7.21
2.92
4.10
5.46
7.56
3.01
4.26
5.71
8.02
3.11
4.46
6.06
8.65
2
2.46
3.24
4.08
5.29
2.49
3.29
4.15
5.42
2.52
3.34
4.24
5.56
2.56
3.41
4.35
5.74
2.61
3.49
4.47
5.95
2.66
3.59
4.63
6.22
2.73
3.71
4.83
6.55
2.81
3.86
5.08
6.99
2.92
4.07
5.42
7.59
3
2.33
3.01
3.73
4.77
2.36
3.06
3.80
4.89
2.39
3.11
3.89
5.04
2.43
3.18
4.00
5.21
2.48
3.26
4.12
5.41
2.54
3.36
4.28
5.67
2.61
3.48
4.47
5.99
2.69
3.63
4.72
6.42
2.81
3.84
5.05
7.01
4
2.24
2.85
3.50
4.44
2.27
2.90
3.58
4.56
2.31
2.96
3.66
4.69
2.35
3.03
3.77
4.86
2.39
3.11
3.89
5.06
2.45
3.20
4.04
5.32
2.52
3.33
4.24
5.64
2.61
3.48
4.48
6.06
2.73
3.69
4.82
6.63
5
2.18
2.74
3.34
4.20
2.21
2.79
3.41
4.32
2.24
2.85
3.50
4.46
2.28
2.92
3.60
4.62
2.33
3.00
3.73
4.82
2.39
3.09
3.88
5.07
2.46
3.22
4.07
5.39
2.55
3.37
4.32
5.80
2.67
3.58
4.65
6.37
6
2.13
2.66
3.22
4.03
2.16
2.71
3.29
4.14
2.19
2.76
3.38
4.28
2.23
2.83
3.48
4.44
2.28
2.91
3.61
4.64
2.34
3.01
3.76
4.89
2.41
3.14
3.95
5.20
2.51
3.29
4.20
5.61
2.62
3.50
4.53
6.18
7
2.09
2.59
3.12
3.89
2.12
2.64
3.20
4.00
2.15
2.70
3.29
4.14
2.20
2.77
3.39
4.30
2.24
2.85
3.51
4.50
2.30
2.95
3.66
4.74
2.38
3.07
3.85
5.06
2.47
3.23
4.10
5.47
2.59
3.44
4.43
6.03
df1
8
9
2.06
2.54
3.05
3.78
2.09
2.59
3.12
3.89
2.12
2.65
3.21
4.03
2.16
2.71
3.31
4.19
2.21
2.80
3.44
4.39
2.27
2.90
3.59
4.63
2.35
3.02
3.78
4.94
2.44
3.18
4.03
5.35
2.56
3.39
4.36
5.91
10
2.03
2.49
2.99
3.69
2.06
2.54
3.06
3.80
2.10
2.60
3.15
3.94
2.14
2.67
3.25
4.10
2.19
2.75
3.37
4.30
2.25
2.85
3.53
4.54
2.32
2.98
3.72
4.85
2.42
3.14
3.96
5.26
2.54
3.35
4.30
5.81
15
1.94
2.35
2.79
3.41
1.97
2.40
2.86
3.52
2.01
2.46
2.95
3.66
2.05
2.53
3.05
3.82
2.10
2.62
3.18
4.01
2.17
2.72
3.33
4.25
2.24
2.85
3.52
4.56
2.34
3.01
3.77
4.96
2.46
3.22
4.10
5.52
25
1.86
2.23
2.61
3.16
1.89
2.28
2.69
3.28
1.93
2.34
2.78
3.41
1.98
2.41
2.88
3.57
2.03
2.50
3.01
3.76
2.10
2.60
3.16
4.01
2.17
2.73
3.35
4.31
2.27
2.89
3.60
4.71
2.40
3.11
3.94
5.26
50
1.79
2.12
2.47
2.97
1.83
2.18
2.55
3.08
1.87
2.24
2.64
3.22
1.92
2.31
2.74
3.38
1.97
2.40
2.87
3.57
2.04
2.51
3.03
3.81
2.12
2.64
3.22
4.12
2.22
2.80
3.47
4.52
2.35
3.02
3.81
5.07
100
1.76
2.07
2.40
2.86
1.79
2.12
2.47
2.98
1.83
2.19
2.56
3.11
1.88
2.26
2.67
3.27
1.94
2.35
2.80
3.47
2.01
2.46
2.96
3.71
2.09
2.59
3.15
4.01
2.19
2.76
3.40
4.41
2.32
2.97
3.74
4.96
500
1.73
2.02
2.33
2.78
1.76
2.08
2.41
2.89
1.80
2.14
2.50
3.03
1.85
2.22
2.61
3.19
1.91
