Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
APPENDIX
Back Matter
B
Answers to Most Odd-Numbered Problems
Chapter 1
1–1.
1–3.
1–5.
1–7.
1–11.
1–13.
1–15.
1–17.
1–19.
1–21.
823
© The McGraw−Hill
Companies, 2009
Appendix B: Answers to
Most Odd−Numbered
Problems
1–23.
1. quantitative/ratio
2. qualitative/nominal
3. quantitative/ratio
4. qualitative/nominal
5. quantitative/ratio
6. quantitative/interval
7. quantitative/ratio
8. quantitative/ratio
9. quantitative/ratio
10. quantitative/ratio
11. quantitative/ordinal
Weakest to strongest:
nominal, ordinal,
interval, ratio.
Ordinal
Nonrandom sample;
frame is random
sample
Ordinal
LQ 121
MQ 128
UQ 133.5
10th percentile 114.8
15th percentile 118.1
65th percentile 131.1
IQR 12.5
Formula (template)
Median 0.15
(0.15)
20th percentile 0.7 (0.7)
30th percentile 0.64 (0.56)
60th percentile 0.16 (0.14)
90th percentile 1.52 (0.88)
Median 51
LQ 31.5
UQ 162.75
IQR 131.25
45th percentile 42.2
mean 126.64
median 128
modes 128, 134, 136
mean 66.955
median 70
mode 45
1–25.
1–27.
1–29.
1–31.
1–33.
1–35.
1–37.
1–39.
1–45.
1–47.
1–49.
1–51.
1–53.
mean 199.875
median 51
mode none
mean 21.75
median 13
mode 12
mean 18.34
median 19.1
Variance, standard
deviation
range 27
var 57.74
s.d. 7.5986
range 60
var 321.38
s.d. 17.927
range 1186
var 110,287.45
s.d. 332.096
Chebyshev holds; data
not mound-shaped,
empirical rule does not
apply
Chebyshev holds; data
not mound-shaped,
empirical rule does not
apply
mean 13.33
median 12.5
5 | 5688
6 | 0123677789
7 | 0222333455667889
8 | 224
Stem-and-leaf is
similar to a histogram
but it retains the
individual data points;
box plot is useful in
identifying outliers
and the shape of the
distribution of
the data.
Data are concentrated
about the median;
2 outliers
127
11.45
2 131.04
Mode 127
1–55.
Suspected outliers:
101, 157
Can use stem-and-leaf
or box plots to identify
outliers; outliers need
to be evaluated
instead of just
eliminated.
1–57.
Mine A:
Mine B:
3 | 2457
2 | 3489
4 | 12355689
3 | 24578
5 | 123
4 | 034789
6|0
5 | 0129
—out values—
7 | 36
8|5
1–59. LW 0.3
LH 0.275
median 0.6
UH 1.15
UW 1.6
1–63. mean 504.688
s.d. 94.547
1–65. range 346
90th percentile 632.7
LQ 419.25
MQ 501.5
UQ 585.75
1–67. 1 | 2456789
2 | 02355
3 | 24
4 | 01
1–69. 1 | 012
—out values—
1|9
2 | 1222334556677889
3 | 02457
—out values—
6|2
1–71. mean 8.067
median 9
mode 10
1–73. mean 33.271
s.d. 16.945
var 287.15
LQ 25.41
MQ 26.71
UQ 35
743
824
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
744
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix B: Answers to
Most Odd−Numbered
Problems
Appendix B
1–75.
1–77.
1–79.
1–81.
1–83.
1–85.
1–89.
1. 3.5
2. Right skewed
3. d
4. Nothing will be
affected
mean 186.7
median 56.2
s.d. 355.6
outliers: 1459, 707.1,
481.9
mean 1720.2
median 930
s.d. 1409.85
var 1987680.96
mean 17.587
var 0.2172
s.d. 0.466
mean 37.17
median 34
s.d. 13.12758
var 172.33
a. VARP 3.5 offset2
b. VARP 3.5
mean 5.148
median 5.35
s.d. 0.6021
var 0.3625
Chapter 2
2–1.
2–3.
2–5.
2–7.
2–9.
2–11.
2–13.
2–15.
Objective and
subjective
The sample space is
the set of all possible
outcomes of an
experiment.
G F: the baby is
either a girl, or is
over 5 pounds (of
either sex). G F: the
baby is a girl over
5 pounds.
0.417
S B: purchase stock
or bonds, or both.
S B: purchase stock
and bonds.
0.12
a. 0.1667
b. 0.0556
c. 0.3889
0.85 is a typical
“very likely”
probability.
2–17.
2–19.
2–21.
2–23.
2–25.
2–27.
2–29.
2–31.
2–33.
2–35.
2–37.
2–39.
2–41.
2–43.
2–45.
2–47.
2–49.
2–51.
2–53.
2–55.
2–57.
2–59.
2–61.
2–63.
2–65.
2–67.
2–69.
2–71.
2–73.
2–75.
2–77.
2–79.
2–81.
2–83.
2–85.
2–87.
2–89.
The team is very likely
to win.
a. Mutually exclusive
b. 0.035
c. 0.985, complements
0.49
0.7909
0.500
0.34
0.60
a. 0.1002
b. 0.2065
c. 0.59
d. 0.144
e. 0.451
f. 0.571
g. 0.168
h. 0.454
i. 0.569
a. 0.484
b. 0.455
c. 0.138
d. 0.199
e. 0.285
f. 0.634
g. 0.801
0.333
0.8143
0.72675
0.99055
0.9989
not independent
0.3686
0.00048
0.0039, 0.684
362,880
120
0.00275
0.0000924
0.86
0.78
0.9944
0.2857
0.8824
0.0248
0.6
0.20
0.60
not independent
0.388
0.59049, 0.40951
0.132, not random
0.6667
a. 0.255
b. 0.8235
2–91. 0.5987
2–93. 0.5825
2–95. Practically speaking,
the probabilities
involved vary only
slightly, and their
role in the outcome
of the game should
be more or less
unnoticeable when
averaged over a span
of games.
2–97. 12
2–99. 0.767
2–101. a. 0.202
b. 1.00
c. 0.0
2–103. P (Pass) 0.0002;
P(Success) 0.00014
2–105. P (A|def) 0.3838
P(B|good) 0.3817
Chapter 3
3–1. a. P (x) 1.0
b. x F (x)
0
1
2
3
4
5
0.3
0.5
0.7
0.8
0.9
1.0
c. 0.3
3–3. a. P (x) 1.0
b. x F(x)
0
10
0.10
0.30
20
30
40
0.65
0.85
0.95
50
1.00
c. 0.35
3–5. x
P(x)
F(x)
2
3
4
5
1/36
2/36
3/36
4/36
1/36
3/36
6/36
10/36
6
7
8
5/36
6/36
5/36
15/36
21/36
26/36
9
10
4/36
3/36
30/36
33/36
11
12
2/36
1/36
35/36
36/36
Most likely sum is 7
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Answers to Most Odd-Numbered Problems
3–7.
a. 0.30
b. x F(x)
400
3–23.
3–25.
0.05
600 0.10
800 0.20
1000 0.30
3–27.
1200 0.60
1500 0.80
1700 1.00
3–9.
3–11.
3–13.
3–15.
c.
d.
a.
b.
c.
0.30
0.70
P(x) 1.0
0.50
x
9
10
11
12
13
14
15
F(x)
0.05
0.20
0.50
0.70
0.85
0.95
1.00
E (X ) 1.8
E (X 2) 6
V (X ) 2.76
SD(X ) 1.661
E (X ) 21.5
E (X 2) 625
V (X ) 162.75
E(sum of 2 dice) 7
x
P(x)
xP(x)
2
3
4
5
6
7
8
9
10
11
12
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
2/36
6/36
12/36
20/36
30/36
42/36
40/36
36/36
30/36
22/36
12/36
252/36 7
3–17.
3–19.
3–21.
mean 1230
var 137100
s.d. 370.27
Three standard
deviations
89 1 132 k 3
a. 2000
b. Yes P(X 0) .6
c. E(X ) 800
d. good measure of
risk is the standard
deviation
E (X 2) 2,800,000
3–29.
3–31.
3–33.
3–35.
3–37.
3–39.
3–41.
3–43.
3–45.
3–47.
3–49.
3–51.
V (X ) 2,160,000
SD(X ) 1,469.69
$868.5 million
Penalty X 2
E (X 2) 12.39
Variance is a measure
of the spread or
uncertainty of the
random variable.
V(Cost) V (aX b) a 2V (X ) 68,687,500
SD(Cost) 8,287.79
3.11
X is binomial if sales
calls are independent.
X is not binomial
because members of the
same family are related
and not independent of
each other.
a. slightly skewed;
becomes more
symmetric as n
increases.
b. Symmetric if p 0.5. Left-skewed if
p 0.5; rightskewed if p 0.5
a. 0.8889
b. 11
c. 0.55
a. 0.9981
b. 0.889, 0.935
c. 4, 5
d. increase the reliability
of each engine
a. mean 6.25,
var 6.77083
b. 61.80%
c. 11
d. p 0.7272
a. mean 2.857,
var 5.306
b. 82.15%
c. 7
d. p 0.5269
a. 0.5000
b. Add 4 more
women or remove
3 men.
a. 0.8430
b. 6
c. 1.972
a. MTBF 5.74 days
b. P (x 1) 0.1599
825
© The McGraw−Hill
Companies, 2009
Appendix B: Answers to
Most Odd−Numbered
Problems
3–55.
3–57.
3–59.
3–61.
3–63.
3–65.
3– 67.
745
c. 0.1599
d. 0.4185
As s or p increases
skewness decreases.
a. 0.7807
b. 0.2858
a. 0.7627
b. 0.2373
a. 0.2119
b. 0.2716
c. 0.1762
a. P(x 5) 0.2061
b. P (x 4) 0.2252
c. P (x 3) 0.2501
d. P (x 2) 0.2903
e. P (x 1) 0.3006
P (x 2) 0.5134
a. P (x) 1.0
x
F(x)
b.
0
1
2
3
4
5
6
7
8
.05
.10
.20
.35
.55
.70
.85
.95
1.00
P (3 x 7) 0.65
P (X 5) 0.70
E (X ) 4.25
E (X 2) 22.25
V (X ) 4.1875
SD(X ) 2.0463
g. [0.1574, 8.3426] vs
P(1 X 8)
0.95
3–69. a. P (x) 1.0
b.
x
F(x)
c.
d.
e.
f.
c.
d.
e.
f.
0
.10
1
2
3
.30
.60
.75
4
5
6
.90
.95
1.00
0.35, 0.40, 0.10
0.20
0.0225
E (X ) 2.4
SD(X ) 1.562
3–71. a. E(X ) 17.56875
profit $31.875
b. SD(X ) 0.3149
a measure of risk
826
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
746
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix B: Answers to
Most Odd−Numbered
Problems
Appendix B
c. the assumption
of stationary and
independence of
the stock prices
3–73. a. Yes.
b. 0.7716
3–75. a. The distribution is
binomial if the cars
are independent of
each other.
b. x P(x)
0
1
2
3
4
5
3–77.
3–79.
3–81.
3–83.
3–85.
3–87.
3–89.
3–91.
3–93.
.5987
.3151
.0746
.0105
.0010
.0001
c. 0.0861
d. 1⁄2 a car
N/n 10
b. 1.00
c. 0.75
0.9945
a. 11.229
b. 2020 : 1.02
G.B.: 0.33
M.C.: 1.48
0.133; 10
0.3935
P(X 5) 0.0489
0.9999; 0.0064; 0.6242
i) MTBF 103.86 hrs
ii) 49.9 hrs
iii) Because it is right
skewed
4–29.
4–31.
4–33.
4–35.
4–37.
4–39.
4–41.
4–43.
4–45.
4–47.
4–49.
4–51.
4–53.
4–55.
4–57.
4–59.
4–61.
4–63.
4–65.
4–67.
4–69.
4–71.
4–73.
4–75.
4–77.
4–79.
4–81.
4–83.
Chapter 4
4–1.
4–3.
4–5.
4–7.
4–9.
4–11.
4–13.
4–15.
4–17.
4–19.
4–21.
4–23.
4–25.
4–27.
0.6826; 0.95; 0.9802
0.1805
0.0215
0.9901
a very small number,
close to 0
0.9544
Not likely,
P 0.00003
z 0.48
z 1.175
z 1.96
0.0164
0.927
0.003
0.8609; 0.2107;
0.6306
0.0931
0.2525
0.8644
0.3759; 0.0135; 0.8766
0.0344
15.67
76.35; 99.65
46.15
832.6; 435.5
[18,130.2; 35,887.8]
74.84
1.54
0.99998
0.3804
0.1068
0.7642
0.1587; 0.9772
791,580
[7.02, 8.98]
1555.52, [1372.64,
3223.36]
8.856 kW
more than 0.26%
u 64.31, s.d.
5.49
6015.6
0.0000
a. N(248, 5.38522)
b. 0.6448
c. 0.0687
a. 0.0873
b. 0.4148
c. 16,764.55
d. 0.0051
i) Yes
ii) Mean $7133;
Std. dev $177.29
iii) 0.7734
5–11.
5–13.
5–15.
5–17.
5–19.
5–21.
5–23.
5–25.
5–27.
5–29.
5–31.
5–33.
5–37.
Chapter 5
5–1.
5–3.
5–5.
Parameters are
numerical measures
of populations.
Sample statistics are
numerical measures of
samples. An estimator
is a sample statistic
used for estimating
a population
parameter.
512 0.41667
mean 4.368
s.d. 0.3486
5–39.
5–41.
5–43.
5–45.
5–47.
The probability
distribution of a
sample statistic; useful
in determining the
accuracy of estimation
results.
E ( X ) 125
SE( X ) 8.944
When the population
distribution is
unknown.
Binomial. Cannot use
normal approximation
since np 1.2
0.075
1.000
0.2308
0.000
0.0497
0.0190
A consistent estimator
means as n → the
probability of getting
close to the parameter
increases. A generous
budget affords a large
sample size, making
this probability high.
Advantage: uses all
information in the
data. Disadvantage:
may be too sensitive
to the influence of
outliers.
a. mean 43.667,
SSD 358,
MSD 44.75
b. use Means: 40.75,
49.667, 40.5
c. SSD 195.917,
MSD 32.6528
d. SSD 719,
MSD 89.875
Yes, we can solve the
equation for the one
unknown amount.
E( X ) 1065
V ( X ) 2500
E( X ) 53
SE( X ) 0.5
E( p̂ ) 0.2
SE( p̂ ) 0.04216
1.000
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Answers to Most Odd-Numbered Problems
5–49.
5–51.
5–55.
5–57.
5–59.
5–61.
5–63.
5–65.
5–67.
5–69.
5–71.
5–73.
5–75.
5–77.
0.9544
a. 8128.08
b. 0.012
The sample median is
unbiased. The sample
mean is more efficient
and is sufficient. Must
assume normality for
using the sample
median to estimate ;
it is more resistant to
outliers.
0.000
1.000
0.9503
No minimum (n 1 is
enough for normality).
This estimator is
consistent, and_is more
efficient than X ,
because 2/n 2 2/n
Relative minimum
sample sizes:
na nb nd ne nc
Use a computer
simulation, draw
repeated samples,
determine the
empirical distribution.
