FORMULAS AND TABLES
STATISTICS
August 2020
Department of Economics and Econometrics
Faculty of Economics and Business
University of Amsterdam
IMPORTANT
During the examinations it is obligatory to have at your disposal
your own ‘FORMULAS AND TABLES’ syllabus. It is prohibited to
insert sentences, symbols, lines, markings or anything else into
this syllabus, so do not write in it! At the examination this will be
checked, and any offence will be regarded as fraud, which induces
sanctions.
CONTENTS
Formulas
Part 1 (Statistics)
Part 2 (Intermediate Statistics)
p. 2
p. 8
Table
Table
Table
Table
Table
Table
Binomial Probabilities
Poisson Probabilities
Standard Normal Curve Areas
Critical Values of t
Critical Values of χ2
Percentage Points of the F
Distribution, α=.05
Percentage Points of the F
Distribution, α=.025
Percentage Points of the F
Distribution, α=.01
Critical Values of the Wilcoxon
Rank Sum Test for Independent
Samples
Critical Values for the Wilcoxon
Signed Rank Sum Test for the
Matched Pairs Experiment
Critical Values of the Spearman
Rank Correlation Coefficient
Critical Values for the DurbinWatson d Statistic, α=.05
Critical Values for the DurbinWatson d Statistic, α=.01
Critical Values of the Lilliefors
Test
p. 13
p. 16
p. 18
p. 19
p. 20
p. 21
1
2
3
4
5
6(a)
Table 6(b)
Table 6(c)
Table 7
Table 8
Table 9
Table 10(a)
Table 10(b)
Table 11
1
p. 23
p. 25
p. 27
p. 28
p. 29
p. 30
p. 31
p. 32
PART 1 (Statistics)
Population coefficient of variation: CV =
Geometric mean
n
X1 × X 2 × ... × X n=
Sample coefficient of variation: cv =
( X1 × X 2 × ... × X n )1/ n
Harmonic mean (weighted):
Location of a Percentile:
P
LP = ( n + 1) ⋅
100
n
H
=
g1 + g 2 + ... + g n
=
g
g1 g 2
+
+ ... + n
x1 x2
xn
∑ gi
i =1
n
g
∑ xi
Sample mean for grouped data:
1 k
x ≈ ∑ mi fi
n i =1
i =1 i
Sample variance for grouped data:
1 k
s2 ≈
( mi − x )2 fi
∑
n − 1 i =1
Descriptive statistics (for one variable)
Sturges’ formula: #classes =+
1 3.3 ⋅ log10 (n)
2
k
 k
 
1
1
2

largest obs. − smallest
obs.
=
∑ mi fi − n  ∑ mi fi  
Class width =

1
n
−
=i 1 =
i 1
 
# classes

1 N
Population mean: µ = ∑ xi
N i =1
Sample mean: x =
1 n
∑ xi
n i =1
Population variance:
1
1 N
2
( x=
=
σ2
∑
i − µ)
N
N i =1
N
∑ xi2 − µ2
i =1
Sample variance:
1 n
( xi −=
=
s2
x )2
∑
n − 1 i =1
2

1  n 2 1 n  
∑ xi − n  ∑ xi  

1
n
−
=i 1 =
i 1  

Population standard deviation: σ =
Sample standard deviation: s =
σ2
s2
2
s
x
σ
µ
Descriptive statistics for two variables
Population covariance:
1 N
=
σ xy
∑ ( xi − µ X )( yi − µY )
N i =1
=
1
N
Counting rules
Factorial:
n ! = 1 ⋅ 2 ⋅ ... ⋅ n (for n = 1, 2,3,...)
N
0! = 1
∑ xi yi − µ X µY
i =1
Number of possible outcomes, if k objects
are drawn out of n objects:
Sample covariance:
1 n
=
s xy
∑ ( xi − x )( yi − y )
n − 1 i =1
=
- with replacement, with regard to order:
nk
1 n
1  n   n 
 ∑ xi yi −  ∑ xi   ∑ yi  
− 1  i 1
n=
n  i 1=
=
  i 1  
- without replacement, with regard to order
(‘permutations’):
n!
= n ( n − 1) ... ( n − k + 1)
( n − k )!
Population coefficient of correlation:
σ xy
ρ=
σ xσ y
- without replacement, without regard to
order (‘combinations’):
n!
n  n  n ( n − 1) ... ( n − k + 1)
C
=
=
k
=

k!
k !( n − k ) !
k 
Sample coefficient of correlation:
s xy
r=
sx s y
Slope and intercept of regression line:
s xy
b1 = 2 , b0= y − b1 x
sx
Discrete probability distributions
=
µ
N
∑ xi ⋅ p ( xi )
E ( X=
)
i =1
Probability
P ( A B)
(X )
=
σ2 V=
P ( A and B )
=
P ( B)
( )
( )
P ( A) = 1 − P A
P ( A and/or B ) =
=
P ( B) ⋅ P ( A B)
P ( X= x=
)
=
A and B are called independent if:
or
P ( A B ) = P ( A)
P ( B A) = P ( B )
P ( A and=
B)
2
i =1
⋅ p ( xi )
( ) ∑ xi2 ⋅ p ( xi )
Binomial distribution: Bin(n, p)
P ( A ) + P ( B ) − P ( A and B )
P ( A) ⋅ P ( B A)
∑ ( xi − µ )
2
=
σ2 E X 2 − µ 2 , where E X
=
C
P ( A and=
B)
N
p ( x )=
n x
n− x
  p (1 − p )
 x
n!
p x (1 − p ) n− x ,=
x
x !( n − x ) !
E ( X ) = µ = np
V ( X ) = σ2 = np (1 − p )
or
P ( A) ⋅ P ( B )
3
0,1,..., n
Approximation of Bin(n,p) by
=
Poi(µ np ) if p < 0.05
Continuous probability distributions
b
P (a ≤ X ≤ b) =
∫ f ( x ) dx
Approximation of Bin(n, p ) by
X c − np
Z =
np (1 − p )
a
∞
~ N (0,1)
=
µ
∫
)
E ( X=
−∞
∞
if np ≥ 5 and n (1 − p ) ≥ 5
2
=
σ
(with continuity correction)
x ⋅ f ( x ) dx
V (=
X)
2
∫ ( x − µ ) f ( x ) dx
−∞
2
Hypergeometric distribution:
Hyp(n, N ,k)
P (=
X x=
)
p =
k 
N −k
  × 

 x
 n−x 
N
 
n
k
N
=
σ
∞
where E ( X 2 ) =
∫
x 2 f ( x ) dx
−∞
Uniform distribution: U(a, b)
=
f ( x)
E(X )
k
=np
N
N −n
np (1 − p ) ⋅
N −1
E ( X ) =n ⋅
V (X )=
E ( X 2 ) − µ2 ,
1
for a ≤ x ≤ b
b−a
a + b
=
2
V (X ) =
Approximation of Hyp(n,N,k) by
Bin(n,p = k/N) if n/N < 0.05
(b − a ) 2
12
Normal distribution: N(µ, σ2)
Standard normal distribution:
Poisson distribution: Poi(µ)
P (=
X x=
)
E(X ) =
e -µ µ x
x 0,1, 2,...
, =
x!
p ( x=
)
V (X ) =
N (µ= 0, σ2= 1) or in short N(0,1)
Standardize : Z =
µ
X −µ
~ N (0,1)
σ
Linear functions
Approximation of Poi(m) by
Z
Xc −m
~ N (0,1)
m
E ( aX + b =
)
if m > 15
aE ( X ) + b
V ( aX + b ) =
a 2V ( X )
(with continuity correction)
If X is normally distributed, then
aX + b is also normally distributed.
4
Discrete bivariate probability
distributions
p ( x=
, y)
P
=
=
Y y)
( X x and
P (=
X x=
Y y=
)
=
P ( X x=
and Y y )
P (Y = y )
X and Y are independent if for all ( x, y ) :
P ( X= x and Y
= y=
) P ( X= x) P(Y= y )
=
σ xy
∑ ( x − µ x ) ( y − µ y ) p ( x, y )
all ( x , y )
=
σ xy
E ( XY ) − µ xµ y ,
∑
where E ( XY ) =
xy p ( x, y )
all ( x , y )
Bivariate probability distributions
(discrete and continuous)
(
)
E ( X − µ x ) Y − µ y 