2.31
2.74
3.38
1.98
2.42
2.90
3.62
2.06
2.55
3.09
3.93
2.17
2.72
3.35
4.33
2.30
2.94
3.68
4.88
Final PDF to printer
TABLES
09/11/20 11:24 PM
jag16309_appB_727-738
735
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
20
21
22
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
19
24
0.10
0.05
0.025
0.01
18
TABLES
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
17
Appendix B
23
α
df2
2.93
4.26
5.72
7.82
2.94
4.28
5.75
7.88
2.95
4.30
5.79
7.95
2.96
4.32
5.83
8.02
2.97
4.35
5.87
8.10
2.99
4.38
5.92
8.18
3.01
4.41
5.98
8.29
3.03
4.45
6.04
8.40
1
2.54
3.40
4.32
5.61
2.55
3.42
4.35
5.66
2.56
3.44
4.38
5.72
2.57
3.47
4.42
5.78
2.59
3.49
4.46
5.85
2.61
3.52
4.51
5.93
2.62
3.55
4.56
6.01
2.64
3.59
4.62
6.11
2
2.33
3.01
3.72
4.72
2.34
3.03
3.75
4.76
2.35
3.05
3.78
4.82
2.36
3.07
3.82
4.87
2.38
3.10
3.86
4.94
2.40
3.13
3.90
5.01
2.42
3.16
3.95
5.09
2.44
3.20
4.01
5.18
3
2.19
2.78
3.38
4.22
2.21
2.80
3.41
4.26
2.22
2.82
3.44
4.31
2.23
2.84
3.48
4.37
2.25
2.87
3.51
4.43
2.27
2.90
3.56
4.50
2.29
2.93
3.61
4.58
2.31
2.96
3.66
4.67
4
2.10
2.62
3.15
3.90
2.11
2.64
3.18
3.94
2.13
2.66
3.22
3.99
2.14
2.68
3.25
4.04
2.16
2.71
3.29
4.10
2.18
2.74
3.33
4.17
2.20
2.77
3.38
4.25
2.22
2.81
3.44
4.34
5
2.04
2.51
2.99
3.67
2.05
2.53
3.02
3.71
2.06
2.55
3.05
3.76
2.08
2.57
3.09
3.81
2.09
2.60
3.13
3.87
2.11
2.63
3.17
3.94
2.13
2.66
3.22
4.01
2.15
2.70
3.28
4.10
6
1.98
2.42
2.87
3.50
1.99
2.44
2.90
3.54
2.01
2.46
2.93
3.59
2.02
2.49
2.97
3.64
2.04
2.51
3.01
3.70
2.06
2.54
3.05
3.77
2.08
2.58
3.10
3.84
2.10
2.61
3.16
3.93
7
1.94
2.36
2.78
3.36
1.95
2.37
2.81
3.41
1.97
2.40
2.84
3.45
1.98
2.42
2.87
3.51
2.00
2.45
2.91
3.56
2.02
2.48
2.96
3.63
2.04
2.51
3.01
3.71
2.06
2.55
3.06
3.79
df1
8
9
1.91
2.30
2.70
3.26
1.92
2.32
2.73
3.30
1.93
2.34
2.76
3.35
1.95
2.37
2.80
3.40
1.96
2.39
2.84
3.46
1.98
2.42
2.88
3.52
2.00
2.46
2.93
3.60
2.03
2.49
2.98
3.68
10
1.88
2.25
2.64
3.17
1.89
2.27
2.67
3.21
1.90
2.30
2.70
3.26
1.92
2.32
2.73
3.31
1.94
2.35
2.77
3.37
1.96
2.38
2.82
3.43
1.98
2.41
2.87
3.51
2.00
2.45
2.92
3.59
15
1.78
2.11
2.44
2.89
1.80
2.13
2.47
2.93
1.81
2.15
2.50
2.98
1.83
2.18
2.53
3.03
1.84
2.20
2.57
3.09
1.86
2.23
2.62
3.15
1.89
2.27
2.67
3.23
1.91
2.31
2.72
3.31
25
1.70
1.97
2.26
2.64
1.71
2.00
2.29
2.69
1.73
2.02
2.32
2.73
1.74
2.05
2.36
2.79
1.76
2.07
2.40
2.84
1.78
2.11
2.44
2.91
1.80
2.14
2.49
2.98
1.83
2.18
2.55
3.07