P(Z 5) 0.0000003 Not
probable
0.923
0.1999
0.0171
Chapter 6
6–5.
6–7.
6–9.
6–11.
6–13.
6–15.
6–17.
6–19.
[86,978.12, 92,368.12]
[31.098, 32.902] m.p.g.
[9.045, 9.555] percent
a. [1513.91, 1886.09]
b. fly the route
95% C.I.: [136.99,
156.51]
90% C.I.: [138.56,
154.94]
99% C.I.: [133.93,
159.57]
[17.4, 22.6] percent
8.393
95% C.I.: [15,684.37,
17,375.63]
99% C.I.: [15,418.6,
17,641.4]
6–21.
6–23.
6–25.
6–27.
6–29.
6–31.
6–33.
6–35.
6–37.
6–39.
6–41.
6–43.
6–45.
6–47.
6–49.
6–51.
6–53.
6–55.
6–57.
6–59.
6–61.
6–63.
6–65.
6–67.
6–69.
6–71.
6–73.
6–75.
6–77.
6–79.
6–81.
6–83.
6–85.
6–87.
6–89.
6–95.
[27.93, 33.19] thousand
miles
[72.599, 89.881]
[4.92368, 12.06382]
[2.344, 2.856] days
[627478.6, 666521.4]
[15.86, 17.14] dollars
[5.44, 7.96] years
[55.85, 67.48] containers
[9.764, 10.380]
[10.76, 12.16]
[0.4658, 0.7695]
[0.3430, 0.4570]
[0.0078, 0.0122]
[0.0375, 0.2702]
[0.5357, 0.6228]
[0.1937, 0.2625]
[61.11, 197.04]
[19.25, 74.92]
[1268.03, 1676.68]
271
39
131
865
[21.507, 25.493]
[0.6211, 0.7989]
[35.81417, 52.18583]
[1.0841, 1.3159]
[0.6974, 0.7746]
did benefit
[7.021, 8.341]
[0.508, 0.692]
[0.902, 0.918]
[0.5695, 0.6304]
75% C.I.: [7.76, 8.24]
95% CI: [0.9859,
0.9981]
99% CI: [0.9837,
1.0003]
[5.147, 6.853]
7–13.
7–15.
7–17.
7–19.
7–21.
7–23.
7–25.
7–27.
7–29.
7–31.
7–33.
7–35.
7–37.
7–39.
Chapter 7
7–1.
7–3.
7–5.
7–11.
H0: p 0.8
H1: p 0.8
H0: 12
H1: 12
H0: $3.75
H1: $3.75
a. left-tailed H1:
10
b. right-tailed H1:
p 0.5
c. left-tailed H1:
100
827
© The McGraw−Hill
Companies, 2009
Appendix B: Answers to
Most Odd−Numbered
Problems
7–41.
7–43.
7–45.
7–47.
7–49.
7–51.
7–53.
747
d. right-tailed H1:
20
e. two-tailed H1:
p 0.22
f. right-tailed H1:
50
g. two-tailed H1:
2 140
a. to the left tail
b. to the right tail
c. either to the left or
to the right tail
a. p-value will
decrease
b. p-value increases
c. p-value decreases
z 1.936,
Do not reject H0
(p-value 0.0528)
z 1.7678, Reject H0
t(15) 1.55, Do not
reject H0 (p-value
0.10)
z 3.269, Reject H0
(p-value 0.0011)
t(24) 2.25, Do not
reject H0 at 0.01,
reject at 0.05
z 6.5385, Reject H0
z 1.539, Do not
reject H0 (p-value 0.1238)
z 1.622, Do not
reject H0 (p-value 0.1048)
t(23) 2.939,
Reject H0
z 16.0, Reject H0
z 20, Reject H0
z 1.304, Do not
reject H0 (p-value 0.0962)
z 9.643, Reject H0
z 3.2332,
Reject H0
standing start: z 2.7368, Reject H0
braking: z 1.8333,
Reject H0
power 0.9092
z 10.30, Reject H0
z 2.711, Reject H0
z 2.3570,
Reject H0
828
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
748
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix B: Answers to
Most Odd−Numbered
Problems
Appendix B
7–55.
7–65.
7–67.
7–69.
7–71.
7–73.
7–75.
7–77.
7–79.
7–81.
7–83.
7–85.
7–87.
7–89.
z 4.249, Reject H0
(p-value 0.00001)
z 4.86, Reject H0,
power 0.5214
2(24) 26.923, Do not
reject H0 ( p-value
0.10)
t(20) 1.06, Do not
reject H0 at 0.10
(p-value 0.15)
t 5.4867,
Reject H0
z 2.53, Reject H0
( p-value 0.0057)
Do not reject H0 at
0.05 level of
significance
Do not reject H0 at
0.05 level of
significance
t 1.1899, Do not
reject H0
Reject H0, p-value 0.0188
b. power 0.4968
c. No.
b. power 0.6779
c. Yes.
1. 54
z 3.2275, Reject H0
8–21.
8–23.
8–25.
8–27.
8–29.
8–31.
8–33.
8–35.
8–37.
8–39.
8–41.
8–43.
8–45.
8–47.
Chapter 8
8–1.
8–3.
8–5.
8–7.
8–9.
8–11.
8–13.
8–15.
8–17.
8–19.
t(24) 3.11, Reject H0
( p-value 0.01)
t 4.4907, Reject H0
t(14) 1.469, Cannot
reject H0
Power P(Z 1.55) 0.0606
t 2.1025,
Reject H0
t 11.101, Reject H0
z 4.24, Reject H0
a. One-tailed: H0:
1 2 0
b. z 1.53
c. At .05, Do not
reject H0
d. 0.063
e. t (19) 0.846, Do not
reject H0
[2.416, 2.664] percent
t(26) 1.132, Do not
reject H0 ( p-value
0.10)
8–49.
8–51.
8–53.
8–55.
8–57.
8–59.
8–61.
8–63.
8–65.
8–67.
8–69.
8–71.
t 1.676, Do not
reject H0
t(13) 1.164, Strongly
reject H0 ( p-value
0.10)
z 2.785, Reject H0
(p-value 0.0026)
t 0.7175, Do not
reject H0
z 2.835, Reject H0
( p-value 0.0023)
z 0.228, Do not
reject H0
[0.0419, 0.0781]
z 1.601, Do not
reject H0 at .05
(p-value 0.0547)
z 2.6112, Reject H0
z 5.33, Strongly
reject H0 (p-value is
very small)
F 1.1025, Do not
reject H0
F(27,20) 1.838, At 0.10, cannot reject H0
[0.652, 4.837]
F(24,24) 1.538, Do not
reject H0
Independent random
sampling from the
populations and
normal population
distributions
[3235.97, 1321.97]
[0.465, 9.737]
[0.0989, 0.3411]
z 1.447, Do not
reject H0 (p-value 0.1478)
t(22) 2.719, Reject
H0 (0.01 p-value
0.02)
z 1.7503, Do not
reject H0
t(26) 2.479, Reject H0
(p-value 0.02)
t(29) 1.08, Do not
reject H0
Since s 21 s 22, Do not
reject H0
t(15) 0.9751, Do not
reject H0
[0.0366, 0.1474]
t 1.5048, Do not
reject H0
8–73.
8–75.
8–77.
8–79.
t 14.2414,
Reject H0
Do not reject H0
[0.1264, 0.5264]
1. t 2.1356,
Reject H0
2. F 5.11,
Reject H0
Chapter 9
9–1.
9–3.
9–5.
9–7.
9–11.
9–19.
H0: All 4 means are
equal
H1: All 4 are different;
or 2 equal, 2 different;
or 3 equal, 1 different;
or 2 equal, other 2
equal but different
from first 2.
Series of paired
t tests are dependent
on each other. No
control over the
probability of a type
I error.
F(3,176) 12.53,
Reject H0
The sum of all the
deviations from a
mean is equal to 0.
Both MSTR and
MSE are sample
statistics given to
natural variation
about their own
means.
Source
SS
Between 187.696
df
3
Within
Total
28
31
152.413
340.108
MS
F
62.565
5.4433
11.494
p-value 0.000,
Reject H0
9–21.
Source
df
Treatment 2
Error
38
SS
91.043
140.529
Total
231.571
40
MS
F
45.521
3.698
12.31
Critical point F(2,38)
for 0.01 is 3.24,
Reject H0
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Answers to Most Odd-Numbered Problems
9–23.
9–25.
9–27.
9–29.
9–31.
9–33.
9–35.
9–37.
9–41.
Performances of the
four different portfolios
are significantly
different.
T 4.738. The mean
for squares is
significantly greater
than those for circles
and for triangles;
circles and triangles
show no significant
difference.
UK UAE, UKOMAN, MEX-OMAN
df factor 2
df error 154
df total 156
No; the 3 prototypes
were not randomly
chosen from a
population.
Fly all 3 planes on
the same route every
time.
Otherwise not a
random sample from
a population of
treatments, and
inference is thus not
valid for the entire
“population.”
If the locations and the
artists are chosen
randomly, we have a
random-effects model.
Since there are interactions, there are
differences in
emotions averaged
over all levels of
advertisements.
9–43. Source
df
145
SS
2
Newstime
160
Interaction
240
Error
6200
2
4
441
Total
449
Network
6745
MS
72.5
F
5.16
80
60
5.69
4.27
14.06
All are significant at
0.01. There are
9–45.
9–47.
9–49.
interactions. There are
Network main effects
averaged over
Newstime levels.
There are Newstime
main effects averaged
over Network
levels.
a. Explained is
treatment Factor
A Factor B (AB)
b. a 3
c. b 2
d. N 150
e. n 25 There are
no exercise-price
main effects.
g. There are time-ofexpiration main
effects at 0.05 but
not at 0.01.
h. There are no
interactions.
i. Some evidence for
time-of-expiration
main effects; no
evidence for
exercise-price
main effects or
interaction effects.
j. For time-ofexpiration main
effects, 0.01 p-value 0.05. For
the other two tests,
the p-values are very
high.
k. Could use a t test
for time-ofexpiration effects:
t 2(144) F(1,144)
Advantages: reduced
experimental errors
and great economy of
sample size.
Disadvantages:
restrictive, because it
requires that number
of treatments number of rows number of columns.
Could use a
randomized blocking
design.
829
© The McGraw−Hill
Companies, 2009
Appendix B: Answers to
Most Odd−Numbered
Problems
9–51.
9–53.
9–55.
9–57.
9–59.
749
Yes; have people of
the same occupation/
age/demographics use
sweaters of the 3 kinds
under study. Each
group of 3 people is a
block.
Group the executives
into blocks according
to some choice of
common
characteristics such as
age, sex, or years
employed at current
firm; these blocks
would then form a
third variable beyond
Location and Type to
use in a 3-way
ANOVA.
F(2,198) 25.84,
Reject H0 ( p-value
very small)
F(7,152) 14.67,
Reject H0 ( p-value
very small)
Source
Software
Computer
Interaction
Error
Total
SS
df
77,645
2
54,521
3
88,699
6
434,557 708
655,422 719
MS
38,822.5
18,173.667
14,783.167
613.78
F
63.25
29.60
24.09
Both main effects and
the interactions are
highly significant.
9–61.
Source
SS
df
Pet
22,245
3
Location 34,551
3
Interaction 31,778
9
Error
554,398 144
Total
642,972 159
MS
7,415
11,517
3,530.89
3,849.99
F
1.93
2.99
0.92
No interactions; no pet
main effects. There are
location main effects at
0.05
830
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
750
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix B: Answers to
Most Odd−Numbered
Problems
Appendix B
9–63.
9–65.
9–67.
b. F(2,58) 11.47,
Reject H0
F(2,98) 0.14958,
Do not reject H0
Rents are equal on
average; no evidence
of differences among
the four cities.
Chapter 10
10–1.
A set of mathematical
formulas and assumptions that describe some
real-world situation.
10–3. 1. a straight-line
relationship
between X and Y
2. the values of X are
fixed
3. the regression errors,
, are identically
normally distributed
random variables,
uncorrelated with
each other through
time.
10–5. It is the population
regression line.
10–7. 1. It captures the
randomness in the
process.
2. It makes the result
(Y ) a random
variable.
3. It captures the
effects on Y of
other unknown
components not
accounted by the
regression model.
10–9. The line is the best
unbiased linear
estimator of the true
regression line. Leastsquares line is
obtained by
minimizing the sum of
the squared deviations
of the data points
about the line.
10–11. b0 6.38,
b1 10.12
10–13. b0 3.057,
b1 0.187
10–15. b0 16.096;
b1 0.9681
10–17. b0 39.6717,
b1 0.06129
10–19. [1.1158, 1.3949]
10–21. s(b0) 2.897,
s(b1) 0.873
10–23. s(b0) 0.971
s(b1) 0.016 Estimate
of error variance is
MSE 0.991
10–25. s 2 gives information
about the variation of
the data points about
the computed
regression line.
10–27. r 0.9890
10–29. t (5) 0.601, Do not
reject H0
10–31. t(8) 5.11, Reject H0
10–35. z 1.297, Do not
reject H0
10–37. t(16) 1.0727, Do not
reject H0
10–39. t (11) 11.69, Strongly
reject H0
10–41. t (58) 5.90, Reject H0
10–43. t (211) 0.0565, Do not
reject H0
10–45. 9% of the variation in
customer satisfaction
can be explained by
the changes in a
customer’s materialism
measurement.
10–47. r 2 0.9781
10–49. r 2 0.067
10– 51. U.K. model explains
31.7% of the variation;
next best models:
Germany, Canada,
Japan, then United
States.
10–53. r 2 0.835
10–57. F(1,11) 129.525
t (11) 11.381 t 2 F
(11.381)2 129.525
10–59. F(1,17) 85.90, Very
strongly reject H0
10–61. F(1,20) 0.3845,
Do not reject H0
10–63. a. Heteroscedasticity
b. No apparent
inadequacy
10–65.
10–69.
10–71.
10–73.
10–75.
10–77.
10–79.
10–81.
10–83.
10–85.
10–87.
c. Data display
curvature, not a
linear relationship
a. no serious
inadequacy
b. Yes. A deviation
from the normaldistribution
assumption is
apparent.
6551.35
P.I.: [5854.4, 7248.3]
[5605.75, 7496.95]
[36.573, 77.387]
[157990, 477990]
a. Y 2.779337X 0.284157. When
X 10, Y 27.5092
b. Y 2.741537X.
When X 10,
Y 27.41537
c. Y 2.825566X 1.12783. When
X 10,
Y 27.12783
d. Y 2X 4.236.
When X 10,
Y 24.236
Reject the null. 19%
of the variation in job
performance can be
explained by
neuroticism.
Mean 13.4%;
s.d. 5.3563%
b0 62.292;
b1 1.8374
b0 2311;
b1 5.2031
b0 713.95;
b1 0.0239
Chapter 11
11–5. 8 equations
11–7. b0 1.134
b1 0.048
b2 10.897
11–11. n 13
11–21. R 2 0.9174, a good
regression, R 2
0.8983
11–23. R 2 0.8907. Do not
include the new
variable.
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Answers to Most Odd-Numbered Problems
11–25. a. Assume n 50;
Regression is:
Return 0.484 0.030 (Siz rnk) 0.017 (Prc rnk)
b. R 2 0.130. 13%
of the variation is
due to the two
independent
variables
c. Adjusted R 2 is quite
low; try regressing
on size alone.