=
σ xy
E ( XY ) − µ xµ y
=
σ xy
ρ =
σ xy
σ xσ y
E ( aX + bY + =
c)
aE ( X ) + bE (Y ) + c
V ( aX + bY + c ) =
= a 2V ( X ) + b 2V (Y ) + 2abσ xy
= a 2V ( X ) + b 2V (Y ) + 2abρσ x σ y
If X and Y are jointly normally distributed,
then aX + bY + c is also normally
distributed.
If X and Y are independent, then
they are uncorrelated, so that:
σ xy = 0, ρ = 0, E ( XY ) = µ X µY ,
V ( aX + bY +=
c)
a 2V ( X ) + b 2V (Y )
5
Sampling distributions
Two independent samples:
Mean:
E ( X1 − X 2 ) = m x1 − x2 = m1 − m 2
E ( X ) =m x =m
2
2
ss
1
s2x1 − x2 =
+ 2
V ( X1 − X 2 ) =
n1
n2
s
V ( X ) = s2x =
n
X −m
~ N (0,1) if X is normally
Z =
s n
distributed (or n is large).
2
Z =
( X1 − X 2 ) − ( m1 − m2 ) ~ N (0,1)
2
2
ss
1 + 2
n1 n2
if X1 and X 2 are normally distributed
If the sample is drawn from a finite
population (without replacement)
and n / N > 0.05 , then use
σ2 N − n
2
V (X ) =
σX = ×
n N −1
(or n1 and n2 are large).
Proportion:
( )
V ( Pˆ )
E Pˆ
Z =
=µ Pˆ =p
=σ2Pˆ =
p (1 − p )
n
Pˆ − p
p (1 − p ) n
~ N (0,1)
if np ≥ 5 and n(1 − p ) ≥ 5
6
Estimating and testing μ (σ known)
Estimating and testing p
Condition: X is normally distributed (or n
is large) and σ is known.
Exact test statistic: X ~ Bin(n, p )
Confidence interval: x ± zα 2
Test statistic: Z =
Pˆ − p
~ N (0,1)
p (1 − p ) n
if np ≥ 5 and n(1 − p ) ≥ 5
Test statistic: Z =
σ
n
pˆ (1 − pˆ )
n
if npˆ ≥ 5 and n(1 − pˆ ) ≥ 5
X− µ
~ N (0,1)
σ n
Confidence interval: pˆ ± zα 2
P-value of the test:
P ( Z > z ) if H1 : µ > µ0
P ( Z < z ) if H1 : µ < µ0
2⋅ P(Z > z
)
Sign test for one variable
if H1 : µ ≠ µ0
Condition: ordinal or quantitative data
Estimating and testing μ (σ unknown)
Sign test is test for the median (M), if the
distribution is continuous
Condition: X is (approximately) normally
distributed (or n is large) and σ is
unknown.
Define differences with respect to a
reference value
n = ’number of nonzero differences’
X = ’number of positive differences’
Test statistic:
X −µ
T=
~ t [df = n − 1]
S n
Confidence interval: x ± tα 2;n−1
Exact test statistic: X ~ Bin(n, p = 0.5)
X c − 0.5n
~ N (0,1)
0.5 n
(with continuity correction)
if n ≥ 10
s
n
Test statistic: Z =
Estimating and testing σ2
Condition: X is normally distributed.
Test statistic:
( n − 1) S 2 ~ χ2 [df = n − 1]
χ2 =
σ2
Confidence interval:
 ( n − 1) s 2 ( n − 1) s 2 


;
 χα2 2;n−1 χ12−α 2;n−1 


7
Confidence interval:
PART 2 (Intermediate Statistics)
( x1 − x2 ) ± tα /2 ⋅
(A)
PARAMETRIC INFERENCE FOR
COMPARISONS BETWEEN TWO
POPULATIONS
(A3)
COMPARING TWO MEANS USING
MATCHED PAIRS
(A1)
COMPARING TWO MEANS USING
INDEPENDENT SAMPLES
WHEN THE POPULATIONS HAVE
EQUAL VARIANCES
Test statistic:
X D − µD
=
T
~ t [df
= nD − 1]
SD
nD
Test statistic:
( X − X 2 ) − (µ1 − µ 2 )
t= 1
~ t [df = n1 + n2 − 2]
1 
2 1
Sp  + 
 n1 n2 
Confidence interval:
s
xD ± tα /2 ⋅ D
nD
Confidence interval:
(A4)
COMPARING TWO VARIANCES
1 1 
( x1 − x2 ) ± tα /2 s 2p  + 
 n1 n2 
P( F > Fα,n1 −1,n2 −1 ) =
α
Where:
(n − 1) s12 + (n2 − 1) s22
s 2p = 1
n1 + n2 − 2
F1−α,n1 −1,n2 −1 =
2
Fα,n2 −1,n1 −1
Confidence interval:
  s2 

 s2 
1
,  12  × Fα /2,n2 −1,n1 −1 
  12  ×
  s  Fα /2,n −1,n −1  s 

 2
 2 

1
2
Test statistic:
( X − X 2 ) − (µ1 − µ 2 )
T= 1
~ t[df ]
S12 S22
+
n1 n2
Where:
df =
1
Test statistic:
S12
F=
~ F [df numerator =
n1 − 1& df denominator =
n2 − 1]
S22
(A2)
COMPARING TWO MEANS USING
INDEPENDENT SAMPLES WHEN
THE POPULATIONS HAVE
UNEQUAL VARIANCES
 s12 s22 
 + 
 n1 n2 
s12 s22
+
n1 n2
2
2
 s12 
 s22 
n
/
(
−
1)
+
 
  / (n2 − 1)
1
n
1
 
 n2 
8
(A5)
COMPARING TWO PROPORTIONS
(B)
INFERENCE FOR NOMINAL
VARIABLES
Test statistic, if the null hypothesis states
a difference equal to 0:
(B1)
GOODNESS-OF-FIT TEST
Z=
( Pˆ1 − Pˆ2 )
Test statistic:
k
( fi − ei ) 2
2
χ = ∑
~ χ 2 [df = k − 1]
ei
i =1
~ N (0,1)
1 1 
Pˆ (1 − Pˆ )  + 
 n1 n2 
n pˆ + n pˆ
Where: pˆ = 1 1 2 2
n1 + n2
n1 pˆ ≥ 5 & n1 (1 − pˆ ) ≥ 5
Where: ei = npi
If: ei ≥ 5
n2 pˆ ≥ 5 & n2 (1 − pˆ ) ≥ 5
(B2)
TEST FOR INDEPENDENCE
Test statistic, if the null hypothesis states
Test statistic:
a difference unequal to 0:
r c ( f − e )2
r×c
( f − e )2
ij
ij
=
c 2 ∑∑= ∑ i i ~ c 2
eij
ei
=i 1 =j 1
=i 1
( Pˆ1 − Pˆ2 ) − ( p1 − p2 )
Z=
~ N (0,1)
[df =(r − 1)(c − 1)]
Pˆ1 (1 − Pˆ1 ) Pˆ2 (1 − Pˆ2 )
+
Where:
n1
n2
Row i total × Column j total fi × f j
=
eij =
n
n
Where: n1 pˆ1 ≥ 5 & n1 (1 − pˆ1 ) ≥ 5
If: eij ≥ 5
n2 pˆ 2 ≥ 5 & n2 (1 − pˆ 2 ) ≥ 5
(C)
NON PARAMETRIC INFERENCE
FOR COMPARISONS BETWEEN
TWO POPULATIONS
Confidence interval:
( pˆ1 − pˆ 2 ) ± zα /2
pˆ1 (1 − pˆ1 ) pˆ 2 (1 − pˆ 2 )
+
n1
n2
(C1)
USING INDEPENDENT SAMPLES
Where: n1 pˆ1 ≥ 5 & n1 (1 − pˆ1 ) ≥ 5
n2 pˆ 2 ≥ 5 & n2 (1 − pˆ 2 ) ≥ 5
Test statistic
Wilcoxon rank sum test:
T = T1
Test statistic:
n (n + n + 1)
T− 1 1 2
2
Z=
~ N (0,1)
n1n2 (n1 + n2 + 1)
12
Where: T = T1
If: n1 > 10, n2 > 10
9
(C2)
USING MATCHED PAIRS
(D)
LILLIEFORS TEST FOR
NORMALITY
CASE 1: ORDINAL DATA
Test statistic:
D = max { D+ , D− }
(ORDINAL LEVEL OF
MEASUREMENT IN THE
POPULATION)
Where :
Test statistic
Sign test:
X ~ Bin(n, p = 0.5)
Where:
X = ‘number of positive differences’
n = ‘number of nonzero differences’
=
D+
=
D−
Where: Y ~ N (m y = x , s2y = s x2 )
# obs ≤ xi
n
# obs < xi
S− ( xi ) =
n
S+ ( xi ) =
(E)
REGRESSION ANALYSIS
(E1)
APPLICABLE FOR k = 1
s xy
b1 = 2
sx
CASE 2: QUANTITATIVE DATA
b0= y − b1 x
(INTERVAL LEVEL OF
MEASUREMENT IN THE
POPULATION)
T =T
1 n
s xy
Where: =
∑ ( xi − x )( yi − y )
n − 1 i =1
sx2
=
1 n
( xi − x ) 2
∑
n − 1 i =1
yˆ=
i b0 + b1 xi
e=
yi − yˆi
i
+
Test statistic:
n(n + 1)
T−
4
=
z
~ N (=
µ 0 &=
σ 1)
n(n + 1)(2n + 1)
24
Where: T = T +
If: n > 30
max F ( xi ) − S− ( xi )
i =1,...,n
F (=
xi ) P (Y ≤ xi )
Test statistic:
X C − 0.5n
~ N (=
Z
=
µ 0 &=
σ 1)
0.5 n
( with continuity correction )
If: n ≥ 10
Test statistic
Wilcoxon signed rank sum test:
max F ( xi ) − S+ ( xi )
i =1,...,n
n
SST= s 2y (n − 1)=
∑ ( yi − y )2
n
n
SSE =
i =1
∑ ( yi − yˆi )2 =
∑ ei2 =
=i 1 =i 1
2