50
1.62
1.86
2.11
2.44
1.64
1.88
2.14
2.48
1.65
1.91
2.17
2.53
1.67
1.94
2.21
2.58
1.69
1.97
2.25
2.64
1.71
2.00
2.30
2.71
1.74
2.04
2.35
2.78
1.76
2.08
2.41
2.87
100
1.58
1.80
2.02
2.33
1.59
1.82
2.06
2.37
1.61
1.85
2.09
2.42
1.63
1.88
2.13
2.48
1.65
1.91
2.17
2.54
1.67
1.94
2.22
2.60
1.70
1.98
2.27
2.68
1.73
2.02
2.33
2.76
500
1.54
1.75
1.95
2.24
1.56
1.77
1.99
2.28
1.58
1.80
2.02
2.33
1.60
1.83
2.06
2.38
1.62
1.86
2.10
2.44
1.64
1.89
2.15
2.51
1.67
1.93
2.20
2.59
1.69
1.97
2.26
2.68
Final PDF to printer
B U S I N E S S S TAT I S T I C S
735
09/11/20 11:24 PM
736
B U S I N E S S S TAT I S T I C S
jag16309_appB_727-738
736
Appendix B
TABLES
09/11/20 11:24 PM
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
0.10
0.05
0.025
0.01
25
26
27
28
29
30
50
100
500
2.72
3.86
5.05
6.69
2.76
3.94
5.18
6.90
2.81
4.03
5.34
7.17
2.88
4.17
5.57
7.56
2.89
4.18
5.59
7.60
2.89
4.20
5.61
7.64
2.90
4.21
5.63
7.68
2.91
4.23
5.66
7.72
2.92
4.24
5.69
7.77
1
Source: F-values calculated with Excel.
α
df2
TABLE 4 (Continued)
2.31
3.01
3.72
4.65
2.36
3.09
3.83
4.82
2.41
3.18
3.97
5.06
2.49
3.32
4.18
5.39
2.50
3.33
4.20
5.42
2.50
3.34
4.22
5.45
2.51
3.35
4.24
5.49
2.52
3.37
4.27
5.53
2.53
3.39
4.29
5.57
2
2.09
2.62
3.14
3.82
2.14
2.70
3.25
3.98
2.20
2.79
3.39
4.20
2.28
2.92
3.59
4.51
2.28
2.93
3.61
4.54
2.29
2.95
3.63
4.57
2.30
2.96
3.65
4.60
2.31
2.98
3.67
4.64
2.32
2.99
3.69
4.68
3
1.96
2.39
2.81
3.36
2.00
2.46
2.92
3.51
2.06
2.56
3.05
3.72
2.14
2.69
3.25
4.02
2.15
2.70
3.27
4.04
2.16
2.71
3.29
4.07
2.17
2.73
3.31
4.11
2.17
2.74
3.33
4.14
2.18
2.76
3.35
4.18
4
1.86
2.23
2.59
3.05
1.91
2.31
2.70
3.21
1.97
2.40
2.83
3.41
2.05
2.53
3.03
3.70
2.06
2.55
3.04
3.73
2.06
2.56
3.06
3.75
2.07
2.57
3.08
3.78
2.08
2.59
3.10
3.82
2.09
2.60
3.13
3.85
5
1.79
2.12
2.43
2.84
1.83
2.19
2.54
2.99
1.90
2.29
2.67
3.19
1.98
2.42
2.87
3.47
1.99
2.43
2.88
3.50
2.00
2.45
2.90
3.53
2.00
2.46
2.92
3.56
2.01
2.47
2.94
3.59
2.02
2.49
2.97
3.63
6
1.73
2.03
2.31
2.68
1.78
2.10
2.42
2.82
1.84
2.20
2.55
3.02
1.93
2.33
2.75
3.30
1.93
2.35
2.76
3.33
1.94
2.36
2.78
3.36
1.95
2.37
2.80
3.39
1.96
2.39
2.82
3.42
1.97
2.40
2.85
3.46
7
1.68
1.96
2.22
2.55
1.73
2.03
2.32
2.69
1.80
2.13
2.46
2.89
1.88
2.27
2.65
3.17
1.89
2.28
2.67
3.20
1.90
2.29
2.69
3.23
1.91
2.31
2.71
3.26
1.92
2.32
2.73
3.29
1.93
2.34
2.75
3.32
df1
8
9
1.64
1.90
2.14
2.44
1.69
1.97
2.24
2.59
1.76
2.07