11–27. F 168.153, adj.
R2 0.7282,
Reject H0
11–29. firm size: z 12.00
(significant)
firm profitability: z 5.533 (significant)
fixed-asset ratio:
z 0.08
growth opportunities:
z 0.72
nondebt tax shield:
z 4.29 (significant)
11–31. 2 [3.052, 8.148]
3 [3.135, 23.835]
4 [1.842, 8.742]
5 [4.995, 3.505]
11–33. Yes
11–35. Lend seems
insignificant because
of collinearity with M1
or price.
11–37. Autocorrelation of the
regression errors.
11–39. b0 0.578053
b1 0.155178
b 2 0.04974
R 2 0.355968
F 1.934515
Regression is not
statistically significant.
11–41. a. Residuals appear
to be normally
distributed.
b. Residuals are not
normally distributed.
11–47. Creates a bias. There is
no reason to force the
regression surface to
go through the origin.
11–51. 363.78
11–53. 0.341. 0.085
11–55. The estimators are the
same although their
standard errors are
different.
11–59. Two-way ANOVA
11–61. Early investment is not
statistically significant
(or may be collinear
with another variable).
Rerun the regression
without it. The dummy
variables are both
significant. Investment
is significant.
11–63. The STEPWISE
routine chooses Price
and M1 *Price as the
best set of explanatory
variables. Exports 1.39 0.0229 Price
0.00248M1 *Price.
t statistics: 2.36, 4.57,
9.08, respectively.
R 2 0.822
11–65. After * Bankdep:
z 11.3714
After * Bankdep *
ROA: z 2.7193
After * ROA:
z 3.00
Bankdep * ROA:
z 3.9178
All interactions
significant.
adj. R2 0.53
11–67. Quadratic regression
(should get a negative
estimated x 2 coefficient)
11–69. Linearizing a model;
finding a more
parsimonious model
than is possible without
a transformation;
stabilizing the variance.
11–71. The transformation
log Y
11–73. A logarithmic model
11–77. No
11–79. Taking reciprocals of
both sides of the
equation.
11–81. No. They minimize
the sum of the squared
831
© The McGraw−Hill
Companies, 2009
Appendix B: Answers to
Most Odd−Numbered
Problems
751
deviations relevant to
the estimated,
transformed model.
11–83. Earn
Prod .867
Prom .882
Book .547
11–85.
11–89.
11–91.
11–93.
11–97.
11–99.
11–103.
11–105.
11–107.
11–109.
Prod Prom
.638
.402
.319
Multicollinearity does
not seem to be serious.
Sample correlation is
0.740
Not true. Predictions
may be good when
carried out within the
same region of the
multicollinearity as
used in the estimation
procedure.
X 2 and X3 are
probably collinear.
Drop some of the other
variables one at a time
and see what happens
to the suspected sign of
the estimate.
1. The test checks only
for first-order
autocorrelation.
2. The test may not
be conclusive.
3. The usual
limitations of a
statistical test owing
to the two possible
types of errors.
DW 2.13 At the
0.10 level, no evidence
of a first-order
autocorrelation.
F (r,n–(k1)) 0.0275,
Cannot reject H0
The STEPWISE
procedure selects all
three variables.
R 2 0.9667
Because a variable
may lose explanatory
power and become
insignificant once
other variables are
added to the model.
No. There may be
several different “best”
models.
832
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
752
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix B: Answers to
Most Odd−Numbered
Problems
Appendix B
11–111. Transforms a nonlinear
model to a linear
model.
11–113. Predictor
Coef
Constant
Sincerity
Excitement
Ruggedness
Sophistication
36.49
0.0983
1.9859
0.5071
0.3664
Only Excitement is
significant. R2 0.946,
adj. R2 0.918
Chapter 12
12–3. 2005: 198.182
2006: 210.748
12–5. No, because of the
seasonality.
12–11. debt 8728083
12–13. forecast 0.33587
12–15. 6.2068 using
trend season
12–17. The w 0.8 forecasts
follow the raw data
much more closely.
12–19. forecast 6037828
12–23. Year Old CPI New CPI
1950
1951
1952
1953
72.1
77.8
79.5
80.1
24.9
26.9
27.5
27.7
12–25. A simple price index
reflects changes in a
single price variable of
time, relative to a
single base time.
12–27. a. 1988
b. Divide each index
number by 163100
c. It fell, from 145% of
the 1988 output
down to 133% of
that output.
12–29. Sales 4.23987 0.03870 Month.
Forecast for July 1997
(month #19) 3.5046
12–33. forecast 6.73
12–35. 7.9
Chapter 13
13–3.
1. Natural, random
variation
2. variation due to
assignable causes
13–9.
13–15.
13–17.
13–21.
13–23.
13–25.
13–27.
13–29.
13–31.
13–33.
13–37.
13–39.
13–41.
a. 77.62%
b. Omissions, Quantity
Entry, Part Numbers
(90.70%)
Random sampling, so
that the observations
are independent.
Process is in control.
Process is in control.
Process is in control.
Process is out of
control (9th sample).
All points are well
within the p chart
limits; process is in
control.
All points are well
within the p chart
limits; process is in
control.
The tenth sample
barely exceeds the
UCL 8.953;
otherwise in
control.
All points within
c chart limits; process
is in control.
The 20th observation
far exceeds the UCL
8.92100; the last
nine observations are
all on one side of the
center line P 3.45100
Last group’s mean is
below the LCL 2.136
X-bar chart shows the
process is out of
control.
Chapter 14
T 3, p-value 0.2266 Accept H0.
14–3. z 1.46 Cannot
reject H0.
14–5. T 9 Cannot reject
H0 (p-value .593).
14–7. z 2.145 Reject H0.
14–9. z 3.756
Reject H0.
14–11. z 3.756
Reject H0.
14–13. U 3.5 Reject H0.
14–17. U 12 Reject H0.
14–1.
14–21. H0 : 1 2
H1 : 1 2
p-value 0.117
14–23. Sign Test
14–25. Wilcoxon SignedRank Test
14–27. H0 : 1 2
H1 : 1 2
p-value 0.1120
14–29. p-value 0.001
Reject H0.
14–31. H 8.97,
p-value 0.0120
14–33. H 29.61
Reject H0.
CKW 11.68
14–35. H 12.37,
p-value 0.002
14–39. The three managers
are not equally
effective.
14–41. No, the 4 baking
processes are not
equally good.
14–43. n 9, rs 0.9289
Reject H0.
14–45. 2 0.586 Do not
reject H0.
14–47. 2 12.193 Reject H0.
14–49. 2 6.94 Do not
reject H0 at 0.05
14–51. 2 50.991 Reject H0.
14–53. 2 109.56 Reject H0.
14–55. 2 16.15 Reject H0.
14–57. 2 24.36 Reject H0.
14–59. 2 94.394 Reject H0.
14–61. 32.5
72.5
2-tailed p-value 0.10
14–63. 0.125. Do not
reject H0
14–67. 2 51.6836
Reject H0.
Chapter 15
15–1.
15–3.
15–5.
0.02531, 0.46544,
0.27247, 0.17691,
0.05262, 0.00697,
0.00028. Credible set
is [0.2, 0.4].
0.0126, 0.5829,
0.3658, 0.0384,
0.0003, 0.0000
0.1129, 0.2407, 0.2275,
0.2909, 0.0751, 0.0529
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Answers to Most Odd-Numbered Problems
15–7.
15–9.
15–11.
15–13.
15–15.
15–17.
15–23.
15–25.
15–27.
15–29.
15–33.
15–35.
15–37.
15–39.
15–47.
0.0633, 0.2216,
0.2928, 0.2364,
0.1286, 0.0465,
0.0099, 0.0009, 0.0000
0.0071, 0.0638, 0.3001,
0.3962, 0.1929, 0.0399
Normal with mean
9,207.3 and standard
deviation 61.58
Normal with mean
95.95 and standard
deviation 0.312
[5892.15, 6553.12]
Governor: D (largest
s.d.) ARCO expert: C
(smallest s.d.) Most
embarrassed: C
Expected payoff $10.55 million. Buy
the ad.
Expected payoff $405 million. Develop
the drug.
Optimal decision is
long—change if
possible. Expected
profit is $698,000
Optimal decision is
invest in wheat futures.
Expected value is
$3,040
Expected payoff $11.19 million. Do the
test.
Sell watches, no
testing: expected
payoff $20,000
Sell watches, with
testing: expected
payoff $127,200
No change in in-flight
sales: expected
payoff $700,000
Do not change their
in-flight sales.
Test and follow the
test’s recommendation.
E (payoff) $587,000
A utility function is a
value-of-money
function of an
individual.
EVPI $290,000.
Buy information if it
is perfect.
15–49. 0.01142, 0.30043,
0.35004, 0.27066,
0.05561, 0.01184
15–51. 0.0026, 0.3844,
0.4589, 0.1480,
0.0060, 0.0001
15–55. [25649.75, 27416.22]
15–61. 1,400 3,000: a risk
taker
15–63. Merge; E(payoff) $2.45 million
15–65. EVPI $1.125 million
15–67. Hire candidate A
On the CD
Chapter 16
x st 33.48%
S.D. 0.823%
[31.87, 35.09]
x st $40.01
16–3.
S.D. 0.6854
[38.88, 41.14]
Data has many zero
values
16–5. $35,604.5 C.I. [30,969.19, 40,239.87]
16–9. a–c. All no. Clusters
need to be
randomly
chosen.
d. Consider the
companies as
strata, ships as
clusters.
Randomly draw
clusters from the
strata.
16–11. Arrange elements in
a circle, randomly
choose a number
from 1 to 28, and then
add 28 to element
number until you
have a sample of 30
sales.
16–13. If k is a multiple of 7,
we would sample on
the same day for
different weeks and
bias the results due
to weekly sales
cycles.
16–17. Yes, with each vehicle
a cluster.
16–1.
a.
b.
c.
a.
b.
c.
d.
833
© The McGraw−Hill
Companies, 2009
Appendix B: Answers to
Most Odd−Numbered
Problems
753
16–19. OK unless a
nonnegligible
fraction of the
businesses in the
community are
unlisted in the Yellow
Pages.
16–21. [0.055, 0.091]
16–25. Sample about 109
children, 744 young
adults, 147 older
people.
16–27. Regression and ratio
estimators
16–29. No; benefits are not
substantial when the
number of strata is
much greater than 6.
Combine some.
Chapter 17
17–1.
17–3.
17–7.
17–9.
17–11.
17–13.
17–19.
17–21.
Statistically classifying
elements into one of
several groups.
Classify as default.
Used the information
in a Bayesian
formulation of a
discriminant function,
using P (G ), P (D|G ),
leading to P (G|D )
Group 3
The discriminant
function is not
statistically significant
and should not be
used.
5 functions; some
may not be
significant
VARIMAX maximizes
the sum of the
variances of the
loadings in the factor
matrix. Other rotation
methods are
QUARTIMAX and
EQUIMAX.
Factor 1 is price
items, factor 2 is
retailing/selling, factor
3 is advertising, and
factor 4 is negative (or
opposite) of product
ratings.
834
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
754
Back Matter
Appendix B: Answers to
Most Odd−Numbered
Problems
© The McGraw−Hill
Companies, 2009
Appendix B
17–23. Pricing policies:
associate with factor 2;
communality 0.501.
Record and reporting
procedures: associate
with factor 2;
commonality 0.077
17–27. 3 functions
17–31. Not a worthwhile
result since the
dimensionality
has not been
reduced.
17–33. Communality of a
variable
17–37. Wilks’ 0.412
F (3,21) 9.987
( p-value 0.000)
Production cost
p-value 0.009.
Number of sponsors
p-value 0.066 (n.s.).
Promotions p-value 0.004. Discriminant
function coefficients:
production cost 0.945, promotions 0.996. 84% correct
prediction.