2 s xy
=
(n − 1)  s y − 2

sx

=
SSR
n
∑ ( yˆi − y )2
i =1
SST
= SSR + SSE
10




Test statistic:
s
where a rank(
rs = ab =
=
x), b rank( y )
sa sb
2
SSR
SSE s xy
R = =
1−
=
SST
SST sx2 s 2y
2
SSE
n−2
MSR=SSR
sε = MSE
sε
=
sb1 =
(n − 1) s x2
MSE =
Test statistic:
=
Z rs n − 1 ~ N (μ=0 & σ=1)
If: n > 30
sε
n
∑ ( xi − x )
Standardized residual:
2
i =1
(Excel uses:
Test statistic:
b −β
T= 1 1 ~ t [df = n − 2]
sb1
)
(E2)
APPLICABLE FOR k ≥ 1
(1 − R2 ) / (n − 2)
~
s=
xj y
F [df numerator= 1& df denominator= n − 2]
s x2 j
=
Prediction interval:
2
1 ( xg − x )
+
n (n − 1) sx2
1 n
∑ ( xij − x j )( yi − y )
n − 1 i =1
1 n
( xij − x j ) 2
∑
n − 1 i =1
k
yˆ=
i b0 + ∑ b j xij
j =1
e=
yi − yˆi
i
Confidence interval:
1 ( xg − x )
+
n (n − 1) sx2
SSE =
2
yˆ ± tα /2;n−2 × sε ×
SSE / ( n − 1)
1 ( x − x )2
se 1 − − i
n (n − 1) s x2
R2
yˆ ± tα /2;n−2 × sε × 1 +
ei
Studentized residual:
ei
Confidence interval:
b1 ± tα /2 × sb1
Test statistic:
MSR
2
=
=
F T=
MSE
ei
se
Coefficient of correlation:
s xy
r=
( r 2 = R2 )
sx s y
n
n
∑ ( yi − yˆi )2 = ∑ ei2
=i 1 =i 1
n
=
SSR
( yˆi − y ) 2
i =1
n
SST= s 2y (n − 1)=
( yi −
i =1
∑
∑
SST
= SSR + SSE
SSR
SSE
R2 =
= 1−
SST
SST
SSE
MSE =
n − k −1
MSR = SSR / k
sε = MSE
Test statistic:
n−2
T=
r⋅
~ t [df =
n − 2]
1− r2
11
y )2
Test statistic:
bj − β j
~ t [df = n − k − 1]
T=
sb j
Multicollinearity
Confidence interval:
b j ± tα /2 × sb j
Outlier
If | ‘Standardized residual’ | ≥ 3
(or | ‘Studentized residual’ | ≥ 3)
If rx h x j ≥ 0.7
Test statistic for testing the model:
MSR
R2 / k
=
F =
~
MSE 1 − R 2 / (n − k − 1)
(
)
F [df numerator = k & df denominator = n − k − 1]
Adjusted R2 = 1 −
MSE
s 2y
Partial F test:
(SSR c − SSR r ) / q
F=
MSE c
or
Rc2 − Rr2 ) / q
(
F=
~F
(1 − Rc2 ) / (n − k − 1)
[df numerator = q & df denominator = n − k − 1]
White heteroscedasticity regression (no
cross terms):
e2 = γ 0 + γ1 x1 + ... + γ k xk
+ γ k +1 x12 + ... + γ 2 k xk2 + ε*
e denotes the OLS residual(s)
First order autocorrelation regression:
=
ei ρ ei −1 + e*i
Durbin-Watson statistic:
n
d=
∑ (ei − ei−1 )2
i =2
n
∑ ei2
i =1
Ramsey RESET linearity regression:
y = β 0 + β1 x1 + ... + β k xk + γ yˆ 2 + ε*
12
Table 1
Binomial Probabilities
k
Tabulated values are P ( X ≤ k ) =
∑ p( x). Values are rounded to three decimal places.
x =0
n=5
p
k
0
1
2
3
4
.01
.951
.999
1.000
1.000
1.000
.05
.774
.977
.999
1.000
1.000
.10
.590
.919
.991
1.000
1.000
.20
.328
.737
.942
.993
1.000
.25
.237
.633
.896
.984
.999
.30
.168
.528
.837
.969
.998
.40
.078
.337
.683
.913
.990
.01
.941
.999
1.000
1.000
1.000
1.000
.05
.735
.967
.998
1.000
1.000
1.000
.10
.531
.886
.984
.999
1.000
1.000
.20
.262
.655
.901
.983
.998
1.000
.25
.178
.534
.831
.962
.995
1.000
.30
.118
.420
.744
.930
.989
.999
.40
.047
.233
.544
.821
.959
.996
.01
.932
.998
1.000
1.000
1.000
1.000
1.000
.05
.698
.956
.996
1.000
1.000
1.000
1.000
.10
.478
.850
.974
.997
1.000
1.000
1.000
.20
.210
.577
.852
.967
.995
1.000
1.000
.25
.133
.445
.756
.929
.987
.999
1.000
.30
.082
.329
.647
.874
.971
.996
1.000
.40
.028
.159
.420
.710
.904
.981
.998
.01
.923
.997
1.000
1.000
1.000
1.000
1.000
1.000
.05
.663
.943
.994
1.000
1.000
1.000
1.000
1.000
.10
.430
.813
.962
.995
1.000
1.000
1.000
1.000
.20
.168
.503
.797
.944
.990
.999
1.000
1.000
.25
.100
.367
.679
.886
.973
.996
1.000
1.000
.30
.058
.255
.552
.806
.942
.989
.999
1.000
.40
.017
.106
.315
.594
.826
.950
.991
.999
.50
.031
.188
.500
.813
.969
.60
.010
.087
.317
.663
.922
.70
.002
.031
.163
.472
.832
.75
.001
.016
.104
.367
.763
.80
.000
.007
.058
.263
.672
.90
.000
.000
.009
.081
.410
.95
.000
.000
.001
.023
.226
.99
.000
.000
.000
.001
.049
.50
.016
.109
.344
.656
.891
.984
.60
.004
.041
.179
.456
.767
.953
.70
.001
.011
.070
.256
.580
.882
.75
.000
.005
.038
.169
.466
.822
.80
.000
.002
.017
.099
.345
.738
.90
.000
.000
.001
.016
.114
.469
.95
.000
.000
.000
.002
.033
.265
.99
.000
.000
.000
.000
.001
.059
.50
.008
.063
.227
.500
.773
.938
.992
.60
.002
.019
.096
.290
.580
.841
.972
.70
.000
.004
.029
.126
.353
.671
.918
.75
.000
.001
.013
.071
.244
.555
.867
.80
.000
.000
.005
.033
.148
.423
.790
.90
.000
.000
.000
.003
.026
.150
.522
.95
.000
.000
.000
.000
.004
.044
.302
.99
.000
.000
.000
.000
.000
.002
.068
.50
.004
.035
.145
.363
.637
.855
.965
.996
.60
.001
.009
.050
.174
.406
.685
.894
.983
.70
.000
.001
.011
.058
.194
.448
.745
.942
.75
.000
.000
.004
.027
.114
.321
.633
.900
.80
.000
.000
.001
.010
.056
.203
.497
.832
.90
.000
.000
.000
.000
.005
.038
.187
.570
.95
.000
.000
.000
.000
.000
.006
.057
.337
.99
.000
.000
.000
.000
.000
.000
.003
.077
n=6
p
k
0
1
2
3
4
5
n=7
p
k
0
1
2
3
4
5
6
n=8
p
k
0
1
2
3
4
5
6
7
13
Table 1 continued
Binomial Probabilities
k
Tabulated values are P ( X ≤ k ) =
∑ p( x). Values are rounded to three decimal places.
x =0
n=9
p
k
0
1
2
3
4
5
6
7
8
.01
.914
.997
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.05
.630
.929
.992
.999
1.000
1.000
1.000
1.000
1.000
.10
.387
.775
.947
.992
.999
1.000
1.000
1.000
1.000
.20
.134
.436
.738
.914
.980
.997
1.000
1.000
1.000
.25
.075
.300
.601
.834
.951
.990
.999
1.000
1.000
.30
.040
.196
.463
.730
.901
.975
.996
1.000
1.000
.40
.010
.071
.232
.483
.733
.901
.975
.996
1.000
.05
.599
.914
.988
.999
1.000
1.000
1.000
1.000
1.000
1.000
.10
.349
.736
.930
.987
.998
1.000
1.000
1.000
1.000
1.000
.20
.107
.376
.678
.879
.967
.994
.999
1.000
1.000
1.000
.25
.056
.244
.526
.776
.922
.980
.996
1.000
1.000
1.000
.30
.028
.149
.383
.650
.850
.953
.989
.998
1.000
1.000
.40
.006
.046
.167
.382
.633
.834
.945
.988
.998
1.000
.05
.463
.829
.964
.995
.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.10
.206
.549
.816
.944
.987
.998
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.20
.035
.167
.398
.648
.836
.939
.982
.996
.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.25
.013
.080
.236
.461
.686
.852
.943
.983
.996
.999
1.000
1.000
1.000
1.000
1.000
1.000
.30
.005
.035
.127
.297
.515
.722
.869
.950
.985
.996
.999
1.000
1.000
1.000
1.000
1.000
.40
.000
.005
.027
.091
.217
.403
.610
.787
.905
.966
.991
.998
1.000
1.000
1.000
1.000
.50
.002
.020
.090
.254
.500
.746
.910
.980
.998
.60
.000
.004
.025
.099
.267
.517
.768
.929
.990
.70
.000
.000
.004
.025
.099
.270
.537
.804
.960
.75
.000
.000
.001
.010
.049
.166
.399
.700
.925
.80
.000
.000
.000
.003
.020
.086
.262
.564
.866
.90
.000
.000
.000
.000
.001
.008
.053
.225
.613
.95
.000
.000
.000
.000
.000
.001
.008
.071
.370
.99
.000
.000
.000
.000
.000
.000
.000
.003
.086
.50
.001
.011
.055
.172
.377
.623
.828
.945
.989
.999
.60
.000
.002
.012
.055
.166
.367
.618
.833
.954
.994