2.38
2.78
1.85
2.21
2.57
3.07
1.86
2.22
2.59
3.09
1.87
2.24
2.61
3.12
1.87
2.25
2.63
3.15
1.88
2.27
2.65
3.18
1.89
2.28
2.68
3.22
10
1.61
1.85
2.07
2.36
1.66
1.93
2.18
2.50
1.73
2.03
2.32
2.70
1.82
2.16
2.51
2.98
1.83
2.18
2.53
3.00
1.84
2.19
2.55
3.03
1.85
2.20
2.57
3.06
1.86
2.22
2.59
3.09
1.87
2.24
2.61
3.13
15
1.50
1.69
1.86
2.07
1.56
1.77
1.97
2.22
1.63
1.87
2.11
2.42
1.72
2.01
2.31
2.70
1.73
2.03
2.32
2.73
1.74
2.04
2.34
2.75
1.75
2.06
2.36
2.78
1.76
2.07
2.39
2.81
1.77
2.09
2.41
2.85
25
1.39
1.53
1.65
1.81
1.45
1.62
1.77
1.97
1.53
1.73
1.92
2.17
1.63
1.88
2.12
2.45
1.64
1.89
2.14
2.48
1.65
1.91
2.16
2.51
1.66
1.92
2.18
2.54
1.67
1.94
2.21
2.57
1.68
1.96
2.23
2.60
50
1.28
1.38
1.46
1.57
1.35
1.48
1.59
1.74
1.44
1.60
1.75
1.95
1.55
1.76
1.97
2.25
1.56
1.77
1.99
2.27
1.57
1.79
2.01
2.30
1.58
1.81
2.03
2.33
1.59
1.82
2.05
2.36
1.61
1.84
2.08
2.40
100
1.21
1.28
1.34
1.41
1.29
1.39
1.48
1.60
1.39
1.52
1.66
1.82
1.51
1.70
1.88
2.13
1.52
1.71
1.90
2.16
1.53
1.73
1.92
2.19
1.54
1.74
1.94
2.22
1.55
1.76
1.97
2.25
1.56
1.78
2.00
2.29
500
1.12
1.16
1.19
1.23
1.23
1.31
1.38
1.47
1.34
1.46
1.57
1.71
1.47
1.64
1.81
2.03
1.48
1.65
1.83
2.06
1.49
1.67
1.85
2.09
1.50
1.69
1.87
2.12
1.51
1.71
1.90
2.16
1.53
1.73
1.92
2.19
Final PDF to printer
Final PDF to printer
TABLE 5 Studentized Range Values qα,(c,nT −c) for Tukey’s HSD Method
The number of means, c
nT − c
α
2
3
4
5
4
0.05
0.01
3.93
6.51
5.04
8.12
5.76
9.17
6.29
9.96
6.71
10.58
7.05
11.10
7.35
11.54
7.60
11.92
5
0.05
0.01
3.64
5.70
4.60
6.98
5.22
7.80
5.67
8.42
6.03
8.91
6.33
9.32
6.58
9.67
6
0.05
0.01
3.46
5.24
4.34
6.33
4.90
7.03
5.30
7.56
5.63
7.97
5.90
8.32
7
0.05
0.01
3.34
4.95
4.16
5.92
4.68
6.54
5.06
7.01
5.36
7.37
8
0.05
0.01
3.26
4.75
4.04
5.64
4.53
6.20
4.89
6.62
9
0.05
0.01
3.20
4.60
3.95
5.43
4.41
5.96
10
0.05
0.01
3.15
4.48
3.88
5.27
11
0.05
0.01
3.11
4.39
12
0.05
0.01
13
6
11
12
7.83
12.26
8.03
12.57
8.21
12.84
6.80
9.97
6.99
10.24
7.17
10.48
7.32
10.70
6.12
8.61
6.32
8.87
6.49
9.10
6.65
9.30
6.79
9.48
5.61
7.68
5.82
7.94
6.00
8.17
6.16
8.37
6.30
8.55
6.43
8.71
5.17
6.96
5.40
7.24
5.60
7.47
5.77
7.68
5.92
7.86
6.05
8.03
6.18
8.18
4.76
6.35
5.02
6.66
5.24
6.91
5.43
7.13
5.59
7.33
5.74
7.49
5.87
7.65
5.98
7.78
4.33
5.77
4.65
6.14
4.91
6.43
5.12
6.67
5.30
6.87
5.46
7.05
5.60
7.21
5.72
7.36
5.83
7.49
3.82
5.15
4.26
5.62
4.57
5.97
4.82
6.25
5.03
6.48
5.20