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
APPENDIX
TABLE 1
C
Statistical Tables
Cumulative Binomial Distribution
F(x) P(X x) a 1ni2 pi(1 p)nI
Example: if p 0.10, n 5, and x 2, then F(x) 0.991
p
i0
n
5
835
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
x
0
.01
.951
.05
.774
.10
.590
.20
.328
.30
.168
.40
.078
.50
.031
.60
.010
.70
.002
.80
.000
.90
.000
.95
.000
.99
.000
1
2
3
4
.999
1.000
1.000
1.000
.977
.999
1.000
1.000
.919
.991
1.000
1.000
.737
.942
.993
1.000
.528
.837
.969
.998
.337
.683
.913
.990
.187
.500
.813
.969
.087
.317
.663
.922
.031
.163
.472
.832
.007
.058
.263
.672
.000
.009
.081
.410
.000
.001
.023
.226
.000
.000
.001
.049
6
0
1
2
3
4
5
.941
.999
1.000
1.000
1.000
1.000
.735
.967
.998
1.000
1.000
1.000
.531
.886
.984
.999
1.000
1.000
.262
.655
.901
.983
.998
1.000
.118
.420
.744
.930
.989
.999
.047
.233
.544
.821
.959
.996
.016
.109
.344
.656
.891
.984
.004
.041
.179
.456
.767
.953
.001
.011
.070
.256
.580
.882
.000
.002
.017
.099
.345
.738
.000
.000
.001
.016
.114
.469
.000
.000
.000
.002
.033
.265
.000
.000
.000
.000
.001
.059
7
0
1
2
3
.932
.998
1.000
1.000
.698
.956
.996
.478
.850
.974
.210
.577
.852
.082
.329
.647
.028
.159
.420
.008
.063
.227
.002
.019
.096
.000
.004
.029
.000
.000
.005
.000
.000
.000
.000
.000
.000
.000
.000
.000
4
5
6
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.997
1.000
1.000
1.000
.967
.995
1.000
1.000
.874
.971
.996
1.000
.710
.904
.981
.998
.500
.773
.937
.992
.290
.580
.841
.972
.126
.353
.671
.918
.033
.148
.423
.790
.003
.026
.150
.522
.000
.004
.044
.302
.000
.000
.002
.068
0
1
2
3
4
5
6
.923
.997
1.000
1.000
1.000
1.000
1.000
.663
.943
.994
1.000
1.000
1.000
1.000
.430
.813
.962
.995
1.000
1.000
1.000
.168
.503
.797
.944
.990
.999
1.000
.058
.255
.552
.806
.942
.989
.999
.017
.106
.315
.594
.826
.950
.991
.004
.035
.145
.363
.637
.855
.965
.001
.009
.050
.174
.406
.685
.894
.000
.001
.011
.058
.194
.448
.745
.000
.000
.001
.010
.056
.203
.497
.000
.000
.000
.000
.005
.038
.187
.000
.000
.000
.000
.000
.006
.057
.000
.000
.000
.000
.000
.000
.003
7
1.000
1.000
1.000
1.000
1.000
.999
.996
.983
.942
.832
.570
.337
.077
0
1
.914
.997
.630
.929
.387
.775
.134
.436
.040
.196
.010
.071
.002
.020
.000
.004
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
2
3
4
1.000
1.000
1.000
.992
.999
.947
.992
.738
.914
.463
.730
.232
.483
.090
.254
.025
.099
.004
.025
.000
.003
.000
.000
.000
.000
.000
.000
5
6
1.000
1.000
1.000
1.000
1.000
.999
1.000
1.000
.980
.997
1.000
.901
.975
.996
.733
.901
.975
.500
.746
.910
.267
.517
.768
.099
.270
.537
.020
.086
.262
.001
.008
.053
.000
.001
.008
.000
.000
.000
7
8
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.996
1.000
.980
.998
.929
.990
.804
.960
.564
.866
.225
.613
.071
.370
.003
.086
0
.904
.599
.349
.107
.028
.006
.001
.000
.000
.000
.000
.000
.000
1
2
3
.996
1.000
1.000
.914
.988
.999
.736
.930
.987
.376
.678
.879
.149
.383
.650
.046
.167
.382
.011
.055
.172
.002
.012
.055
.000
.002
.011
.000
.000
.001
.000
.000
.000
.000
.000
.000
.000
.000
.000
4
1.000
1.000
.998
.967
.850
.633
.377
.166
.047
.006
.000
.000
.000
5
6
1.000
1.000
1.000
1.000
1.000
1.000
.994
.999
.953
.989
.834
.945
.623
.828
.367
.618
.150
.350
.033
.121
.002
.013
.000
.001
.000
.000
8
9
10
755
836
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
756
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 1 (continued)
Cumulative Binomial Distribution
p
n
15
20
x
7
8
.01
1.000
1.000
.05
1.000
1.000
.10
1.000
1.000
.20
1.000
1.000
.30
.998
1.000
.40
.988
.998
.50
.945
.989
.60
.833
.954
.70
.617
.851
.80
.322
.624
.90
.070
.264
.95
.012
.086
.99
.000
.004
9
1.000
1.000
1.000
1.000
1.000
1.000
.999
.994
.972
.893
.651
.401
.096
0
1
.860
.990
.463
.829
.206
.549
.035
.167
.005
.035
.000
.005
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
2
3
4
5
6
7
8
9
10
11
12
13
14
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
.964
.995
.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.816
.944
.987
.998
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.398
.648
.836
.939
.982
.996
.999
1.000
1.000
1.000
1.000
1.000
1.000
.127
.297
.515
.722
.869
.950
.985
.996
.999
1.000
1.000
1.000
1.000
.027
.091
.217
.403
.610
.787
.905
.966
.991
.998
1.000
1.000
1.000
.004
.018
.059
.151
.304
.500
.696
.849
.941
.982
.996
1.000
1.000
.000
.002
.009
.034
.095
.213
.390
.597
.783
.909
.973
.995
1.000
.000
.000
.001
.004
.015
.050
.131
.278
.485
.703
.873
.965
.995
.000
.000
.000
.000
.001
.004
.018
.061
.164
.352
.602
.833
.965
.000
.000
.000
.000
.000
.000
.000
.002
.013
.056
.184
.451
.794
.000
.000
.000
.000
.000
.000
.000
.000
.001
.005
.036
.171
.537
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.010
.140
0
1
2
3
4
5
6
7
8
9
10
.818
.983
.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.358
.736
.925
.984
.997
1.000
1.000
1.000
1.000
1.000
1.000
.122
.392
.677
.867
.957
.989
.998
1.000
1.000
1.000
1.000
.012
.069
.206
.411
.630
.804
.913
.968
.990
.997
.999
.001
.008
.035
.107
.238
.416
.608
.772
.887
.952
.983
.000
.001
.004
.016
.051
.126
.250
.416
.596
.755
.872
.000
.000
.000
.001
.006
.021
.058
.132
.252
.412
.588
.000
.000
.000
.000
.000
.002
.006
.021
.057
.128
.245
.000
.000
.000
.000
.000
.000
.000
.001
.005
.017
.048
.000
.000
.000
.000
.000
.000
.000
.000
.000
.001
.003
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
11
12
1.000
1.000
1.000
1.000
.995
.943
.748
.404
.113
.010
.000
.000
.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
.999
1.000
1.000
.979
.994
.998
.868
.942
.979
.584
.750
.874
.228
.392
.584
.032
.087
.196
.000
.002
.011
.000
.000
.000
.000
.000
.000
15
16
17
18
19
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.994
.999
.949
.984
.762
.893
.370
.589
.043
.133
.003
.016
.000
.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
.996
.999
.965
.992
.794
.931
.323
.608
.075
.264
.001
.017
1.000
1.000
1.000
.999
.988
.878
.642
.182
0
1
2
.778
.974
.998
.277
.642
.873
.072
.271
.537
.004
.027
.098
.000
.002
.009
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
3
4
5
1.000
1.000
1.000
.966
.993
.999
.764
.902
.967
.234
.421
.617
.033
.090
.193
.002
.009
.029
.000
.000
.002
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
6
7
8
1.000
1.000
1.000
1.000
1.000
1.000
.991
.998
1.000
.780
.891
.953
.341
.512
.677
.074
.154
.274
.007
.022
.054
.000
.001
.004
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
.000
9
10
11
1.000
1.000
1.000
1.000
1.000
1.000
.983
.994
.811
.902
.425
.586
.115
.212
.013
.034
.000
.002
.000
.000
.000
.000
.000
.000
.000
.000
1.000
1.000
1.000
.998
.956
.732
.345
.078
.006
.000
.000
.000
.000
13
14
25
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 1 (concluded)
837
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
757
Cumulative Binomial Distribution
p
n
x
12
13
.01
1.000
1.000
.05
1.000
1.000
.10
1.000
1.000
.20
1.000
1.000
.30
.983
.994
.40
.846
.922
.50
.500
.655
.60
.154
.268
.70
.017
.044
.80
.000
.002
.90
.000
.000
.95
.000
.000
.99
.000
.000
14
15
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.998
1.000
1.000
.966
.987
.996
.788
.885
.946
.414
.575
.726
.098
.189
.323
.006
.017
.047
.000
.000
.000
.000
.000
.000
.000
.000
.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
.999
1.000
1.000
.978
.993
.998
.846
.926
.971
.488
.659
.807
.109
.220
.383
.002
.009
.033
.000
.000
.001
.000
.000
.000
20
21
22
23
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
1.000
.991
.998
1.000
.910
.967
.991
.579
.766
.902
.098
.236
.463
.007
.034
.127
.000
.000
.002
1.000
1.000
1.000
1.000
1.000
1.000
.998
.973
.729
.358
.026
24
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
.996
.928
.723
.222
16
17
18
19
838
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
758
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 2
Areas of the Standard Normal Distribution
Table area for z
0
z
The table areas are probabilities that the standard normal random variable is between 0 and z.
Second Decimal Place in z
z
0.0
0.1
0.2
0.3
0.4
0.5
0.00
0.0000
0.0398
0.0793
0.1179
0.1554
0.1915
0.01
0.0040
0.0438
0.0832
0.1217
0.1591
0.1950
0.02
0.0080
0.0478
0.0871
0.1255
0.1628
0.1985
0.03
0.0120
0.0517
0.0910
0.1293
0.1664
0.2019
0.04
0.0160
0.0557
0.0948
0.1331
0.1700
0.2054
0.05
0.0199
0.0596
0.0987
0.1368
0.1736
0.2088
0.06
0.0239
0.0636
0.1026
0.1406
0.1772
0.2123
0.07
0.0279
0.0675
0.1064
0.1443
0.1808
0.2157
0.08
0.0319
0.0714
0.1103
0.1480
0.1844
0.2190
0.09
0.0359
0.0753
0.1141
0.1517
0.1879
0.2224
0.6
0.7
0.8
0.2257
0.2580
0.2881
0.2291
0.2611
0.2910
0.2324
0.2642
0.2939
0.2357
0.2673
0.2967
0.2389
0.2704
0.2995
0.2422
0.2734
0.3023
0.2454
0.2764
0.3051
0.2486
0.2794
0.2517 0.2549
0.2823 0.2852
0.9
1.0
0.3159
0.3413
0.3186
0.3438
0.3212
0.3461
0.3238
0.3485
0.3264
0.3508
0.3289
0.3531
0.3315
0.3554
0.3078
0.3340
0.3577
0.3106 0.3133
0.3365 0.3389
0.3599 0.3621
1.1
1.2
1.3
1.4
1.5
0.3643
0.3849
0.4032
0.4192
0.4332
0.3665
0.3869
0.4049
0.4207
0.4345
0.3686
0.3888
0.4066
0.4222
0.4357
0.3708
0.3907
0.4082
0.4236
0.4370
0.3729
0.3925
0.4099
0.4251
0.4382
0.3749
0.3944
0.4115
0.4265
0.4394
0.3770
0.3962
0.4131
0.4279
0.4406
0.3790
0.3980
0.4147
0.4292
0.4418
0.3810
0.3997
0.4162
0.4306
0.4429
1.6
1.7
0.4452
0.4554
0.4463
0.4564
0.4474
0.4573
0.4484
0.4582
1.8
1.9
2.0
0.4641
0.4713
0.4772
0.4649
0.4719
0.4778
0.4656
0.4726
0.4783
2.1
2.2
2.3
2.4
0.4821
0.4861
0.4893
0.4918
0.4826
0.4864
0.4896
0.4920
0.4830
0.4868
0.4898
0.4922
2.5
0.4938
0.4940
0.4941
0.4943
0.4945
0.4946
0.4948
0.4949
0.4951 0.4952
2.6
0.4953
0.4955
2.7
0.4965
0.4966
0.4956
0.4967
0.4957
0.4968
0.4959
0.4969
0.4960
0.4970
0.4961
0.4971
0.4962
0.4972
0.4963 0.4964
0.4973 0.4974
2.8
2.9
0.4974
0.4981
0.4975
0.4982
0.4976
0.4982
0.4977
0.4983
0.4977
0.4984
0.4978
0.4984
0.4979
0.4985
0.4979
0.4985
0.4980 0.4981
0.4986 0.4986
3.0
0.4987
0.4987
0.4987
0.4988
0.4988
0.4989
0.4989
0.4989
0.4990 0.4990
3.1
3.2
0.4990
0.4993
0.4991
0.4993
0.4991
0.4994
0.4991
0.4994
0.4992
0.4994
0.4992
0.4994
0.4992
0.4994
0.4992
0.4995
0.4993 0.4993
0.4995 0.4995
3.3
3.4
0.4995
0.4997
0.4995
0.4997
0.4995
0.4997
0.4996
0.4997
0.4996
0.4997
0.4996
0.4997
0.4996
0.4997
0.4996
0.4997
0.4996 0.4997
0.4997 0.4998
3.5
4.0
0.4998
0.49997
4.5
5.0
0.499997
0.4999997
6.0
0.49999999
0.3830
0.4015
0.4177
0.4319
0.4441
0.4495
0.4505
0.4515
0.4525
0.4535 0.4545
0.4664
0.4732
0.4788
0.4591
0.4671
0.4738
0.4793
0.4599
0.4678
0.4744
0.4798
0.4608
0.4686
0.4750
0.4803
0.4616
0.4693
0.4756
0.4808
0.4625
0.4699
0.4761
0.4812
0.4633
0.4706
0.4767
0.4817
0.4834
0.4871
0.4901
0.4925
0.4838
0.4875
0.4904
0.4927
0.4842
0.4878
0.4906
0.4929
0.4846
0.4881
0.4909
0.4931
0.4850
0.4884
0.4911
0.4932
0.4854
0.4887
0.4913
0.4934
0.4857
0.4890
0.4916
0.4936
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 3
839
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Critical Values of the t Distribution
t
Degrees of
Freedom
1
2
3
4
t.100
3.078
1.886
1.638
1.533
t.050
6.314
2.920
2.353
2.132
t.025
12.706
4.303
3.182
2.776
t.010
31.821
6.965
4.541
3.747
t.005
63.657
9.925
5.841
4.604
5
6
7
8
9
1.476
1.440
1.415
1.397
1.383
2.015
1.943
1.895
1.860
1.833
2.571
2.447
2.365
2.306
2.262
3.365
3.143
2.998
2.896
2.821
4.032
3.707
3.499
3.355
3.250
10
1.372
1.812
11
12
13
14
1.363
1.356
1.350
1.345
1.796
1.782
1.771
1.761
2.228
2.201
2.179
2.160
2.145
2.764
2.718
2.681
2.650
2.624
3.169
3.106
3.055
3.012
2.977
15
16
17
18
19
1.341
1.337
1.333
1.330
1.328
1.753
1.746
1.740
1.734
1.729
2.131
2.120
2.110
2.101
2.093
2.602
2.583
2.567
2.552
2.539
2.947
2.921
2.898
2.878
2.861
20
21
22
23
1.325
1.323
1.321
1.319
1.725
1.721
1.717
1.714
2.086
2.080
2.074
2.069
2.528
2.518
2.508
2.500
2.845
2.831
2.819
2.807
24
1.318
1.711
2.064
2.492
2.797
25
26
1.316
1.315
1.708
1.706
2.060
2.056
2.485
2.479
2.787
2.779
27
28
29
1.314
1.313
1.311
1.703
1.701
1.699
2.052
2.048
2.045
2.473
2.467
2.462
2.771
2.763
2.756
30
1.310
1.697
2.042
2.457
2.750
40
60
1.303
1.296
1.684
1.671
2.021
2.000
2.423
2.390
2.704
2.660
120
1.289
1.658
1.980
2.358
2.617
1.282
1.645
1.960
2.326
2.576
Source: M. Merrington, “Table of Percentage Points of the t-Distribution,” Biometrika 32 (1941), p. 300.
Reproduced by permission of the Biometrika trustees.