.70
.000
.000
.002
.011
.047
.150
.350
.617
.851
.972
.75
.000
.000
.000
.004
.020
.078
.224
.474
.756
.944
.80
.000
.000
.000
.001
.006
.033
.121
.322
.624
.893
.90
.000
.000
.000
.000
.000
.002
.013
.070
.264
.651
.95
.000
.000
.000
.000
.000
.000
.001
.012
.086
.401
.99
.000
.000
.000
.000
.000
.000
.000
.000
.004
.096
.50
.000
.000
.004
.018
.059
.151
.304
.500
.696
.849
.941
.982
.996
1.000
1.000
1.000
.60
.000
.000
.000
.002
.009
.034
.095
.213
.390
.597
.783
.909
.973
.995
1.000
1.000
.70
.000
.000
.000
.000
.001
.004
.015
.050
.131
.278
.485
.703
.873
.965
.995
1.000
.75
.000
.000
.000
.000
.000
.001
.004
.017
.057
.148
.314
.539
.764
.920
.987
1.000
.80
.000
.000
.000
.000
.000
.000
.001
.004
.018
.061
.164
.352
.602
.833
.965
1.000
.90
.000
.000
.000
.000
.000
.000
.000
.000
.000
.002
.013
.056
.184
.451
.794
1.000
.95
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.001
.005
.036
.171
.537
1.000
.99
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.010
.140
1.000
n = 10
p
k
0
1
2
3
4
5
6
7
8
9
.01
.904
.996
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
n = 15
p
k
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
.01
.860
.990
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
14
Table 1 continued
Binomial Probabilities
k
Tabulated values are P ( X ≤ k ) =
∑ p( x). Values are rounded to three decimal places.
x =0
n = 20
p
k
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
.01
.818
.983
.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.05
.358
.736
.925
.984
.997
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.10
.122
.392
.677
.867
.957
.989
.998
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.20
.012
.069
.206
.411
.630
.804
.913
.968
.990
.997
.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.25
.003
.024
.091
.225
.415
.617
.786
.898
.959
.986
.996
.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.30
.001
.008
.035
.107
.238
.416
.608
.772
.887
.952
.983
.995
.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.40
.000
.001
.004
.016
.051
.126
.250
.416
.596
.755
.872
.943
.979
.994
.998
1.000
1.000
1.000
1.000
1.000
.05
.277
.642
.873
.966
.993
.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.10
.072
.271
.537
.764
.902
.967
.991
.998
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.20
.004
.027
.098
.234
.421
.617
.780
.891
.953
.983
.994
.998
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.25
.001
.007
.032
.096
.214
.378
.561
.727
.851
.929
.970
.989
.997
.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.30
.000
.002
.009
.033
.090
.193
.341
.512
.677
.811
.902
.956
.983
.994
.998
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.40
.000
.000
.000
.002
.009
.029
.074
.154
.274
.425
.586
.732
.846
.922
.966
.987
.996
.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.50
.000
.000
.000
.001
.006
.021
.058
.132
.252
.412
.588
.748
.868
.942
.979
.994
.999
1.000
1.000
1.000
.60
.000
.000
.000
.000
.000
.002
.006
.021
.057
.128
.245
.404
.584
.750
.874
.949
.984
.996
.999
1.000
.70
.000
.000
.000
.000
.000
.000
.000
.001
.005
.017
.048
.113
.228
.392
.584
.762
.893
.965
.992
.999
.75
.000
.000
.000
.000
.000
.000
.000
.000
.001
.004
.014
.041
.102
.214
.383
.585
.775
.909
.976
.997
.80
.000
.000
.000
.000
.000
.000
.000
.000
.000
.001
.003
.010
.032
.087
.196
.370
.589
.794
.931
.988
.90
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.002
.011
.043
.133
.323
.608
.878
.95
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.003
.016
.075
.264
.642
.99
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.001
.017
.182
.50
.000
.000
.000
.000
.000
.002
.007
.022
.054
.115
.212
.345
.500
.655
.788
.885
.946
.978
.993
.998
1.000
1.000
1.000
1.000
1.000
.60
.000
.000
.000
.000
.000
.000
.000
.001
.004
.013
.034
.078
.154
.268
.414
.575
.726
.846
.926
.971
.991
.998
1.000
1.000
1.000
.70
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.002
.006
.017
.044
.098
.189
.323
.488
.659
.807
.910
.967
.991
.998
1.000
.75
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.001
.003
.011
.030
.071
.149
.273
.439
.622
.786
.904
.968
.993
.999
.80
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.002
.006
.017
.047
.109
.220
.383
.579
.766
.902
.973
.996
.90
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.002
.009
.033
.098
.236
.463
.729
.928
.95
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.001
.007
.034
.127
.358
.723
.99
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.002
.026
.222
n = 25
p
k
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
.01
.778
.974
.998
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
15
Table 2
Poisson Probabilities
k
Tabulated values are P ( X ≤ k ) =
∑ p( x). Values are rounded to three decimal places.
x =0
µ
k
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
.1
.905
.995
1.000
.2
.819
.982
.999
1.000
.3
.741
.963
.996
1.000
.4
.670
.938
.992
.999
1.000
.5
.607
.910
.986
.998
1.000
1.0
.368
.736
.920
.981
.996
.999
1.000
1.5
.223
.558
.809
.934
.981
.996
.999
1.000
2.0
.135
.406
.677
.857
.947
.983
.995
.999
1.000
16
2.5
.082
.287
.544
.758
.891
.958
.986
.996
.999
1.000
3.0
.050
.199
.423
.647
.815
.916
.966
.988
.996
.999
1.000
3.5
.030
.136
.321
.537
.725
.858
.935
.973
.990
.997
.999
1.000
4.0
.018
.092
.238
.433
.629
.785
.889
.949
.979
.992
.997
.999
1.000
4.5
.011
.061
.174
.342
.532
.703
.831
.913
.960
.983
.993
.998
.999
1.000
5.0
.007
.040
.125
.265
.440
.616
.762
.867
.932
.968
.986
.995
.998
.999
1.000
5.5
.004
.027
.088
.202
.358
.529
.686
.809
.894
.946
.975
.989
.996
.998
.999
1.000
Table 2 continued
Poisson Probabilities
k
Tabulated values are P ( X ≤ k ) =
∑ p( x). Values are rounded to three decimal places.
x =0
µ
k
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
6.0
.002
.017
.062
.151
.285
.446
.606
.744
.847
.916
.957
.980
.991
.996
.999
.999
1.000
6.5
.002
.011
.043
.112
.224
.369
.527
.673
.792
.877
.933
.966
.984
.993
.997
.999
1.000
7.0
.001
.007
.030
.082
.173
.301
.450
.599
.729
.830
.901
.947
.973
.987
.994
.998
.999
1.000
7.5
.001
.005
.020
.059
.132
.241
.378
.525
.662
.776
.862
.921
.957
.978
.990
.995
.998
.999
1.000
8.0
.000
.003
.014
.042
.100
.191
.313
.453
.593
.717
.816
.888
.936
.966
.983
.992
.996
.998
.999
1.000
8.5
.000
.002
.009
.030
.074
.150
.256
.386
.523
.653
.763
.849
.909
.949
.973
.986
.993
.997
.999
.999
1.000
9.0
.000
.001
.006
.021
.055
.116
.207
.324
.456
.587
.706
.803
.876
.926
.959
.978
.989
.995
.998
.999
1.000
9.5
.000
.001
.004
.015
.040
.089
.165
.269
.392