6.67
5.35
6.84
5.49
6.99
5.61
7.13
5.71
7.25
3.08
4.32
3.77
5.05
4.20
5.50
4.51
5.84
4.75
6.10
4.95
6.32
5.12
6.51
5.27
6.67
5.39
6.81
5.51
6.94
5.61
7.06
0.05
0.01
3.06
4.26
3.73
4.96
4.15
5.40
4.45
5.73
4.69
5.98
4.88
6.19
5.05
6.37
5.19
6.53
5.32
6.67
5.43
6.79
5.53
6.90
14
0.05
0.01
3.03
4.21
3.70
4.89
4.11
5.32
4.41
5.63
4.64
5.88
4.83
6.08
4.99
6.26
5.13
6.41
5.25
6.54
5.36
6.66
5.46
6.77
15
0.05
0.01
3.01
4.17
3.67
4.84
4.08
5.25
4.37
5.56
4.59
5.80
4.78
5.99
4.94
6.16
5.08
6.31
5.20
6.44
5.31
6.55
5.40
6.66
16
0.05
0.01
3.00
4.13
3.65
4.79
4.05
5.19
4.33
5.49
4.56
5.72
4.74
5.92
4.90
6.08
5.03
6.22
5.15
6.35
5.26
6.46
5.35
6.56
17
0.05
0.01
2.98
4.10
3.63
4.74
4.02
5.14
4.30
5.43
4.52
5.66
4.70
5.85
4.86
6.01
4.99
6.15
5.11
6.27
5.21
6.38
5.31
6.48
18
0.05
0.01
2.97
4.07
3.61
4.70
4.00
5.09
4.28
5.38
4.49
5.60
4.67
5.79
4.82
5.94
4.96
6.08
5.07
6.20
5.17
6.31
5.27
6.41
19
0.05
0.01
2.96
4.05
3.59
4.67
3.98
5.05
4.25
5.33
4.47
5.55
4.65
5.73
4.79
5.89
4.92
6.02
5.04
6.14
5.14
6.25
5.23
6.34
Appendix B
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7
8
TABLES
9
10
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Final PDF to printer
TABLE 5 (Continued)
The number of means, c
nT − c
α
2
3
4
5
6
7
8
9
10
11
12
20
0.05
0.01
2.95
4.02
3.58
4.64
3.96
5.02
4.23
5.29
4.45
5.51
4.62
5.69
4.77
5.84
4.90
5.97
5.01
6.09
5.11
6.19
5.20
6.28
24
0.05
0.01
2.92
3.96
3.53
4.55
3.90
4.91
4.17
5.17
4.37
5.37
4.54
5.54
4.68
5.69
4.81
5.81
4.92
5.92
5.01
6.02
5.10
6.11
30
0.05
0.01
2.89
3.89
3.49
4.45
3.85
4.80
4.10
5.05
4.30
5.24
4.46
5.40
4.60
5.54
4.72
5.65
4.82
5.76
4.92
5.85
5.00
5.93
40
0.05
0.01
2.86
3.82
3.44
4.37
3.79
4.70
4.04
4.93
4.23
5.11
4.39
5.26
4.52
5.39
4.63
5.50
4.73
5.60
4.82
5.69
4.90
5.76
60
0.05
0.01
2.83
3.76
3.40
4.28
3.74
4.59
3.98
4.82
4.16
4.99
4.31
5.13
4.44
5.25
4.55
5.36
4.65
5.45
4.73
5.53
4.81
5.60
120
0.05
0.01
2.80
3.70
3.36
4.20
3.68
4.50
3.92
4.71
4.10
4.87
4.24
5.01
4.36
5.12
4.47
5.21
4.56
5.30
4.64
5.37
4.71
5.44
∞
0.05
0.01
2.77
3.64
3.31
4.12
3.63
4.40
3.86
4.60
4.03
4.76
4.17
4.88
4.29
4.99
4.39
5.08
4.47
5.16
4.55
5.23
4.62
5.29
Source: E. S. Pearson and H. O. Hartley, Biometrika Tables for Statisticians, vol. 1 (Cambridge: Cambridge University Press, 1966).
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TABLES
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