759
840
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
760
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 4
Critical Values of the Chi-Square Distribution
2
0
Degrees of
Freedom
1
2
3
4
2
.995
2
.990
2
.975
2
.950
2
.900
0.0000393
0.0100251
0.0717212
0.206990
0.0001571
0.0201007
0.114832
0.297110
0.0009821
0.0506356
0.215795
0.484419
0.0039321
0.102587
0.351846
0.710721
0.0157908
0.210720
0.584375
1.063623
5
6
7
8
9
0.411740
0.675727
0.989265
1.344419
1.734926
0.554300
0.872085
1.239043
1.646482
2.087912
0.831211
1.237347
1.68987
2.17973
2.70039
1.145476
1.63539
2.16735
2.73264
3.32511
1.61031
2.20413
2.83311
3.48954
4.16816
10
2.15585
2.55821
11
12
13
14
2.60321
3.07382
3.56503
4.07468
3.05347
3.57056
4.10691
4.66043
3.24697
3.81575
4.40379
5.00874
5.62872
3.94030
4.57481
5.22603
5.89186
6.57063
4.86518
5.57779
6.30380
7.04150
7.78953
15
16
17
18
19
4.60094
5.14224
5.69724
6.26481
6.84398
5.22935
5.81221
6.40776
7.01491
7.63273
6.26214
6.90766
7.56418
8.23075
8.90655
7.26094
7.96164
8.67176
9.39046
10.1170
8.54675
9.31223
10.0852
10.8649
11.6509
20
21
22
7.43386
8.03366
8.64272
8.26040
8.89720
9.54249
9.59083
10.28293
10.9823
10.8508
11.5913
12.3380
12.4426
13.2396
14.0415
23
24
9.26042
9.88623
10.19567
10.8564
11.6885
12.4011
13.0905
13.8484
14.8479
15.6587
25
10.5197
11.5240
13.1197
14.6114
16.4734
26
27
28
11.1603
11.8076
12.4613
12.1981
12.8786
13.5648
13.8439
14.5733
15.3079
15.3791
16.1513
16.9279
17.2919
18.1138
29
13.1211
14.2565
16.0471
17.7083
18.9392
19.7677
30
40
13.7867
20.7065
14.9535
22.1643
16.7908
24.4331
18.4926
26.5093
20.5992
29.0505
50
60
27.9907
35.5346
29.7067
37.4848
32.3574
40.4817
34.7642
43.1879
37.6886
46.4589
70
43.2752
45.4418
48.7576
51.7393
55.3290
80
90
100
51.1720
59.1963
67.3276
53.5400
61.7541
70.0648
57.1532
65.6466
74.2219
60.3915
69.1260
77.9295
64.2778
73.2912
82.3581
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 4 (concluded)
Critical Values of the Chi-Square Distribution
Degrees of
Freedom
1
2
2.70554
4.60517
3.84146
5.99147
5.02389
7.37776
3
4
6.25139
7.77944
7.81473
9.48773
9.34840
11.1433
11.3449
13.2767
12.8381
14.8602
5
6
9.23635
10.6446
11.0705
12.5916
12.8325
14.4494
15.0863
16.8119
16.7496
18.5476
7
8
9
12.0170
13.3616
14.6837
14.0671
15.5073
16.9190
16.0128
17.5346
19.0228
18.4753
20.0902
21.6660
20.2777
21.9550
23.5893
10
15.9871
18.3070
11
12
13
14
17.2750
18.5494
19.8119
21.0642
19.6751
21.0261
22.3621
23.6848
20.4831
21.9200
23.3367
24.7356
26.1190
23.2093
24.7250
26.2170
27.6883
29.1413
25.1882
26.7569
28.2995
29.8194
31.3193
15
16
17
18
19
22.3072
23.5418
24.7690
25.9894
27.2036
24.9958
26.2962
27.5871
28.8693
30.1435
27.4884
28.8454
30.1910
31.5264
32.8523
30.5779
31.9999
33.4087
34.8053
36.1908
32.8013
34.2672
35.7185
37.1564
38.5822
20
21
22
23
24
28.4120
29.6151
30.8133
32.0069
33.1963
31.4104
32.6705
33.9244
35.1725
36.4151
34.1696
35.4789
36.7807
38.0757
39.3641
37.5662
38.9321
40.2894
41.6384
42.9798
39.9968
41.4010
42.7956
44.1813
45.5585
25
26
27
28
29
34.3816
35.5631
36.7412
37.9159
39.0875
37.6525
38.8852
40.1133
41.3372
42.5569
40.6465
41.9232
43.1944
44.4607
45.7222
44.3141
45.6417
46.9630
48.2782
49.5879
46.9278
48.2899
49.6449
50.9933
52.3356
30
40
40.2560
51.8050
43.7729
55.7585
46.9792
59.3417
50.8922
63.6907
53.6720
66.7659
50
63.1671
67.5048
71.4202
76.1539
79.4900
60
74.3970
79.0819
83.2976
88.3794
70
85.5271
90.5312
95.0231
80
90
96.5782
107.565
101.879
113.145
100
118.498
124.342
2
.100
2
.050
2
.025
841
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
2
.010
2
.005
6.63490
9.21034
7.87944
10.5966
91.9517
100.425
104.215
106.629
118.136
112.329
124.116
116.321
128.299
129.561
135.807
140.169
Source: C. M. Thompson, “Tables of the Percentage Points of the 2-Distribution,” Biometrika 32 (1941),
pp. 188–89. Reproduced by permission of the Biometrika Trustees.
761
842
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
762
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 5
Critical Values of the F Distribution for 0.10
0
Denominator
Degrees of
Freedom (k2)
1
2
3
4
F
Numerator Degrees of Freedom (k1)
1
39.86
8.53
5.54
4.54
2
49.50
9.00
5.46
4.32
3
53.59
9.16
5.39
4.19
5
6
4.06
3.78
3.78
3.46
3.62
3.29
7
8
9
3.59
3.46
3.36
3.26
3.11
3.01
10
11
12
13
14
3.29
3.23
3.18
3.14
3.10
15
16
17
4
55.83
9.24
5.34
4.11
5
57.24
9.29
5.31
4.05
6
58.20
9.33
5.28
4.01
7
58.91
9.35
5.27
3.98
8
59.44
9.37
5.25
3.95
9
59.86
9.38
5.24
3.94
3.52
3.45
3.40
3.37
3.34
3.32
3.07
2.92
2.81
3.18
2.96
2.81
2.69
3.11
2.88
2.73
2.61
3.05
2.83
2.67
2.55
3.01
2.78
2.62
2.51
2.98
2.75
2.59
2.47
2.96
2.72
2.56
2.44
2.92
2.86
2.81
2.76
2.73
2.73
2.66
2.61
2.56
2.52
2.61
2.54
2.48
2.43
2.39
2.52
2.45
2.39
2.35
2.31
2.46
2.39
2.33
2.28
2.24
2.41
2.34
2.28
2.23
2.19
2.38
2.30
2.24
2.20
2.15
2.35
2.27
2.21
2.16
2.12
3.07
3.05
2.70
2.67
2.49
2.46
2.36
2.33
2.27
2.24
2.21
2.18
2.16
2.13
2.12
2.09
2.09
2.06
18
19
3.03
3.01
2.99
2.64
2.62
2.61
2.44
2.42
2.40
2.31
2.29
2.27
2.22
2.20
2.18
2.15
2.13
2.11
2.10
2.08
2.06
2.06
2.04
2.02
2.03
2.00
1.98
20
21
22
23
2.97
2.96
2.95
2.94
2.59
2.57
2.56
2.55
2.38
2.36
2.35
2.34
2.25
2.23
2.22
2.21
2.16
2.14
2.13
2.11
2.09
2.08
2.06
2.05
2.04
2.02
2.01
1.99
2.00
1.98
1.97
1.95
1.96
1.95
1.93
1.92
24
2.93
2.54
2.33
2.19
2.10
2.04
1.98
1.94
1.91
25
26
2.92
2.91
2.53
2.52
2.32
2.31
2.18
2.17
2.09
2.08
2.02
2.01
27
28
2.90
2.89
2.51
2.50
2.30
2.29
2.17
2.16
2.07
2.06
2.00
2.00
1.97
1.96
1.95
1.94
1.93
1.92
1.91
1.90
1.89
1.88
1.87
1.87
29
2.89
2.50
2.28
2.15
2.06
1.99
1.93
1.89
1.86
30
40
2.88
2.84
2.49
2.44
2.28
2.23
2.14
2.09
2.05
2.00
1.98
1.93
1.93
1.87
1.88
1.83
1.85
1.79
60
120
2.79
2.39
2.18
2.04
1.95
1.87
1.82
1.77
1.74
2.75
2.71
2.35
2.30
2.13
2.08
1.99
1.94
1.90
1.85
1.82
1.77
1.77
1.72
1.72
1.67
1.68
1.63
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 5 (continued)
843
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
763
Critical Values of the F Distribution for 0.10
Denominator
Degrees of
Freedom (k2)
1
10
60.19
12
60.71
15
61.22
20
61.74
24
62.00
30
62.26
40
62.53
60
62.79
120
63.06
63.33
2
3
4
9.39
5.23
3.92
9.41
5.22
3.90
9.42
5.20
3.87
9.44
5.18
3.84
9.45
5.18
3.83
9.46
5.17
3.82
9.47
5.16
3.80
9.47
5.15
3.79
9.48
5.14
3.78
9.49
5.13
3.76
5
3.30
3.27
3.24
3.21
3.19
3.17
3.16
3.14
6
2.94
2.90
2.87
2.84
2.82
2.80
2.78
2.76
3.12
2.74
3.10
2.72
7
8
9
2.70
2.54
2.42
2.67
2.50
2.38
2.63
2.46
2.34
2.59
2.42
2.30
2.58
2.40
2.28
2.56
2.38
2.25
2.54
2.36
2.23
2.51
2.34
2.21
2.49
2.32
2.18
2.47
2.29
2.16
10
11
12
13
14
2.32
2.25
2.19
2.14
2.10
2.28
2.21
2.15
2.10
2.05
2.24
2.17
2.10
2.05
2.01
2.20
2.12
2.06
2.01
1.96
2.18
2.10
2.04
1.98
1.94
2.16
2.08
2.01
1.96
1.91
2.13
2.05
1.99
1.93
1.89
2.11
2.03
1.96
1.90
1.86
2.08
2.00
1.93
1.88
1.83
2.06
1.97
1.90
1.85
1.80
Numerator Degrees of Freedom (k1)
15
2.06
2.02
1.97
1.92
16
17
18
19
2.03
2.00
1.98
1.96
1.99
1.96
1.93
1.91
1.94
1.91
1.89
1.86
1.89
1.86
1.84
1.81
1.90
1.87
1.84
1.81
1.79
1.87
1.84
1.81
1.78
1.76
1.85
1.81
1.78
1.75
1.73
1.82
1.78
1.75
1.72
1.70
1.79
1.75
1.72
1.69
1.67
1.76
1.72
1.69
1.66
1.63
20
1.94
1.89
21
22
23
24
1.92
1.90
1.89
1.88
1.87
1.86
1.84
1.83
1.84
1.83
1.81
1.80
1.78
1.79
1.78
1.76
1.74
1.73
1.77
1.75
1.73
1.72
1.70
1.74
1.72
1.70
1.69
1.67
1.71
1.69
1.67
1.66
1.64
1.68
1.66
1.64
1.62
1.61
1.64
1.62
1.60
1.59
1.57
1.61
1.59
1.57
1.55
1.53
25
26
27
28
29
1.87
1.86
1.85
1.84
1.83
1.82
1.81
1.80
1.79
1.78
1.77
1.76
1.75
1.74
1.73
1.72
1.71
1.70
1.69
1.68
1.69
1.68
1.67
1.66
1.65
1.66
1.65
1.64
1.63
1.62
1.63
1.61
1.60
1.59
1.58
1.59
1.58
1.57
1.56
1.55
1.56
1.54
1.53
1.52
1.51
1.52
1.50
1.49
1.48
1.47
30
1.82
1.77
1.72
1.67
1.64
1.61
1.57
1.54
1.50
1.46
40
60
120
1.76
1.71
1.65
1.71
1.66
1.60
1.66
1.60
1.55
1.61
1.54
1.48
1.57
1.51
1.45
1.54
1.48
1.41
1.51
1.44
1.37
1.47
1.40
1.32
1.42
1.35
1.26
1.38
1.29
1.19
1.60
1.55
1.49
1.42
1.38
1.34
1.30
1.24
1.17
1.00
844
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
764
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 5 (continued)
Denominator
Degrees of
Freedom (k2)
1
Critical Values of the F Distribution for 0.05
Numerator Degrees of Freedom (k1)
1
161.4
2
199.5
3
215.7
4
224.6
19.25
9.12
6.39
5
230.2
19.30
9.01
6.26
6
234.0
19.33
8.94
6.16
7
236.8
19.35
8.89
6.09
8
238.9
19.37
8.85
6.04
9
240.5
2
3
4
18.51
10.13
7.71
19.00
9.55
6.94
19.16
9.28
6.59
19.38
8.81
6.00
5
6.61
5.79
5.41
5.19
5.05
4.95
4.88
4.82
4.77
6
5.99
5.14
4.76
7
8
9
5.59
5.32
5.12
4.74
4.46
4.26
4.35
4.07
3.86
4.53
4.12
3.84
3.63
4.39
3.97
3.69
3.48
4.28
3.87
3.58
3.37
4.21
3.79
3.50
3.29
4.15
3.73
3.44
3.23
4.10
3.68
3.39
3.18
10
11
4.96
4.84
4.10
3.98
3.71
3.59
3.48
3.36
3.33
3.20
3.22
3.09
12
13
14
4.75
4.67
4.60
3.89
3.81
3.74
3.49
3.41
3.34
3.26
3.18
3.11
3.11
3.03
2.96
15
16
17
18
19
4.54
4.49
4.45
4.41
4.38
3.68
3.63
3.59
3.55
3.52
3.29
3.24
3.20
3.16
3.13
3.06
3.01
2.96
2.93
2.90
20
21
22
4.35
4.32
4.30
3.49
3.47
3.44
3.10
3.07
3.05
23
24
4.28
4.26
3.42
3.40
25
26
27
28
29
4.24
4.23
4.21
4.20
4.18
30
40
3.14
3.07
3.02
3.00
2.92
2.85
3.01
2.91
2.83
2.76
2.95
2.85
2.77
2.70
2.90
2.80
2.71
2.65
2.90
2.85
2.81
2.77
2.74
2.79
2.74
2.70
2.66
2.63
2.71
2.66
2.61
2.58
2.54
2.64
2.59
2.55
2.51
2.48
2.59
2.54
2.49
2.46
2.42
2.87
2.84
2.71
2.68
2.60
2.57
2.51
2.49
2.45
2.42
2.39
2.37
3.03
3.01
2.82
2.80
2.78
2.66
2.64
2.62
2.55
2.53
2.51
2.46
2.44
2.42
2.40
2.37
2.36
2.34
2.32
2.30
3.39
3.37
3.35
3.34
3.33
2.99
2.98
2.96
2.95
2.93
2.76
2.74
2.73
2.71
2.70
2.60
2.59
2.57
2.56
2.55
2.49
2.47
2.46
2.45
2.43
2.40
2.39
2.37
2.36
2.35
2.34
2.32
2.31
2.29
2.28
2.28
2.27
2.25
2.24
2.22
4.17
3.32
2.92
2.69
2.53
2.42
2.33
2.27
2.21
4.08
3.23
2.84
2.61
2.45
2.34
2.25
2.18
2.12
60
120
4.00
3.15
2.76
2.53
2.37
2.25
2.17
2.10
2.04
3.92
3.07
2.68
2.45
2.29
2.17
2.09
2.02
1.96
3.84
3.00
2.60
2.37
2.21
2.10
2.01
1.94
1.88
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 5 (continued)
Denominator
Degrees of
Freedom (k2)
1
845
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
765
Critical Values of the F Distribution for 0.05
Numerator Degrees of Freedom (k1)
10
241.9
12
243.9
15
245.9
20
248.0
24
249.1
30
250.1
40