.522
.645
.752
.836
.898
.940
.967
.982
.991
.996
.998
.999
1.000
17
10.0
.000
.000
.003
.010
.029
.067
.130
.220
.333
.458
.583
.697
.792
.864
.917
.951
.973
.986
.993
.997
.998
.999
1.000
11.0
.000
.000
.001
.005
.015
.038
.079
.143
.232
.341
.460
.579
.689
.781
.854
.907
.944
.968
.982
.991
.995
.998
.999
1.000
12.0
.000
.000
.001
.002
.008
.020
.046
.090
.155
.242
.347
.462
.576
.682
.772
.844
.899
.937
.963
.979
.988
.994
.997
.999
.999
1.000
13.0
.000
.000
.000
.001
.004
.011
.026
.054
.100
.166
.252
.353
.463
.573
.675
.764
.835
.890
.930
.957
.975
.986
.992
.996
.998
.999
1.000
14.0
.000
.000
.000
.000
.002
.006
.014
.032
.062
.109
.176
.260
.358
.464
.570
.669
.756
.827
.883
.923
.952
.971
.983
.991
.995
.997
.999
.999
1.000
15.0
.000
.000
.000
.000
.001
.003
.008
.018
.037
.070
.118
.185
.268
.363
.466
.568
.664
.749
.819
.875
.917
.947
.967
.981
.989
.994
.997
.998
.999
1.000
Table 3
Standard Normal Curve Areas
Tabulated values are P( Z ≤ z) with Z ~ N (0,1).
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
.00
.5000
.5398
.5793
.6179
.6554
.6915
.7257
.7580
.7881
.8159
.8413
.8643
.8849
.9032
.9192
.9332
.9452
.9554
.9641
.9713
.9772
.9821
.9861
.9893
.9918
.9938
.9953
.9965
.9974
.9981
.9987
.01
.5040
.5438
.5832
.6217
.6591
.6950
.7291
.7611
.7910
.8186
.8438
.8665
.8869
.9049
.9207
.9345
.9463
.9564
.9649
.9719
.9778
.9826
.9864
.9896
.9920
.9940
.9955
.9966
.9975
.9982
.9987
.02
.5080
.5478
.5871
.6255
.6628
.6985
.7324
.7642
.7939
.8212
.8461
.8686
.8888
.9066
.9222
.9357
.9474
.9573
.9656
.9726
.9783
.9830
.9868
.9898
.9922
.9941
.9956
.9967
.9976
.9982
.9987
.03
.5120
.5517
.5910
.6293
.6664
.7019
.7357
.7673
.7967
.8238
.8485
.8708
.8907
.9082
.9236
.9370
.9484
.9582
.9664
.9732
.9788
.9834
.9871
.9901
.9925
.9943
.9957
.9968
.9977
.9983
.9988
.04
.5160
.5557
.5948
.6331
.6700
.7054
.7389
.7704
.7995
.8264
.8508
.8729
.8925
.9099
.9251
.9382
.9495
.9591
.9671
.9738
.9793
.9838
.9875
.9904
.9927
.9945
.9959
.9969
.9977
.9984
.9988
18
.05
.5199
.5596
.5987
.6368
.6736
.7088
.7422
.7734
.8023
.8289
.8531
.8749
.8944
.9115
.9265
.9394
.9505
.9599
.9678
.9744
.9798
.9842
.9878
.9906
.9929
.9946
.9960
.9970
.9978
.9984
.9989
.06
.5239
.5636
.6026
.6406
.6772
.7123
.7454
.7764
.8051
.8315
.8554
.8770
.8962
.9131
.9279
.9406
.9515
.9608
.9686
.9750
.9803
.9846
.9881
.9909
.9931
.9948
.9961
.9971
.9979
.9985
.9989
.07
.5279
.5675
.6064
.6443
.6808
.7157
.7486
.7794
.8078
.8340
.8577
.8790
.8980
.9147
.9292
.9418
.9525
.9616
.9693
.9756
.9808
.9850
.9884
.9911
.9932
.9949
.9962
.9972
.9979
.9985
.9989
.08
.5319
.5714
.6103
.6480
.6844
.7190
.7517
.7823
.8106
.8365
.8599
.8810
.8997
.9162
.9306
.9429
.9535
.9625
.9699
.9761
.9812
.9854
.9887
.9913
.9934
.9951
.9963
.9973
.9980
.9986
.9990
.09
.5359
.5753
.6141
.6517
.6879
.7224
.7549
.7852
.8133
.8389
.8621
.8830
.9015
.9177
.9319
.9441
.9545
.9633
.9706
.9767
.9817
.9857
.9890
.9916
.9936
.9952
.9964
.9974
.9981
.9986
.9990
Table 4
Critical Values of t
Degrees of
freedom
t .10
t .05
t .025
t .01
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
3.078
1.886
1.638
1.533
1.476
1.440
1.415
1.397
1.383
1.372
1.363
1.356
1.350
1.345
1.341
1.337
1.333
1.330
1.328
1.325
1.323
1.321
1.319
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
31.821
6.965
4.541
3.747
3.365
3.143
2.998
2.896
2.821
2.764
2.718
2.681
2.650
2.624
2.602
2.583
2.567
2.552
2.539
2.528
2.518
2.508
2.500
t .005 Degrees of
freedom
63.657
9.925
5.841
4.604
4.032
3.707
3.499
3.355
3.250
3.169
3.106
3.055
3.012
2.977
2.947
2.921
2.898
2.878
2.861
2.845
2.831
2.819
2.807
19
24
25
26
27
28
29
30
35
40
45
50
60
70
80
90
100
120
140
160
180
200
∞
t .10
t .05
t .025
t .01
t .005
1.318
1.316
1.315
1.314
1.313
1.311
1.310
1.306
1.303
1.301
1.299
1.296
1.294
1.292
1.291
1.290
1.289
1.288
1.287
1.286
1.286
1.282
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.690
1.684
1.679
1.676
1.671
1.667
1.664
1.662
1.660
1.658
1.656
1.654
1.653
1.653
1.645
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.030
2.021
2.014
2.009
2.000
1.994
1.990
1.987
1.984
1.980
1.977
1.975
1.973
1.972
1.960
2.492
2.485
2.479
2.473
2.467
2.462
2.457
2.438
2.423
2.412
2.403
2.390
2.381
2.374
2.368
2.364
2.358
2.353
2.350
2.347
2.345
2.326
2.797
2.787
2.779
2.771
2.763
2.756
2.750
2.724
2.704
2.690
2.678
2.660
2.648
2.639
2.632
2.626
2.617
2.611
2.607
2.603
2.601
2.576
Table 5
Critical Values of χ2
Degrees of
freedom
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
χ 2.995
χ 2.99
χ 2.975
χ 2.95
χ 2.90
χ 2.10
χ 2.05
χ 2.025
χ 2.01
χ 2.005
.0000393
.0100
.0717
.207
.412
.676
.989
1.344
1.735
2.156
2.603
3.074
3.565
4.075
4.601
5.142
5.697
6.265
6.844
7.434
8.034
8.643
9.260
9.886
10.520
11.160
11.808
12.461
13.121
13.787
.000157
.0201
.115
.297
.554
.872
1.239
1.646
2.088
2.558
3.053
3.571
4.107
4.660
5.229
5.812
6.408
7.015
7.633
8.260
8.897
9.542
10.196
10.856
11.524
12.198
12.879
13.565
14.256
14.953
.000982
.0506
.216
.484
.831
1.237
1.690
2.180
2.700
3.247
3.816
4.404
5.009
5.629
6.262
6.908
7.564
8.231
8.907
9.591
10.283
10.982
11.689
12.401
13.120
13.844
14.573
15.308
16.047
16.791
.003932
.103
.352
.711
1.145
1.635
2.167
2.733
3.325
3.940
4.575
5.226
5.892
6.571
7.261
7.962
8.672
9.390
10.117
10.851
11.591
12.338
13.091
13.848
14.611
15.379
16.151
16.928
17.708
18.493
.0158
.211
.584
1.064
1.610
2.204
2.833
3.490
4.168
4.865
5.578
6.304
7.042
7.790
8.547
9.312
10.085
10.865
11.651
12.443
13.240
14.041
14.848
15.659
16.473
17.292
18.114
18.939
19.768
20.599
2.706
4.605
6.251
7.779
9.236
10.645
12.017
13.362
14.684
15.987
17.275
18.549
19.812
21.064
22.307
23.542
24.769
25.989
27.204
28.412
29.615
30.813
32.007
33.196
34.382
35.563
36.741
37.916
39.087
40.256
3.841
5.991
7.815
9.488
11.070
12.592
14.067
15.507
16.919
18.307
19.675
21.026
22.362
23.685
24.996
26.296
27.587
28.869
30.144
31.410
32.671
33.924
35.172
36.415
37.652
38.885
40.113
41.337
42.557
43.773
5.024
7.378
9.348
11.143
12.833
14.449
16.013
17.535
19.023
20.483
21.920
23.337
24.736
26.119
27.488
28.845
30.191
31.526
32.852
34.170
35.479
36.781
38.076
39.364
40.646
41.923
43.195
44.461
45.722
46.979
6.635
9.210
11.345
13.277
15.086
16.812
18.475
20.090
21.666
23.209
24.725
26.217
27.688
29.141
30.578
32.000
33.409
34.805
36.191
37.566
38.932
40.289
41.638
42.980
44.314
45.642
46.963
48.278
49.588
50.892
7.879
10.597
12.838
14.860
16.750
18.548
20.278
21.955
23.589
25.188
26.757
28.300
29.819
31.319
32.801
34.267
35.718
37.156
38.582
39.997
41.401
42.796
44.181
45.559
46.928
48.290
49.645
50.993
52.336
53.672
3

2
2 
2
For df ≥ 30 ,
) ≈ df 1 −
− zα
 and χα (df ) ≈ df
9df 
 9df
where zα is from N (0,1) and α ≤ 0.5.
χ12−α (df
3
E.g.
2
(30)
χ0.95

2
2
+ zα
1 −
9df
 9df
3

 ,

3


2
2 
2
2 
2
+ 1.645
≈ 30 1 −
− 1.645
 ≈ 43.77
 ≈ 18.49 & χ0.05 (30) ≈ 30 1 −
270
270
270
270




20
Table 6(a)
Critical Values of F
α=.05
dfnum
D E N O MI N A T O R
D E G R E E S
O F
F R E E D O M
dfden
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120
∞
1
161.4
18.51
10.13
7.71
6.61
5.99
5.59
5.32
5.12
4.96
4.84
4.75
4.67
4.60
4.54
4.49
4.45
4.41
4.38
4.35
4.32
4.30
4.28
4.26
4.24
4.23
4.21
4.20
4.18
4.17
4.08
4.00
3.92
3.84
2
199.5
19.00
9.55
6.94
5.79
5.14
4.74
4.46
4.26
4.10
3.98
3.89
3.81
3.74
3.68
3.63
3.59
3.55
3.52
3.49
3.47
3.44
3.42
3.40
3.39
3.37
3.35
3.34
3.33
3.32
3.23
3.15
3.07
3.00
0.05
NUMERATOR DEGREES OF FREEDOM
3
4
5
6
215.7
224.6
230.2
234.0
19.16
19.25
19.30
19.33
9.28
9.12
9.01
8.94
6.59
6.39
6.26
6.16
5.41
5.19
5.05
4.95
4.76
4.53
4.39
4.28
4.35
4.12
3.97
3.87
4.07
3.84
3.69
3.58
3.86
3.63
3.48
3.37
3.71
3.48
3.33
3.22
3.59
3.36
3.20
3.09
3.49
3.26
3.11
3.00
3.41
3.18
3.03
2.92
3.34
3.11
2.96
2.85
3.29
3.06
2.90
2.79
3.24
3.01
2.85
2.74
3.20
2.96
2.81
2.70
3.16
2.93
2.77
2.66
3.13
2.90
2.74
2.63
3.10
2.87
2.71
2.60
3.07
2.84
2.68
2.57
3.05
2.82
2.66
2.55
3.03
2.80
2.64
2.53
3.01
2.78
2.62
2.51
2.99
2.76
2.60
2.49
2.98
2.74
2.59
2.47
2.96
2.73
2.57
2.46
2.95
2.71
2.56
2.45
2.93
2.70
2.55
2.43
2.92
2.69
2.53
2.42
2.84
2.61
2.45
2.34
2.76
2.53
2.37
2.25
2.68
2.45
2.29
2.18
2.60
2.37
2.21
2.10
21
7
236.8
19.35
8.89
6.09
4.88
4.21
3.79
3.50
3.29
3.14
3.01
2.91
2.83
2.76
2.71
2.66
2.61
2.58
2.54
2.51
2.49
2.46
2.44
2.42
2.40
2.39
2.37
2.36
2.35
2.33
2.25
2.17
2.09
2.01
8
238.9
19.37
8.85
6.04
4.82
4.15
3.73
3.44
3.23
3.07
2.95
2.85
2.77
2.70
2.64
2.59
2.55
2.51
2.48
2.45
2.42
2.40
2.37
2.36
2.34
2.32
2.31
2.29
2.28
2.27
2.18
2.10
2.02
1.94
9
240.5
19.38
8.81
6.00
4.77
4.10
3.68
3.39
3.18
3.02
2.90
2.80
2.71
2.65
2.59
2.54
2.49
2.46
2.42
2.39
2.37
2.34
2.32
2.30
2.28
2.27
2.25
2.24
2.22
2.21
2.12
2.04
1.96
1.88
Table 6(a) continued
Critical Values of F
α=.05
dfnum
D E N O MI N A T O R
D E G R E E S
O F
F R E E D O M
dfden
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120
∞
10
241.9
19.40
8.79
5.96
4.74
4.06
3.64
3.35
3.14
2.98
2.85
2.75
2.67
2.60
2.54
2.49
2.45
2.41
2.38
2.35
2.32
2.30
2.27
2.25
2.24
2.22
2.20
2.19
2.18
2.16
2.08
1.99
1.91
1.83
12
243.9
19.41
8.74
5.91
4.68
4.00
3.57
3.28
3.07
2.91
2.79
2.69
2.60
2.53
2.48
2.42
2.38
2.34
2.31
2.28
2.25
2.23
2.20
2.18
2.16
2.15
2.13
2.12
2.10
2.09
2.00
1.92
1.83
1.75
0.05
NUMERATOR DEGREES OF FREEDOM
15
20
24
30
246.0
248.0
249.1
250.1
19.43
19.45
19.45
19.46
8.70
8.66
8.64
8.62
5.86
5.80
5.77
5.75
4.62
4.56
4.53
4.50
3.94
3.87
3.84
3.81
3.51
3.44
3.41
3.38
3.22
3.15
3.12
3.08
3.01
2.94
2.90
2.86
2.85
2.77
2.74
2.70
2.72
2.65
2.61
2.57
2.62
2.54
2.51
2.47
2.53
2.46
2.42
2.38
2.46
2.39
2.35
2.31
2.40
2.33
2.29
2.25
2.35
2.28
2.24
2.19
2.31
2.23
2.19
2.15
2.27
2.19
2.15
2.11
2.23
2.16
2.11
2.07
2.20
2.12
2.08
2.04
2.18
2.10
2.05
2.01
2.15
2.07
2.03
1.98
2.13
2.05
2.01
1.96
2.11
2.03
1.98
1.94
2.09
2.01
1.96
1.92
2.07
1.99
1.95
1.90
2.06
1.97
1.93
1.88
2.04
1.96
1.91
1.87
2.03
1.94
1.90
1.85
2.01
1.93
1.89
1.84
1.92
1.84
1.79
1.74
1.84
1.75
1.70
1.65
1.75
1.66
1.61
1.55
1.67
1.57
1.52
1.46
22
40
251.1
19.47
8.59
5.72
4.46
3.77
3.34
3.04
2.83
2.66
2.53
2.43
2.34
2.27
2.20
2.15
2.10
2.06
2.03
1.99
1.96
1.94
1.91
1.89
1.87
1.85
1.84
1.82
1.81
1.79
1.69
1.59
1.50
1.39
60
252.2
19.48
8.57
5.69
4.43
3.74
3.30
3.01
2.79
2.62
2.49
2.38
2.30
2.22
2.16
2.11
2.06
2.02
1.98
1.95
1.92
1.89
1.86
1.84
1.82
1.80
1.79
1.77
1.75
1.74
1.64
1.53
1.43
1.32
120
253.3
19.49
8.55
5.66
4.40
3.70
3.27
2.97
2.75
2.58
2.45
2.34
2.25
2.18
2.11
2.06
2.01
1.97
1.93
1.90
1.87
1.84
1.81
1.79
1.77
1.75
1.73
1.71
1.70
1.68
1.58
1.47
1.35
1.22
Table 6(b)
Critical Values of F
α=.025
dfnum
D E N O MI N A T O R
D E G R E E S
O F
F R E E D O M
dfden
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120
∞
1
647.8
38.51
17.44
12.22
10.01
8.81
8.07
7.57
7.21
6.94
6.72
6.55
6.41
6.30
6.20
6.12
6.04
5.98
5.92
5.87
5.83
5.79
5.75
5.72
5.69
5.66
5.63
5.61
5.59
5.57
5.42
5.29
5.15
5.02
2
799.5
39.00
16.04
10.65
8.43
7.26
6.54
6.06
5.71
5.46
5.26
5.10
4.97
4.86
4.77
4.69
4.62
4.56
4.51
4.46
4.42
4.38
4.35
4.32
4.29
4.27
4.24
4.22
4.20
4.18
4.05
3.93
3.80
3.69
0.025
NUMERATOR DEGREES OF FREEDOM
3
4
5
6
864.2
899.6
921.8
937.1
39.17
39.25
39.30
39.33
15.44
15.10
14.88
14.73
9.98
9.60
9.36
9.20
7.76
7.39
7.15
6.98
6.60
6.23
5.99
5.82
5.89
5.52
5.29
5.12
5.42
5.05
4.82
4.65
5.08
4.72
4.48
4.32
4.83
4.47
4.24
4.07
4.63
4.28
4.04
3.88
4.47
4.12
3.89
3.73
4.35
4.00
3.77
3.60
4.24
3.89
3.66
3.50
4.15
3.80
3.58
3.41
4.08
3.73
3.50
3.34
4.01
3.66
3.44
3.28
3.95
3.61
3.38
3.22
3.90
3.56
3.33
3.17
3.86
3.51
3.29
3.13
3.82
3.48
3.25
3.09
3.78
3.44
3.22
3.05
3.75
3.41
3.18
3.02
3.72
3.38
3.15
2.99
3.69
3.35
3.13
2.97
3.67
3.33
3.10
2.94
3.65
3.31
3.08
2.92
3.63
3.29
3.06
2.90
3.61
3.27
3.04
2.88
3.59
3.25
3.03
2.87
3.46
3.13
2.90
2.74
3.34
3.01
2.79
2.63
3.23
2.89
2.67
2.52
3.12
2.79
2.57
2.41
23
7
948.2
39.36
14.62
9.07
6.85
5.70
4.99
4.53
4.20
3.95
3.76
3.61
3.48
3.38
3.29
3.22
3.16
3.10
3.05
3.01
2.97
2.93
2.90
2.87
2.85
2.82
2.80
2.78
2.76
2.75
2.62
2.51
2.39
2.29
8
956.7
39.37
14.54
8.98
6.76
5.60
4.90
4.43
4.10
3.85
3.66
3.51
3.39
3.29
3.20
3.12
3.06
3.01
2.96
2.91
2.87
2.84
2.81
2.78
2.75
2.73
2.71
2.69
2.67
2.65
2.53
2.41
2.30
2.19
9
963.3
39.39
14.47
8.90
6.68
5.52
4.82
4.36
4.03
3.78
3.59
3.44
3.31
3.21
3.12
3.05
2.98
2.93
2.88
2.84
2.80
2.76
2.73
2.70
2.68
2.65
2.63
2.61
2.59
2.57
2.45
2.33
2.22
2.11
Table 6(b) continued
Critical Values of F
α=.025
dfnum
D E N O MI N A T O R
D E G R E E S
O F
F R E E D O M
dfden
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120
∞
10
968.6
39.40
14.42
8.84
6.62
5.46
4.76
4.30
3.96
3.72
3.53
3.37
3.25
3.15
3.06
2.99
2.92
2.87
2.82
2.77
2.73
2.70
2.67
2.64
2.61
2.59
2.57
2.55
2.53
2.51
2.39
2.27
2.16
2.05
12
976.7
39.41
14.34
8.75
6.52
5.37
4.67
4.20
3.87
3.62
3.43
3.28
3.15
3.05
2.96
2.89
2.82
2.77
2.72
2.68
2.64
2.60
2.57
2.54
2.51
2.49
2.47
2.45
2.43
2.41
2.29
2.17
2.05
1.94
0.025
NUMERATOR DEGREES OF FREEDOM
15
20
24
30
984.9
993.1
997.2
1001.4
39.43
39.45
39.46
39.46
14.25
14.17
14.12
14.08
8.66
8.56
8.51
8.46
6.43
6.33
6.28
6.23
5.27