251.1
60
252.2
120
253.3
254.3
2
3
4
19.40
8.79
5.96
19.41
8.74
5.91
19.43
8.70
5.86
19.45
8.66
5.80
19.45
8.64
5.77
19.46
8.62
5.75
19.47
8.59
5.72
19.48
8.57
5.69
19.49
8.55
5.66
19.50
8.53
5.63
5
4.74
4.68
4.62
4.56
4.53
4.50
4.46
4.43
6
4.06
4.00
3.94
3.87
3.84
3.81
3.77
3.74
4.40
3.70
4.36
3.67
7
8
9
3.64
3.35
3.14
3.57
3.28
3.07
3.51
3.22
3.01
3.44
3.15
2.94
3.41
3.12
2.90
3.38
3.08
2.86
3.34
3.04
2.83
3.30
3.01
2.79
3.27
2.97
2.75
3.23
2.93
2.71
10
11
12
13
14
2.98
2.85
2.75
2.67
2.60
2.91
2.79
2.69
2.60
2.53
2.85
2.72
2.62
2.53
2.46
2.77
2.65
2.54
2.46
2.39
2.74
2.61
2.51
2.42
2.35
2.70
2.57
2.47
2.38
2.31
2.66
2.53
2.43
2.34
2.27
2.62
2.49
2.38
2.30
2.22
2.58
2.45
2.34
2.25
2.18
2.54
2.40
2.30
2.21
2.13
15
2.54
2.48
2.40
2.33
16
17
18
19
2.49
2.45
2.41
2.38
2.42
2.38
2.34
2.31
2.35
2.31
2.27
2.23
2.28
2.23
2.19
2.16
2.29
2.24
2.19
2.15
2.11
2.25
2.19
2.15
2.11
2.07
2.20
2.15
2.10
2.06
2.03
2.16
2.11
2.06
2.02
1.98
2.11
2.06
2.01
1.97
1.93
2.07
2.01
1.96
1.92
1.88
20
2.35
2.28
21
22
23
24
2.32
2.30
2.27
2.25
2.25
2.23
2.20
2.18
2.20
2.18
2.15
2.13
2.11
2.12
2.10
2.07
2.05
2.03
2.08
2.05
2.03
2.01
1.98
2.04
2.01
1.98
1.96
1.94
1.99
1.96
1.94
1.91
1.89
1.95
1.92
1.89
1.86
1.84
1.90
1.87
1.84
1.81
1.79
1.84
1.81
1.78
1.76
1.73
25
26
27
28
29
2.24
2.22
2.20
2.19
2.18
2.16
2.15
2.13
2.12
2.10
2.09
2.07
2.06
2.04
2.03
2.01
1.99
1.97
1.96
1.94
1.96
1.95
1.93
1.91
1.90
1.92
1.90
1.88
1.87
1.85
1.87
1.85
1.84
1.82
1.81
1.82
1.80
1.79
1.77
1.75
1.77
1.75
1.73
1.71
1.70
1.71
1.69
1.67
1.65
1.64
30
2.16
2.09
2.01
1.93
1.89
1.84
1.79
1.74
1.68
1.62
40
60
120
2.08
1.99
1.91
2.00
1.92
1.83
1.92
1.84
1.75
1.84
1.75
1.66
1.79
1.70
1.61
1.74
1.65
1.55
1.69
1.59
1.50
1.64
1.53
1.43
1.58
1.47
1.35
1.51
1.39
1.25
1.83
1.75
1.67
1.57
1.52
1.46
1.39
1.32
1.22
1.00
846
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
766
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 5 (continued)
Denominator
Degrees of
Freedom (k2)
1
Critical Values of the F Distribution for 0.025
Numerator Degrees of Freedom (k1)
1
647.8
2
799.5
3
864.2
4
899.6
39.25
15.10
9.60
5
921.8
39.30
14.88
9.36
6
937.1
39.33
14.73
9.20
7
948.2
39.36
14.62
9.07
8
956.7
39.37
14.54
8.98
9
963.3
2
3
4
38.51
17.44
12.22
39.00
16.04
10.65
39.17
15.44
9.98
39.39
14.47
8.90
5
10.01
8.43
7.76
7.39
7.15
6.98
6.85
6.76
6.68
6
8.81
7.26
6.60
7
8
9
8.07
7.57
7.21
6.54
6.06
5.71
5.89
5.42
5.08
6.23
5.52
5.05
4.72
5.99
5.29
4.82
4.48
5.82
5.12
4.65
4.32
5.70
4.99
4.53
4.20
5.60
4.90
4.43
4.10
5.52
4.82
4.36
4.03
10
11
6.94
6.72
5.46
5.26
4.83
4.63
4.47
4.28
4.24
4.04
4.07
3.88
12
13
14
6.55
6.41
6.30
5.10
4.97
4.86
4.47
4.35
4.24
4.12
4.00
3.89
3.89
3.77
3.66
15
16
17
18
19
6.20
6.12
6.04
5.98
5.92
4.77
4.69
4.62
4.56
4.51
4.15
4.08
4.01
3.95
3.90
3.80
3.73
3.66
3.61
3.56
20
21
22
5.87
5.83
5.79
4.46
4.42
4.38
3.86
3.82
3.78
23
24
5.75
5.72
4.35
4.32
25
26
27
28
29
5.69
5.66
5.63
5.61
5.59
30
40
3.95
3.85
3.78
3.73
3.60
3.50
3.76
3.61
3.48
3.38
3.66
3.51
3.39
3.29
3.59
3.44
3.31
3.21
3.58
3.50
3.44
3.38
3.33
3.41
3.34
3.28
3.22
3.17
3.29
3.22
3.16
3.10
3.05
3.20
3.12
3.06
3.01
2.96
3.12
3.05
2.98
2.93
2.88
3.51
3.48
3.29
3.25
3.13
3.09
3.01
2.97
2.91
2.87
2.84
2.80
3.75
3.72
3.44
3.41
3.38
3.22
3.18
3.15
3.05
3.02
2.99
2.93
2.90
2.87
2.84
2.81
2.78
2.76
2.73
2.70
4.29
4.27
4.24
4.22
4.20
3.69
3.67
3.65
3.63
3.61
3.35
3.33
3.31
3.29
3.27
3.13
3.10
3.08
3.06
3.04
2.97
2.94
2.92
2.90
2.88
2.85
2.82
2.80
2.78
2.76
2.75
2.73
2.71
2.69
2.67
2.68
2.65
2.63
2.61
2.59
5.57
4.18
3.59
3.25
3.03
2.87
2.75
2.65
2.57
5.42
4.05
3.46
3.13
2.90
2.74
2.62
2.53
2.45
60
120
5.29
3.93
3.34
3.01
2.79
2.63
2.51
2.41
2.33
5.15
3.80
3.23
2.89
2.67
2.52
2.39
2.30
2.22
5.02
3.69
3.12
2.79
2.57
2.41
2.29
2.19
2.11
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 5 (continued)
Denominator
Degrees of
Freedom (k2)
1
847
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
767
Critical Values of the F Distribution for 0.025
Numerator Degrees of Freedom (k1)
10
968.6
12
976.7
15
984.9
20
993.1
39.45
14.17
8.56
24
997.2
39.46
14.12
8.51
30
1001
39.46
14.08
8.46
40
1006
39.47
14.04
8.41
60
1010
39.48
13.99
8.36
120
1014
39.49
13.95
8.31
1018
2
3
4
39.40
14.42
8.84
39.41
14.34
8.75
39.43
14.25
8.66
39.50
13.90
8.26
5
6.62
6.52
6.43
6.33
6.28
6.23
6.18
6.12
6.07
6.02
6
5.46
5.37
5.27
7
8
9
4.76
4.30
3.96
4.67
4.20
3.87
4.57
4.10
3.77
5.17
4.47
4.00
3.67
5.12
4.42
3.95
3.61
5.07
4.36
3.89
3.56
5.01
4.31
3.84
3.51
4.96
4.25
3.78
3.45
4.90
4.20
3.73
3.39
4.85
4.14
3.67
3.33
10
11
3.72
3.53
12
13
14
15
16
17
18
19
3.62
3.52
3.42
3.37
3.31
3.26
3.20
3.14
3.08
3.37
3.25
3.15
3.43
3.28
3.15
3.05
3.33
3.18
3.05
2.95
3.23
3.07
2.95
2.84
3.17
3.02
2.89
2.79
3.12
2.96
2.84
2.73
3.06
2.91
2.78
2.67
3.00
2.85
2.72
2.61
2.94
2.79
2.66
2.55
2.88
2.72
2.60
2.49
3.06
2.99
2.92
2.87
2.82
2.96
2.89
2.82
2.77
2.72
2.86
2.79
2.72
2.67
2.62
2.76
2.68
2.62
2.56
2.51
2.70
2.63
2.56
2.50
2.45
2.64
2.57
2.50
2.44
2.39
2.59
2.51
2.44
2.38
2.33
2.52
2.45
2.38
2.32
2.27
2.46
2.38
2.32
2.26
2.20
2.40
2.32
2.25
2.19
2.13
20
2.77
2.68
2.57
2.46
2.41
2.35
2.29
21
22
23
24
2.73
2.70
2.67
2.64
2.64
2.60
2.57
2.54
2.53
2.50
2.47
2.44
2.42
2.39
2.36
2.33
2.37
2.33
2.30
2.27
2.31
2.27
2.24
2.21
2.25
2.21
2.18
2.15
2.22
2.18
2.14
2.11
2.08
2.16
2.11
2.08
2.04
2.01
2.09
2.04
2.00
1.97
1.94
25
26
27
28
2.61
2.59
2.57
2.55
2.51
2.49
2.47
2.45
2.41
2.39
2.36
2.34
2.30
2.28
2.25
2.23
2.24
2.22
2.19
2.17
2.18
2.16
2.13
2.11
2.12
2.09
2.07
2.05
2.05
2.03
2.00
1.98
1.98
1.95
1.93
1.91
1.91
1.88
1.85
1.83
29
2.53
2.43
2.32
2.21
2.15
2.09
2.03
1.96
1.89
1.81
30
2.51
2.41
2.31
40
2.39
2.29
2.18
2.20
2.07
2.14
2.01
2.07
1.94
2.01
1.88
1.94
1.80
1.87
1.72
1.79
1.64
60
120
2.27
2.16
2.05
2.17
2.05
1.94
2.06
1.94
1.83
1.94
1.82
1.71
1.88
1.76
1.64
1.82
1.69
1.57
1.74
1.61
1.48
1.67
1.53
1.39
1.58
1.43
1.27
1.48
1.31
1.00
848
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
768
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
Critical Values of the F Distribution for 0.01
TABLE 5 (continued)
Denominator
Degrees of
Freedom (k2)
1
Numerator Degrees of Freedom (k1)
1
4,052
2
4,999.5
3
5,403
4
5,625
5
5,764
6
5,859
7
5,928
8
5,982
9
6,022
2
3
4
98.50
34.12
21.20
99.00
30.82
18.00
99.17
29.46
16.69
99.25
28.71
15.98
99.30
28.24
15.52
99.33
27.91
15.21
99.36
27.67
14.98
99.37
27.49
14.80
99.39
27.35
14.66
5
16.26
13.27
12.06
11.39
10.97
10.67
10.46
10.29
10.16
6
7
13.75
12.25
10.92
9.55
9.78
9.15
8.75
8.47
8.26
8.10
7.98
8
9
11.26
10.56
8.65
8.02
8.45
7.59
6.99
7.85
7.01
6.42
7.46
6.63
6.06
7.19
6.37
5.80
6.99
6.18
5.61
6.84
6.03
5.47
6.72
5.91
5.35
10
11
12
13
14
10.04
9.65
9.33
9.07
8.86
7.56
7.21
6.93
6.70
6.51
6.55
6.22
5.95
5.74
5.56
5.99
5.67
5.41
5.21
5.04
5.64
5.32
5.06
4.86
4.69
5.39
5.07
4.82
4.62
4.46
5.20
4.89
4.64
4.44
4.28
5.06
4.74
4.50
4.30
4.14
4.94
4.63
4.39
4.19
4.03
15
16
17
8.68
8.53
8.40
6.36
6.23
6.11
5.42
5.29
5.18
4.89
4.77
4.67
4.56
4.44
4.34
4.32
4.20
4.10
4.14
4.03
3.93
4.00
3.89
3.79
3.89
3.78
18
19
8.29
8.18
6.01
5.93
5.09
5.01
4.58
4.50
4.25
4.17
4.01
3.94
3.84
3.77
3.71
3.63
3.68
3.60
3.52
20
21
22
23
8.10
8.02
7.95
7.88
5.85
5.78
5.72
4.94
4.87
4.82
4.43
4.37
4.31
4.10
4.04
3.99
3.87
3.81
3.76
3.70
3.64
3.59
3.56
3.51
3.45
3.46
3.40
3.35
24
7.82
5.66
5.61
4.76
4.72
4.26
4.22
3.94
3.90
3.71
3.67
3.54
3.50
3.41
3.36
3.30
3.26
25
26
27
28
29
7.77
7.72
7.68
7.64
7.60
5.57
5.53
5.49
5.45
5.42
4.68
4.64
4.60
4.57
4.54
4.18
4.14
4.11
4.07
4.04
3.85
3.82
3.78
3.75
3.63
3.59
3.56
3.53
3.46
3.42
3.39
3.36
3.32
3.29
3.26
3.23
3.22
3.18
3.15
3.12
3.73
3.50
3.33
3.20
3.09
30
7.56
5.39
4.51
4.02
3.70
3.47
3.30
3.17
3.07
40
60
120
7.31
7.08
6.85
5.18
4.98
4.79
4.31
4.13
3.95
3.83
3.65
3.48
3.51
3.34
3.17
3.29
3.12
2.96
3.12
2.95
2.79
2.99
2.82
2.89
2.72
6.63
4.61
3.78
3.32
3.02
2.80
2.64
2.66
2.51
2.56
2.41
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
849
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Statistical Tables
769
Critical Values of the F Distribution for 0.01
TABLE 5 (continued)
Denominator
Degrees of
10
Freedom (k2)
1
6,056
Numerator Degrees of Freedom (k1)
12
6,106
15
6,157
20
6,209
24
6,235
30
6,261
40
6,287
60
6,313
120
6,339
6,366
2
3
4
99.40
27.23
14.55
99.42
27.05
14.37
99.43
26.87
14.20
99.45
26.69
14.02
99.46
26.60
13.93
99.47
26.50
13.84
99.47
26.41
13.75
99.48
26.32
13.65
99.49
26.22
13.56
99.50
26.13
13.46
5
10.05
9.89
9.72
9.55
9.47
9.38
9.29
9.20
9.11
9.02
6
7.87
7.72
7.56
7.40
7.31
7.23
7.14
7.06
7
8
9
6.62
5.81
5.26
6.47
5.67
5.11
6.31
5.52
4.96
6.16
5.36
4.81
6.07
5.28
4.73
5.99
5.20
4.65
5.91
5.12
4.57
5.82
5.03
4.48
6.97
5.74
4.95
4.40
6.88
5.65
4.86
4.31
10
11
4.85
4.54
4.71
4.40
4.56
4.25
4.41
4.10
4.33
4.02
4.25
3.94
12
13
14
4.30
4.10
3.94
4.16
3.96
3.80
4.01
3.82
3.66
3.86
3.66
3.51
3.78
3.59
3.43
15
16
17
18
19
3.80
3.69
3.59
3.51
3.43
3.67
3.55
3.46
3.37
3.30
3.52
3.41
3.31
3.23
3.15
3.37
3.26
3.16
3.08
3.00
3.29
3.18
3.08
3.00
2.92
20
21
3.37
3.31
3.23
3.17
22
23
24
3.26
3.21
3.17
25
26
27
28
4.17
4.08
4.00
3.91
3.70
3.51
3.35
3.86
3.62
3.43
3.27
3.78
3.54
3.34
3.18
3.69
3.45
3.25
3.09
3.60
3.36
3.17
3.00
3.21
3.10
3.00
2.92
2.84
3.13
3.02
2.92
2.84
2.76
3.05
2.93
2.83
2.75
2.67
2.96
2.84
2.75
2.66
2.58
2.87
2.75
2.65
2.57
2.49
3.09
2.94
2.86
2.78
2.69
2.61
2.52
2.42
3.12
3.07
3.03
3.03
2.98
2.93
2.89
2.88
2.83
2.78
2.74
2.80
2.75
2.70
2.66
2.72
2.67
2.62
2.58
2.64
2.58
2.54
2.49
2.55
2.50
2.45
2.40
2.46
2.40
2.35
2.31
2.36
2.31
2.26
2.21
3.13
3.09
3.06
3.03
2.99
2.96
2.93
2.90
2.85
2.81
2.78
2.75
2.70
2.66
2.63
2.60
2.62
2.58
2.55
2.52
2.54
2.50
2.47
2.44
2.45
2.42
2.38
2.35
2.36
2.33
2.29
2.26
2.27
2.23
2.20
2.17
2.17
2.13
2.10
2.06
29
3.00
2.87
2.73
2.57
2.49
2.41
2.33
2.23
2.14
2.03
30
2.98
2.84
2.70
2.55
2.47
2.39
2.30
2.21
40
2.80
2.66
2.52
2.37
2.29
2.20
2.11
2.02
2.11
1.92
2.01
1.80
60
120
2.63
2.47
2.50
2.34
2.35
2.19
2.20
2.03
2.12
1.95
2.03
1.86
1.94
1.76
1.84
1.66
1.73
1.53
1.60
1.38
2.32
2.18
2.04
1.88
1.79
1.70
1.59
1.47
1.32
1.00
Source: M. Merrington and C. M. Thompson, “Tables of Percentage Points of the Inverted Beta (F )-Distribution,” Biometrika 33 (1943), pp. 73–88.