5.17
5.12
5.07
4.57
4.47
4.42
4.36
4.10
4.00
3.95
3.89
3.77
3.67
3.61
3.56
3.52
3.42
3.37
3.31
3.33
3.23
3.17
3.12
3.18
3.07
3.02
2.96
3.05
2.95
2.89
2.84
2.95
2.84
2.79
2.73
2.86
2.76
2.70
2.64
2.79
2.68
2.63
2.57
2.72
2.62
2.56
2.50
2.67
2.56
2.50
2.44
2.62
2.51
2.45
2.39
2.57
2.46
2.41
2.35
2.53
2.42
2.37
2.31
2.50
2.39
2.33
2.27
2.47
2.36
2.30
2.24
2.44
2.33
2.27
2.21
2.41
2.30
2.24
2.18
2.39
2.28
2.22
2.16
2.36
2.25
2.19
2.13
2.34
2.23
2.17
2.11
2.32
2.21
2.15
2.09
2.31
2.20
2.14
2.07
2.18
2.07
2.01
1.94
2.06
1.94
1.88
1.82
1.94
1.82
1.76
1.69
1.83
1.71
1.64
1.57
24
40
1005.6
39.47
14.04
8.41
6.18
5.01
4.31
3.84
3.51
3.26
3.06
2.91
2.78
2.67
2.59
2.51
2.44
2.38
2.33
2.29
2.25
2.21
2.18
2.15
2.12
2.09
2.07
2.05
2.03
2.01
1.88
1.74
1.61
1.48
60
1009.8
39.48
13.99
8.36
6.12
4.96
4.25
3.78
3.45
3.20
3.00
2.85
2.72
2.61
2.52
2.45
2.38
2.32
2.27
2.22
2.18
2.14
2.11
2.08
2.05
2.03
2.00
1.98
1.96
1.94
1.80
1.67
1.53
1.39
120
1014.0
39.49
13.95
8.31
6.07
4.90
4.20
3.73
3.39
3.14
2.94
2.79
2.66
2.55
2.46
2.38
2.32
2.26
2.20
2.16
2.11
2.08
2.04
2.01
1.98
1.95
1.93
1.91
1.89
1.87
1.72
1.58
1.43
1.27
Table 6(c)
Critical Values of F
α=.01
dfnum
D E N O MI N A T O R
D E G R E E S
O F
F R E E D O M
dfden
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120
∞
1
4052.2
98.50
34.12
21.20
16.26
13.75
12.25
11.26
10.56
10.04
9.65
9.33
9.07
8.86
8.68
8.53
8.40
8.29
8.18
8.10
8.02
7.95
7.88
7.82
7.77
7.72
7.68
7.64
7.60
7.56
7.31
7.08
6.85
6.63
2
4999.5
99.00
30.82
18.00
13.27
10.92
9.55
8.65
8.02
7.56
7.21
6.93
6.70
6.51
6.36
6.23
6.11
6.01
5.93
5.85
5.78
5.72
5.66
5.61
5.57
5.53
5.49
5.45
5.42
5.39
5.18
4.98
4.79
4.61
0.01
NUMERATOR DEGREES OF FREEDOM
3
4
5
6
5403.4
5624.6
5763.7
5859.0
99.17
99.25
99.30
99.33
29.46
28.71
28.24
27.91
16.69
15.98
15.52
15.21
12.06
11.39
10.97
10.67
9.78
9.15
8.75
8.47
8.45
7.85
7.46
7.19
7.59
7.01
6.63
6.37
6.99
6.42
6.06
5.80
6.55
5.99
5.64
5.39
6.22
5.67
5.32
5.07
5.95
5.41
5.06
4.82
5.74
5.21
4.86
4.62
5.56
5.04
4.69
4.46
5.42
4.89
4.56
4.32
5.29
4.77
4.44
4.20
5.19
4.67
4.34
4.10
5.09
4.58
4.25
4.01
5.01
4.50
4.17
3.94
4.94
4.43
4.10
3.87
4.87
4.37
4.04
3.81
4.82
4.31
3.99
3.76
4.76
4.26
3.94
3.71
4.72
4.22
3.90
3.67
4.68
4.18
3.85
3.63
4.64
4.14
3.82
3.59
4.60
4.11
3.78
3.56
4.57
4.07
3.75
3.53
4.54
4.04
3.73
3.50
4.51
4.02
3.70
3.47
4.31
3.83
3.51
3.29
4.13
3.65
3.34
3.12
3.95
3.48
3.17
2.96
3.78
3.32
3.02
2.80
25
7
5928.4
99.36
27.67
14.98
10.46
8.26
6.99
6.18
5.61
5.20
4.89
4.64
4.44
4.28
4.14
4.03
3.93
3.84
3.77
3.70
3.64
3.59
3.54
3.50
3.46
3.42
3.39
3.36
3.33
3.30
3.12
2.95
2.79
2.64
8
5981.1
99.37
27.49
14.80
10.29
8.10
6.84
6.03
5.47
5.06
4.74
4.50
4.30
4.14
4.00
3.89
3.79
3.71
3.63
3.56
3.51
3.45
3.41
3.36
3.32
3.29
3.26
3.23
3.20
3.17
2.99
2.82
2.66
2.51
9
6022.5
99.39
27.35
14.66
10.16
7.98
6.72
5.91
5.35
4.94
4.63
4.39
4.19
4.03
3.89
3.78
3.68
3.60
3.52
3.46
3.40
3.35
3.30
3.26
3.22
3.18
3.15
3.12
3.09
3.07
2.89
2.72
2.56
2.41
Table 6(c) continued
Critical Values of F
α=.01
dfnum
D E N O MI N A T O R
D E G R E E S
O F
F R E E D O M
dfden
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
60
120
∞
10
6055.9
99.40
27.23
14.55
10.05
7.87
6.62
5.81
5.26
4.85
4.54
4.30
4.10
3.94
3.80
3.69
3.59
3.51
3.43
3.37
3.31
3.26
3.21
3.17
3.13
3.09
3.06
3.03
3.00
2.98
2.80
2.63
2.47
2.32
12
6106.3
99.42
27.05
14.37
9.89
7.72
6.47
5.67
5.11
4.71
4.40
4.16
3.96
3.80
3.67
3.55
3.46
3.37
3.30
3.23
3.17
3.12
3.07
3.03
2.99
2.96
2.93
2.90
2.87
2.84
2.66
2.50
2.34
2.18
0.01
NUMERATOR DEGREES OF FREEDOM
15
20
24
30
6157.3
6208.7
6234.6
6260.7
99.43
99.45
99.46
99.47
26.87
26.69
26.60
26.50
14.20
14.02
13.93
13.84
9.72
9.55
9.47
9.38
7.56
7.40
7.31
7.23
6.31
6.16
6.07
5.99
5.52
5.36
5.28
5.20
4.96
4.81
4.73
4.65
4.56
4.41
4.33
4.25
4.25
4.10
4.02
3.94
4.01
3.86
3.78
3.70
3.82
3.66
3.59
3.51
3.66
3.51
3.43
3.35
3.52
3.37
3.29
3.21
3.41
3.26
3.18
3.10
3.31
3.16
3.08
3.00
3.23
3.08
3.00
2.92
3.15
3.00
2.92
2.84
3.09
2.94
2.86
2.78
3.03
2.88
2.80
2.72
2.98
2.83
2.75
2.67
2.93
2.78
2.70
2.62
2.89
2.74
2.66
2.58
2.85
2.70
2.62
2.54
2.81
2.66
2.58
2.50
2.78
2.63
2.55
2.47
2.75
2.60
2.52
2.44
2.73
2.57
2.49
2.41
2.70
2.55
2.47
2.39
2.52
2.37
2.29
2.20
2.35
2.20
2.12
2.03
2.19
2.03
1.95
1.86
2.04
1.88
1.79
1.70
26
40
6286.8
99.47
26.41
13.75
9.29
7.14
5.91
5.12
4.57
4.17
3.86
3.62
3.43
3.27
3.13
3.02
2.92
2.84
2.76
2.69
2.64
2.58
2.54
2.49
2.45
2.42
2.38
2.35
2.33
2.30
2.11
1.94
1.76
1.59
60
6313.0
99.48
26.32
13.65
9.20
7.06
5.82
5.03
4.48
4.08
3.78
3.54
3.34
3.18
3.05
2.93
2.83
2.75
2.67
2.61
2.55
2.50
2.45
2.40
2.36
2.33
2.29
2.26
2.23
2.21
2.02
1.84
1.66
1.47
120
6339.4
99.49
26.22
13.56
9.11
6.97
5.74
4.95
4.40
4.00
3.69
3.45
3.25
3.09
2.96
2.84
2.75
2.66
2.58
2.52
2.46
2.40
2.35
2.31
2.27
2.23
2.20
2.17
2.14
2.11
1.92
1.73
1.53
1.32
Table 7
Critical Values of the Wilcoxon Rank Sum Test for Independent Samples
α=.025 one tail; α=.05 two-tail
n1
3
n 2 TL TU
4
5
6
7
8
9
10
-6
7
7
8
8
9
-21
23
26
28
31
33
4
5
TL TU
10
11
12
13
14
14
15
26
29
32
35
38
42
45
6
TL TU
16
17
18
20
21
22
23
34
38
42
45
49
53
57
7
TL TU
23
24
26
27
29
31
32
8
TL TU
43
48
52
57
61
65
70
31
33
34
36
38
40
42
53
58
64
69
74
79
84
9
TL TU
40
42
44
46
49
51
53
64
70
76
82
87
93
99
TL TU
49 77
52 83
55 89
57 96
60 102
62 109
65 115
10
TL TU
60
63
66
69
72
75
78
90
97
104
111
118
125
132
α=.05 one tail; α=.1 two-tail
n1
3
n 2 TL TU
3
4
5
6
7
8
9
10
6
6
7
8
9
9
10
11
15
18
20
22
24
27
29
31
4
5
TL TU
11
11
12
13
14
15
16
17
21
25
28
31
34
37
40
43
6
TL TU
16
17
19
20
21
23
24
26
29
33
36
40
44
47
51
54
7
TL TU
23
24
26
28
29
31
33
35
8
TL TU
37
42
46
50
55
59
63
67
31
32
34
36
39
41
43
45
27
46
52
57
62
66
71
76
81
9
TL TU
39
41
44
46
49
51
54
56
57
63
68
74
79
85
90
96
TL TU
49 68
51 75
54 81
57 87
60 93
63 99
66 105
69 111
10
TL TU
60
62
66
69
72
75
79
82
80
88
94
101
108
115
121
128
Table 8
Critical Values for the Wilcoxon Signed Rank Sum Test
for the Matched Pairs Experiment
n
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
α = .025 one-tail
α = .05 two-tail
TL
TU
1
2
4
6
8
11
14
17
21
25
30
35
40
46
52
59
66
73
81
90
98
107
117
127
137
20
26
32
39
47
55
64
74
84
95
106
118
131
144
158
172
187
203
219
235
253
271
289
308
328
α = .05 one-tail
α = .10 two-tail
TL
TU
2
4
6
8
11
14
17
21
26
30
36
41
47
54
60
68
75
83
92
101
110
120
130
141
152
28
19
24
30
37
44
52
61
70
79
90
100