Reproduced by permission of the Biometrika Trustees.
3
216
4
225
5
230
6
234
7
237
8
239
9
241
10
242
11
243
12
244
14
245
16
246
20
248
24
249
30
250
40
251
50
252
75
253
100
253
200 500
254 254
254
9.01
8.94
8.88 8.84
8.81
8.78
8.76 8.74
8.71 8.69
8.66 8.64
8.62
8.60
8.58 8.57
8.56
8.54
8.54
8.53
4.96
13
4.75
12
4.67
9.07
3.80 3.41
6.70 5.74
3.88 3.49
6.93 5.95
3.98 3.59
7.20 6.22
4.10 3.71
7.56 6.55
3.18
5.20
3.26
5.41
3.36
5.67
3.48
5.99
3.63
6.42
3.02
4.86
3.11
5.06
3.20
5.32
3.33
5.64
3.48
6.06
2.92
4.62
3.00
4.82
3.09
5.07
3.22
5.39
3.37
5.80
2.84 2.77
4.44 4.30
2.92 2.85
4.65 4.50
3.01 2.95
4.88 4.74
3.14 3.07
5.21 5.06
3.29 3.23
5.62 5.47
3.50 3.44
6.19 6.03
2.72
4.19
2.80
4.39
2.90
4.63
3.02
4.95
3.18
5.35
3.39
5.91
2.67
4.10
2.76
4.30
2.86
4.54
2.97
4.85
3.13
5.26
3.34
5.82
3.63
6.62
2.63 2.60
4.02 3.96
2.72 2.69
4.22 4.16
2.82 2.79
4.46 4.40
2.94 2.91
4.78 4.71
3.10 3.07
5.18 5.11
3.31 3.28
5.74 5.67
3.60 3.57
6.54 6.47
2.55 2.51
3.85 3.78
2.64 2.60
4.05 3.98
2.74 2.70
4.29 4.21
2.86 2.82
4.60 4.52
3.02 2.98
5.00 4.92
3.23 3.20
5.56 5.48
3.52 3.49
6.35 6.27
3.96 3.92
7.60 7.52
2.46 2.42
3.67 3.59
2.54 2.50
3.86 3.78
2.65 2.61
4.10 4.02
2.77 2.74
4.41 4.33
2.93 2.90
4.80 4.73
3.15 3.12
5.36 5.28
3.44 3.41
6.15 6.07
3.87 3.84
7.39 7.31
2.38
3.51
2.46
3.70
2.57
3.94
2.70
4.25
2.86
4.64
3.08
5.20
3.38
5.98
3.81
7.23
2.34
3.42
2.42
3.61
2.53
3.86
2.67
4.17
2.82
4.56
3.05
5.11
3.34
5.90
3.77
7.14
2.32 2.28
3.37 3.30
2.40 2.36
3.56 3.49
2.50 2.47
3.80 3.74
2.64 2.61
4.12 4.05
2.80 2.77
4.51 4.45
3.03 3.00
5.06 5.00
3.32 3.29
5.85 5.78
3.75 3.72
7.09 7.02
2.26
3.27
2.35
3.46
2.45
3.70
2.59
4.01
2.76
4.41
2.98
4.96
3.28
5.75
3.71
6.99
2.24
3.21
2.32
3.41
2.42
3.66
2.56
3.96
2.73
4.36
2.96
4.91
3.25
5.70
3.69
6.94
2.22
3.18
2.31
3.38
2.41
3.62
2.55
3.93
2.72
4.33
2.94
4.88
3.24
5.67
3.68
6.90
2.21
3.16
2.30
3.36
2.40
3.60
2.54
3.91
2.71
4.31
2.93
4.86
3.23
5.65
3.67
6.88
Appendix C: Statistical
Tables
9.33
4.84
9.65
11
10.04
10
4.26 3.86
8.02 6.99
3.58
6.37
3.68
6.71
4.03 4.00
7.79 7.72
4.36
9.02
5.12
10.56
3.69
6.63
3.79 3.73
7.00 6.84
4.06
7.87
4.37
9.04
9
3.84
7.01
4.46 4.07
8.65 7.59
3.87
7.19
4.10
7.98
4.38
9.07
5.32
11.26
3.97
7.46
4.21 4.15
8.26 8.10
4.40
9.13
8
4.12
7.85
4.74 4.35
9.55 8.45
4.28
8.47
4.44 4.42
9.24 9.17
5.59
12.25
4.39
8.75
4.46
9.29
7
4.53
9.15
4.50
9.38
5.99 5.14 4.76
13.74 10.92 9.78
4.56 4.53
9.55 9.47
6
4.64 4.60
9.77 9.68
6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.78 4.74
16.26 13.27 12.06 11.39 10.97 10.67 10.45 10.29 10.15 10.05
5
Back Matter
4.70 4.68
9.96 9.89
7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.93 5.91 5.87 5.84 5.80 5.77 5.74 5.71 5.70 5.68 5.66 5.65 5.64 5.63
21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.54 14.45 14.37 14.24 14.15 14.02 13.93 13.83 13.74 13.69 13.61 13.57 13.52 13.48 13.46
4
34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.34 27.23 27.13 27.05 26.92 26.83 26.69 26.60 26.50 26.41 26.35 26.27 26.23 26.18 26.14 26.12
9.12
10.13
3
9.55 9.28
18.51 19.00 19.16 19.25 19.30 19.33 19.36 19.37 19.38 19.39 19.40 19.41 19.42 19.43 19.44 19.45 19.46 19.47 19.47 19.48 19.49 19.49 19.50 19.50
98.49 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.41 99.42 99.43 99.44 99.45 99.46 99.47 99.48 99.48 99.49 99.49 99.49 99.50 99.50
2
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
4,052 4,999 5,403 5,625 5,764 5,859 5,928 5,981 6,022 6,056 6,082 6,106 6,142 6,169 6,208 6,234 6,261 6,286 6,302 6,323 6,334 6,352 6,361 6,366
2
200
Numerator Degrees of Freedom (k1)
The F Distribution for 0.05 and 0.01 for Many Possible Degrees of Freedom
Denominator
Degrees of
Freedom (k2) 1
1
161
TABLE 5A
850
© The McGraw−Hill
Companies, 2009
4.41
18
3.37 2.98
5.53 4.64
4.30
7.94
4.28
7.88
4.26
7.82
4.24
7.77
4.22
7.72
22
23
24
25
26
3.38 2.99
5.57 4.68
3.40 3.01
5.61 4.72
3.42 3.03
5.66 4.76
3.44 3.05
5.72 4.82
4.14
2.74
2.76
4.18
2.78
4.22
2.59
3.82
2.60
3.86
2.62
3.90
2.64
3.94
2.66
3.99
2.82
4.31
2.80
4.26
2.68
4.04
2.84
4.37
2.71
4.10
2.74
4.17
2.77
4.25
2.81
4.34
2.85
4.44
4.56
2.90
2.96
4.69
2.47
3.59
2.49
3.63
2.51
3.67
2.53
3.71
2.55
3.76
2.57
3.81
2.60
3.87
2.63
3.94
3.66
4.01
2.70
4.10
2.74
4.20
4.32
2.79
2.85
4.46
2.39 2.32
3.42 3.29
2.41 2.34
3.46 3.32
2.43 2.36
3.50 3.36
2.45 2.38
3.54 3.41
2.47 2.40
3.59 3.45
2.49 2.42
3.65 3.51
2.52 2.45
3.71 3.56
2.55 2.48
3.77 3.63
2.58 2.51
3.85 3.71
2.62 2.55
3.93 3.79
2.66 2.59
4.03 3.89
4.14 4.00
2.70 2.64
2.77 2.70
4.28 4.14
2.27
3.17
2.28
3.21
2.30
3.25
2.32
3.30
2.35
3.35
2.37
3.40
2.40
3.45
2.43
3.52
2.46
3.60
2.50
3.68
2.54
3.78
3.89
2.59
2.65
4.03
2.22
3.09
2.24
3.13
2.26
3.17
2.28
3.21
2.30
3.26
2.32
3.31
2.35
3.37
2.38
3.43
2.41
3.51
2.45
3.59
2.49
3.69
3.80
2.55
2.60
3.94
2.18 2.15
3.02 2.96
2.20 2.16
3.05 2.99
2.22 2.18
3.09 3.03
2.24 2.20
3.14 3.07
2.26 2.23
3.18 3.12
2.28 2.25
3.24 3.17
2.31 2.28
3.30 3.23
2.34 2.31
3.36 3.30
2.37 2.34
3.44 3.37
2.41 2.38
3.52 3.45
2.45 2.42
3.61 3.55
3.73 3.67
2.51 2.48
2.56 2.53
3.86 3.80
2.10 2.05
2.86 2.77
2.11 2.06
2.89 2.81
2.13 2.09
2.93 2.85
2.14 2.10
2.97 2.89
2.18 2.13
3.02 2.94
2.20 2.15
3.07 2.99
2.23 2.18
3.13 3.05
2.26 2.21
3.19 3.12
2.29 2.25
3.27 3.19
2.33 2.29
3.35 3.27
2.37 2.33
3.45 3.37
3.56 3.48
2.43 2.39
2.48 2.44
3.70 3.62
1.99 1.95
2.66 2.58
2.00 1.96
2.70 2.62
2.02 1.98
2.74 2.66
2.04 2.00
2.78 2.70
2.07 2.03
2.83 2.75
2.09 2.05
2.88 2.80
2.12 2.08
2.94 2.86
2.15 2.11
3.00 2.92
2.19 2.15
3.07 3.00
2.23 2.19
3.16 3.08
2.28 2.24
3.25 3.18
3.36 3.29
2.33 2.29
2.39 2.35
3.51 3.43
1.90
2.50
1.92
2.54
1.94
2.58
1.96
2.62
1.98
2.67
2.00
2.72
2.04
2.77
2.07
2.84
2.11
2.91
2.15
3.00
2.20
3.10
3.20
2.25
2.31
3.34
1.85
2.41
1.87
2.45
1.89
2.49
1.91
2.53
1.93
2.58
1.96
2.63
1.99
2.69
2.02
2.76
2.07
2.83
2.11
2.92
2.16
3.01
3.12
2.21
2.27
3.26
1.82 1.78
2.36 2.28
1.84 1.80
2.40 2.32
1.86 1.82
2.44 2.36
1.88 1.84
2.48 2.41
1.91 1.87
2.53 2.46
1.93 1.89
2.58 2.51
1.96 1.92
2.63 2.56
2.00 1.96
2.70 2.63
2.04 2.00
2.78 2.71
2.08 2.04
2.86 2.79
2.13 2.09
2.96 2.98
3.07 3.00
2.18 2.15
2.24 2.21
3.21 3.14
1.76
2.25
1.77
2.29
1.80
2.33
1.82
2.37
1.84
2.42
1.87
2.47
1.90
2.53
1.94
2.60
1.98
2.68
2.02
2.76
2.07
2.86
2.97
2.12
2.19
3.11
1.72
2.19
1.74
2.23
1.76
2.27
1.79
2.32
1.81
2.37
1.84
2.42
1.87
2.47
1.91
2.54
1.95
2.62
1.99
2.70
2.04
2.80
2.92
2.10
2.16
3.06
1.70
2.15
1.72
2.19
1.74
2.23
1.77
2.28
1.80
2.33
1.82
2.38
1.85
2.44
1.90
2.51
1.93
2.59
1.97
2.67
2.02
2.77
2.89
2.08
2.14
3.02
1.69
2.13
1.71
2.17
1.73
2.21
1.76
2.26
1.78
2.31
1.81
2.36
1.84
2.42
1.88
2.49
1.92
2.57
1.96
2.65
2.01
2.75
2.87
2.07
2.13
3.00
Back Matter
3.47 3.07
5.78 4.87
4.32
8.02
21
2.87
4.43
3.49 3.10
5.85 4.94
4.35
8.10
20
2.90
4.50
2.93
4.58
2.96
4.67
3.52 3.13
5.93 5.01
4.38
8.18
3.55 3.16
6.01 5.09
3.59 3.20
6.11 5.18
19
8.28
4.45
8.40
17
3.01
4.77
3.63 3.24
6.23 5.29
4.49
8.53
16
3.06
4.89
3.68 3.29
6.36 5.42
4.54
8.68
3.11
5.03
15
3.74 3.34
6.51 5.56
4.60
8.86
14
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Appendix C: Statistical
Tables
© The McGraw−Hill
Companies, 2009
851
4.18
7.60
4.17
7.56
4.15
7.50
4.13
7.44
4.11
29
30
32
34
36
3.19
5.08
4.07
7.27
4.06
7.24
4.05
7.21
4.04
7.19
42
44
46
48
3.20
5.10
3.21
5.12
3.22
5.15
3.23
5.18
4.08
7.31
40
3.25
5.21
4.10
7.35
38
3.26
5.25
4.22
2.56
3.74
2.57
3.76
2.58
3.78
2.41
3.42
2.42
3.44
2.43
3.46
2.44
3.49
2.45
3.51
2.46
3.54
2.48
3.58
2.49
3.61
2.51
3.66
2.53
3.70
2.54
3.73
2.56
3.76
3.79
5
2.57
2.30
3.20
2.30
3.22
2.31
3.24
2.32
3.26
2.34
3.29
2.35
3.32
2.36
3.35
2.38
3.38
2.40
3.42
2.42
3.47
2.43
3.50
2.44
3.53
3.56
6
2.46
2.21 2.14
3.04 2.90
2.22 2.14
3.05 2.92
2.23 2.16
3.07 2.94
2.24 2.17
3.10 2.96
2.25 2.18
3.12 2.99
2.26 2.19
3.15 3.02
2.28 2.21
3.18 3.04
2.30 2.23
3.21 3.08
2.32 2.25
3.25 3.12
2.34 2.27
3.30 3.17
2.35 2.28
3.33 3.20
2.36 2.29
3.36 3.23
3.39 3.26
7
8
2.37 2.30
2.08
2.80
2.09
2.82
2.10
2.84
2.11
2.86
2.12
2.88
2.14
2.91
2.15
2.94
2.17
2.97
2.19
3.01
2.21
3.06
2.22
3.08
2.24
3.11
3.14
9
2.25
2.03
2.71
2.04
2.73
2.05
2.75