112
124
136
150
163
178
193
208
224
241
258
276
294
313
Table 9
Critical Value for the Spearman Rank
Correlation Coefficient
The α values correspond to a one-tail test
of H0 : ρs=0. The value should be doubled for
two-tail tests.
n
α = .05 α = .025 α = .01 α = .005
5
.900
--
--
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
.829
.714
.643
.600
.564
.536
.503
.484
.464
.446
.429
.414
.401
.391
.380
.370
.361
.353
.344
.337
.331
.324
.317
.312
.306
.886
.786
.738
.700
.648
.618
.587
.560
.538
.521
.503
.485
.472
.460
.447
.435
.425
.415
.406
.398
.390
.382
.375
.368
.362
.943
.893
.833
.783
.745
.709
.678
.648
.626
.604
.582
.566
.550
.535
.520
.508
.496
.486
.476
.466
.457
.448
.440
.433
.425
---
.929
.881
.833
.794
.755
.727
.703
.679
.654
.635
.615
.600
.584
.570
.556
.554
.532
.521
.511
.501
.491
.483
.475
.467
29
Table 10(a)
Critical Values for the Durbin-Watson d Statistics, α = .05
k =1
n
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
45
50
55
60
65
70
75
80
85
90
95
100
k =2
k =3
k =4
k =5
dL
dU
dL
dU
dL
dU
dL
dU
dL
dU
1.08
1.10
1.13
1.16
1.18
1.20
1.22
1.24
1.26
1.27
1.29
1.30
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.40
1.41
1.42
1.43
1.43
1.44
1.48
1.50
1.53
1.55
1.57
1.58
1.60
1.61
1.62
1.63
1.64
1.65
1.36
1.37
1.38
1.39
1.40
1.41
1.42
1.43
1.44
1.45
1.45
1.46
1.47
1.48
1.48
1.49
1.50
1.50
1.51
1.51
1.52
1.52
1.53
1.54
1.54
1.54
1.57
1.59
1.60
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.69
0.95
0.98
1.02
1.05
1.08
1.10
1.13
1.15
1.17
1.19
1.21
1.22
1.24
1.26
1.27
1.28
1.30
1.31
1.32
1.33
1.34
1.35
1.36
1.37
1.38
1.39
1.43
1.46
1.49
1.51
1.54
1.55
1.57
1.59
1.60
1.61
1.62
1.63
1.54
1.54
1.54
1.53
1.53
1.54
1.54
1.54
1.54
1.55
1.55
1.55
1.56
1.56
1.56
1.57
1.57
1.57
1.58
1.58
1.58
1.59
1.59
1.59
1.60
1.60
1.62
1.63
1.64
1.65
1.66
1.67
1.68
1.69
1.70
1.70
1.71
1.72
0.82
0.86
0.90
0.93
0.97
1.00
1.03
1.05
1.08
1.10
1.12
1.14
1.16
1.18
1.20
1.21
1.23
1.24
1.26
1.27
1.28
1.29
1.31
1.32
1.33
1.34
1.38
1.42
1.45
1.48
1.50
1.52
1.54
1.56
1.57
1.59
1.60
1.61
1.75
1.73
1.71
1.69
1.68
1.68
1.67
1.66
1.66
1.66
1.66
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.65
1.66
1.66
1.66
1.66
1.67
1.67
1.68
1.69
1.70
1.70
1.71
1.72
1.72
1.73
1.73
1.74
0.69
0.74
0.78
0.82
0.86
0.90
0.93
0.96
0.99
1.01
1.04
1.06
1.08
1.10
1.12
1.14
1.16
1.18
1.19
1.21
1.22
1.24
1.25
1.26
1.27
1.29
1.34
1.38
1.41
1.44
1.47
1.49
1.51
1.53
1.55
1.57
1.58
1.59
1.97
1.93
1.90
1.87
1.85
1.83
1.81
1.80
1.79
1.78
1.77
1.76
1.76
1.75
1.74
1.74
1.74
1.73
1.73
1.73
1.73
1.73
1.72
1.72
1.72
1.72
1.72
1.72
1.72
1.73
1.73
1.74
1.74
1.74
1.75
1.75
1.75
1.76
0.56
0.62
0.67
0.71
0.75
0.79
0.83
0.86
0.90
0.93
0.95
0.98
1.01
1.03
1.05
1.07
1.09
1.11
1.13
1.15
1.16
1.18
1.19
1.21
1.22
1.23
1.29
1.34
1.38
1.41
1.44
1.46
1.49
1.51
1.52
1.54
1.56
1.57
2.21
2.15
2.10
2.06
2.02
1.99
1.96
1.94
1.92
1.90
1.89
1.88
1.86
1.85
1.84
1.83
1.83
1.82
1.81
1.81
1.80
1.80
1.80
1.79
1.79
1.79
1.78
1.77
1.77
1.77
1.77
1.77
1.77
1.77
1.77
1.78
1.78
1.78
k does not include the constant
30
Table 10(b)
Critical Values for the Durbin-Watson d Statistics, α = .01
k =1
n
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
45
50
55
60
65
70
75
80
85
90
95
100
k =2
k =3
k =4
k =5
dL
dU
dL
dU
dL
dU
dL
dU
dL
dU
0.81
0.84
0.87
0.90
0.93
0.95
0.97
1.00
1.02
1.04
1.05
1.07
1.09
1.10
1.12
1.13
1.15
1.16
1.17
1.18
1.19
1.21
1.22
1.23
1.24
1.25
1.29
1.32
1.36
1.38
1.41
1.43
1.45
1.47
1.48
1.50
1.51
1.52
1.07
1.09
1.10
1.12
1.13
1.15
1.16
1.17
1.19
1.20
1.21
1.22
1.23
1.24
1.25
1.26
1.27
1.28
1.29
1.30
1.31
1.32
1.32
1.33
1.34
1.34
1.38
1.40
1.43
1.45
1.47
1.49
1.50
1.52
1.53
1.54
1.55
1.56
0.70
0.74
0.77
0.80
0.83
0.86
0.89
0.91
0.94
0.96
0.98
1.00
1.02
1.04
1.05
1.07
1.08
1.10
1.11
1.13
1.14
1.15
1.16
1.18
1.19
1.20
1.24
1.28
1.32
1.35
1.38
1.40
1.42
1.44
1.46
1.47
1.49
1.50
1.25
1.25
1.25
1.26
1.26
1.27
1.27
1.28
1.29
1.30
1.30
1.31
1.32
1.32
1.33
1.34
1.34
1.35
1.36
1.36
1.37
1.38
1.38
1.39
1.39
1.40
1.42
1.45
1.47
1.48
1.50
1.52
1.53
1.54
1.55
1.56
1.57
1.58
0.59
0.63
0.67
0.71
0.74
0.77
0.80
0.83
0.86
0.88
0.90
0.93
0.95
0.97
0.99
1.01
1.02
1.04
1.05
1.07
1.08
1.10
1.11
1.12
1.14
1.15
1.20
1.24
1.28
1.32
1.35
1.37
1.39
1.42
1.43
1.45
1.47
1.48
1.46
1.44
1.43
1.42
1.41
1.41
1.41
1.40
1.40
1.41
1.41
1.41
1.41
1.41
1.42
1.42
1.42
1.43
1.43
1.43
1.44
1.44
1.45
1.45
1.45
1.46
1.48
1.49
1.51
1.52
1.53
1.55
1.56
1.57
1.58
1.59
1.60
1.60
0.49
0.53
0.57
0.61
0.65
0.68
0.72
0.75
0.77
0.80
0.83
0.85
0.88
0.90
0.92
0.94
0.96
0.98
1.00
1.01
1.03
1.04
1.06
1.07
1.09
1.10
1.16
1.20
1.25
1.28
1.31
1.34
1.37
1.39
1.41
1.43
1.45
1.46
1.70
1.66
1.63
1.60
1.58
1.57
1.55
1.54
1.53
1.53
1.52
1.52
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.51
1.52
1.52
1.52
1.53
1.54
1.55
1.56
1.57
1.58
1.59
1.60
1.60
1.61
1.62
1.63
0.39
0.44
0.48
0.52
0.56
0.60
0.63
0.66
0.70
0.72
0.75
0.78
0.81
0.83
0.85
0.88
0.90
0.92
0.94
0.95
0.97
0.99
1.00
1.02
1.03
1.05
1.11
1.16
1.21
1.25
1.28
1.31
1.34
1.36
1.39
1.41
1.42
1.44
1.96
1.90
1.85
1.80
1.77
1.74
1.71
1.69
1.67
1.66
1.65
1.64
1.63
1.62
1.61
1.61
1.60
1.60
1.59
1.59
1.59
1.59
1.59
1.58
1.58
1.58
1.58
1.59
1.59
1.60
1.61
1.61
1.62
1.62
1.63
1.64
1.64
1.65
k does not include the constant
31
Table 11
Critical Values of the Lilliefors Test
n
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Over 40
.20
.303
.289
.269
.252
.239
.227
.217
.208
.200
.193
.187
.181
.176
.171
.167
.163
.159
.155
.152
.149
.146
.143
.140
.138
.135
.133
.131
.129
.127
.125
.124
.122
.121
.119
.118
.116
.115
.739
n
Significance level α
.15
.10
.321
.346
.303
.319
.281
.297
.264
.280
.250
.265
.238
.252
.228
.241
.218
.231
.210
.222
.202
.215
.196
.208
.190
.201
.184
.195
.179
.190
.175
.185
.170
.181
.166
.176
.162
.172
.159
.168
.156
.165
.153
.162
.150
.159
.147
.156
.145
.153
.142
.151
.140
.148
.138
.146
.136
.144
.134
.142
.132
.140
.130
.138
.128
.136
.126
.134
.125
.133
.123
.131
.121
.129
.120
.128
.772
.822
n
n
.05
.376
.343
.323
.304
.288
.274
.262
.251
.242
.234
.226
.219
.213
.207
.202
.197
.192
.188
.184
.180
.176
.173
.170
.167
.164
.161
.159
.157
.154
.152
.150
.148
.146
.144
.142
.141
.139
.892
n
32
.01
.413
.397
.371
.351
.333
.317
.304
.291
.281
.271
.262
.254
.247
.240
.234
.228
.223
.218
.213
.209
.205
.201
.197
.194
.191
.188
.185
.182
.180
.177
.175
.172
.170
.168
.166
.164
.162
1.039
n