2.06
2.77
2.07
2.80
2.09
2.82
2.10
2.86
2.12
2.89
2.14
2.94
2.16
2.98
2.18
3.00
2.19
3.03
3.06
10
2.20
1.99 1.96
2.64 2.58
2.00 1.97
2.66 2.60
2.01 1.98
2.68 2.62
2.02 1.99
2.70 2.64
2.04 2.00
2.73 2.66
2.05 2.02
2.75 2.69
2.06 2.03
2.78 2.72
2.08 2.05
2.82 2.76
2.10 2.07
2.86 2.80
2.12 2.09
2.90 2.84
2.14 2.10
2.92 2.87
2.15 2.12
2.95 2.90
2.98 2.93
11
12
2.16 2.13
1.90 1.86
2.48 2.40
1.91 1.87
2.50 2.42
1.92 1.88
2.52 2.44
1.94 1.89
2.54 2.46
1.95 1.90
2.56 2.49
1.96 1.92
2.59 2.51
1.98 1.93
2.62 2.54
2.00 1.95
2.66 2.58
2.02 1.97
2.70 2.62
2.04 1.99
2.74 2.66
2.05 2.00
2.77 2.68
2.06 2.02
2.80 2.71
2.83 2.74
14
16
2.08 2.03
1.79 1.74
2.28 2.20
1.80 1.75
2.30 2.22
1.81 1.76
2.32 2.24
1.82 1.78
2.35 2.26
1.84 1.79
2.37 2.29
1.85 1.80
2.40 2.32
1.87 1.82
2.43 2.35
1.89 1.84
2.47 2.38
1.91 1.86
2.51 2.42
1.93 1.89
2.55 2.47
1.94 1.90
2.57 2.49
1.96 1.91
2.60 2.52
2.63 2.55
20
24
1.97 1.93
1.70
2.11
1.71
2.13
1.72
2.15
1.73
2.17
1.74
2.20
1.76
2.22
1.78
2.26
1.80
2.30
1.82
2.34
1.84
2.38
1.85
2.41
1.87
2.44
2.47
30
1.88
1.64
2.02
1.65
2.04
1.66
2.06
1.68
2.08
1.69
2.11
1.71
2.14
1.72
2.17
1.74
2.21
1.76
2.25
1.79
2.29
1.80
2.32
1.81
2.35
2.38
40
1.84
1.61 1.56
1.96 1.88
1.62 1.57
1.98 1.90
1.63 1.58
2.00 1.92
1.64 1.60
2.02 1.94
1.66 1.61
2.05 1.97
1.67 1.63
2.08 2.00
1.69 1.65
2.12 2.04
1.71 1.67
2.15 2.08
1.74 1.69
2.20 2.12
1.76 1.72
2.24 2.16
1.77 1.73
2.27 2.19
1.78 1.75
2.30 2.22
2.33 2.25
50
75
1.80 1.76
1.53 1.50
1.84 1.78
1.54 1.51
1.86 1.80
1.56 1.52
1.88 1.82
1.57 1.54
1.91 1.85
1.59 1.55
1.94 1.88
1.60 1.57
1.97 1.90
1.62 1.59
2.00 1.94
1.64 1.61
2.04 1.98
1.67 1.64
2.08 2.02
1.69 1.66
2.13 2.07
1.71 1.68
2.15 2.10
1.72 1.69
2.18 2.13
2.21 2.16
100 200
1.74 1.71
1.47
1.73
1.48
1.76
1.50
1.78
1.51
1.80
1.53
1.84
1.54
1.86
1.56
1.90
1.59
1.94
1.61
1.98
1.64
2.03
1.65
2.06
1.67
2.09
2.12
500
1.68
1.67
1.45
1.70
1.46
1.72
1.48
1.75
1.49
1.78
1.51
1.81
1.53
1.84
1.55
1.87
1.57
1.91
1.59
1.96
1.62
2.01
1.64
2.03
1.65
2.06
2.10
Appendix C: Statistical
Tables
2.80
2.81
4.24
2.82
4.26
2.59
3.80
2.61
3.83
2.84
4.31
2.83
4.29
2.62
3.86
2.63
3.89
2.65
3.93
2.67
3.97
2.69
4.02
2.70
4.04
2.71
4.07
4.11
2.85
4.34
2.86
4.38
2.88
4.42
2.90
4.46
3.30
5.34
3.28
5.29
2.92
4.51
2.93
4.54
2.95
4.57
3.32
5.39
3.33
5.42
3.34
5.45
4.60
4
2.73
Back Matter
7.39
4.20
7.64
5.49
7.68
3
2.96
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
28
2
3.35
Numerator Degrees of Freedom (k1)
The F Distribution for 0.05 and 0.01 for Many Possible Degrees of Freedom
Denominator
Degrees of
1
Freedom (k2)
27
4.21
TABLE 5A (continued)
852
© The McGraw−Hill
Companies, 2009
3.89
6.76
3.86
6.70
3.85
6.66
3.84
6.63
200
400
1,000
2.99
4.60
3.00
4.62
3.02
4.66
3.04
4.71
3.06
4.75
3.07
4.78
2.60
3.78
2.61
3.80
2.62
3.83
2.65
3.88
2.67
3.91
2.68
3.94
2.70
3.98
2.72
4.04
2.37
3.32
2.38
3.34
2.39
3.36
2.21
3.02
2.22
3.04
2.23
3.06
2.26
3.11
2.27
3.14
2.43
3.44
2.41
3.41
2.29
3.17
2.30
3.20
2.33
3.25
2.35
3.29
2.44
3.47
2.46
3.51
2.48
3.56
2.50
3.60
2.74
4.08
2.36
3.31
2.37
3.34
3.37
2.38
2.40
3.41
2.09
2.80
2.10
2.82
2.12
2.85
2.14
2.90
2.16
2.92
2.17
2.95
2.19
2.99
2.21
3.04
2.23
3.07
2.24
3.09
2.25
3.12
3.15
2.27
2.29
3.18
2.01 1.94
2.64 2.51
2.02 1.95
2.66 2.53
2.03 1.96
2.69 2.55
2.05 1.98
2.73 2.60
2.07 2.00
2.76 2.62
2.08 2.01
2.79 2.65
2.10 2.03
2.82 2.69
2.12 2.05
2.87 2.74
2.14 2.07
2.91 2.77
2.15 2.08
2.93 2.79
2.17 2.10
2.95 2.82
2.98 2.85
2.18 2.11
2.20 2.13
3.02 2.88
1.88
2.41
1.89
2.43
1.90
2.46
1.92
2.50
1.94
2.53
1.95
2.56
1.97
2.59
1.99
2.64
2.01
2.67
2.02
2.70
2.04
2.72
2.75
2.05
2.07
2.78
1.83
2.32
1.84
2.34
1.85
2.37
1.87
2.41
1.89
2.44
1.90
2.47
1.92
2.51
1.95
2.55
1.97
2.59
1.98
2.61
1.99
2.63
2.66
2.00
2.02
2.70
1.79 1.75
2.24 2.18
1.80 1.76
2.26 2.20
1.81 1.78
2.29 2.23
1.83 1.80
2.34 2.28
1.85 1.82
2.37 2.30
1.86 1.83
2.40 2.33
1.88 1.85
2.43 2.36
1.91 1.88
2.48 2.41
1.93 1.89
2.51 2.45
1.94 1.90
2.54 2.47
1.95 1.92
2.56 2.50
2.59 2.53
1.97 1.93
1.98 1.95
2.62 2.56
1.69 1.64
2.07 1.99
1.70 1.65
2.09 2.01
1.72 1.67
2.12 2.04
1.74 1.69
2.17 2.09
1.76 1.71
2.20 2.12
1.77 1.72
2.23 2.15
1.79 1.75
2.26 2.19
1.82 1.77
2.32 2.24
1.84 1.79
2.35 2.28
1.85 1.80
2.37 2.30
1.86 1.81
2.40 2.32
2.43 2.35
1.88 1.83
1.90 1.85
2.46 2.39
1.57 1.52
1.87 1.79
1.58 1.53
1.89 1.81
1.60 1.54
1.92 1.84
1.62 1.57
1.97 1.88
1.64 1.59
2.00 1.91
1.65 1.60
2.03 1.94
1.68 1.63
2.06 1.98
1.70 1.65
2.11 2.03
1.72 1.67
2.15 2.07
1.73 1.68
2.18 2.09
1.75 1.70
2.20 2.12
2.23 2.15
1.76 1.72
1.78 1.74
2.26 2.18
1.46
1.69
1.47
1.71
1.49
1.74
1.52
1.79
1.54
1.83
1.55
1.85
1.57
1.89
1.60
1.94
1.62
1.98
1.63
2.00
1.65
2.03
2.06
1.67
1.69
2.10
1.40
1.59
1.41
1.61
1.42
1.64
1.45
1.69
1.47
1.72
1.49
1.75
1.51
1.79
1.54
1.84
1.56
1.88
1.57
1.90
1.59
1.93
1.96
1.61
1.63
2.00
1.35 1.28
1.52 1.41
1.36 1.30
1.54 1.44
1.38 1.32
1.57 1.47
1.42 1.35
1.62 1.53
1.44 1.37
1.66 1.56
1.45 1.39
1.68 1.59
1.48 1.42
1.73 1.64
1.51 1.45
1.78 1.70
1.53 1.47
1.82 1.74
1.54 1.49
1.84 1.76
1.56 1.50
1.87 1.79
1.90 1.82
1.58 1.52
1.60 1.55
1.94 1.86
1.24 1.17
1.36 1.25
1.26 1.19
1.38 1.28
1.28 1.22
1.42 1.32
1.32 1.26
1.48 1.39
1.34 1.29
1.51 1.43
1.36 1.31
1.54 1.46
1.39 1.34
1.59 1.51
1.42 1.38
1.65 1.57
1.45 1.40
1.69 1.62
1.46 1.42
1.71 1.64
1.48 1.44
1.74 1.68
1.78 1.71
1.50 1.46
1.52 1.48
1.82 1.76
1.11
1.15
1.13
1.19
1.16
1.24
1.22
1.33
1.25
1.37
1.27
1.40
1.30
1.46
1.35
1.52
1.37
1.56
1.39
1.60
1.41
1.63
1.66
1.43
1.46
1.71
1.00
1.00
1.08
1.11
1.13
1.19
1.19
1.28
1.22
1.33
1.25
1.37
1.28
1.43
1.32
1.49
1.35
1.53
1.37
1.56
1.39
1.60
1.64
1.41
1.44
1.68
Appendix C: Statistical
Tables
Source: Reprinted by permission from Statistical Methods, 7th ed., by George W. Snedecor and William G. Cochran, © 1980 by the Iowa State University Press, Ames, Iowa, 50010.
3.91
6.81
150
6.90
3.92
6.84
4.82
3.94
100
3.11
4.88
2.51
3.62
2.52
3.65
3.68
2.54
2.56
3.72
2.75
4.10
2.76
4.13
4.16
2.78
2.79
4.20
Back Matter
125
3.09
3.96
6.96
80
3.13
4.92
3.98
7.01
70
3.14
4.95
3.15
4.98
3.99
7.04
7.08
4.00
65
60
3.17
5.01
4.02
7.12
55
3.18
5.06
4.03
7.17
50
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
© The McGraw−Hill
Companies, 2009
853
6.08
4.50
3.93
3.64
3.46
3.34
3.26
3.20
3.15
3.11
3.08
3.06
3.03
3.01
3.00
2.98
2.97
2.96
2.95
2.92
2.89
2.86
2.83
2.80
2.77
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
24
30
40
60
120
3.74
3.68
3.63
3.85
3.79
3.96
3.90
4.08
4.05
4.02
4.00
3.98
4.11
4.20
4.15
3.98
3.92
3.86
4.10
4.04
4.23
4.17
4.37
4.33
4.30
4.28
4.25
4.41
4.51
4.45
4.65
4.57
4.89
4.76
4.46
4.39
4.31
4.24
4.17
4.16
4.10
4.03
4.62
4.54
4.78
4.74
4.70
4.67
4.65
4.44
4.36
4.29
4.60
4.52
4.77
4.68
4.94
4.90
4.86
4.82
4.79
4.99
5.20
5.12
5.05
4.83
5.30
5.03
4.95
4.88
5.60
5.43
6.12
5.82
6.58
13.0
8.85
7.35
8
45.4
5.12
5.40
5.24
5.90
5.61
6.33
12.4
8.48
7.05
7
43.1
4.30
4.23
4.45
4.37
4.59
4.56
4.52
4.49
4.47
4.64
4.75
4.69
4.91
4.82
5.17
5.02
5.63
5.36
6.03
11.7
8.04
6.71
6
40.4
4.55
4.47
4.39
4.72
4.63
4.90
4.81
5.08
5.03
4.99
4.96
4.92
5.13
5.35
5.27
5.19
5.46
5.77
5.59
6.32
6.00
6.80
13.5
9.18
7.60
9
47.4
4.65
4.56
4.47
4.82
4.73
5.01
4.92
5.20
5.15
5.11
5.07
5.04
5.25
5.49
5.39
5.32
5.60
5.92
5.74
6.49
6.16
6.99
14.0
9.46
7.83
10
49.1
4.73
4.64
4.55
4.92
4.82
5.11
5.01
5.31
5.26
5.21
5.17
5.14
5.36
5.61
5.51
5.43
5.72
6.05
5.87
6.65
6.30
7.17
14.4
9.72
8.03
11
50.6
4.81
4.71
4.62
5.00
4.90
5.20
5.10
5.40
5.35
5.31
5.27
5.23
5.46
5.71
5.61
5.53
5.83
6.18
5.98
6.79
6.43
7.32
14.7
9.95
8.21
12
52.0
4.88
4.78
4.68
5.08
4.98
5.28
5.18
5.49
5.44
5.39
5.35
5.31
5.55
5.81
5.71
5.63
5.93
6.29
6.09
6.92
6.55
7.47
15.1
10.2
8.37
13
53.2
5.00
4.90
4.80
5.32
5.21
5.11
4.94
4.84
4.74
5.43
5.25
5.15
5.04
5.65
5.59
5.54
5.50
5.46
5.71
5.98
5.88
5.79
6.11
6.48
6.28
7.14
6.76
7.72
15.7
10.5
8.66
15
55.4
5.36
5.57
5.52
5.47
5.43
5.39
5.64
5.90
5.80
5.71
6.03
6.39
6.19
7.03
6.66
7.60
15.4
10.3
8.52
14
54.3
5.06
4.95
4.85
5.38
5.27
5.16
5.49
5.72
5.66
5.61
5.57
5.53
5.79
6.06
5.95
5.86
6.19
6.57
6.36
7.24
6.85
7.83
15.9
10.7
8.79
16
56.3
5.11
5.00
4.89
5.44
5.33
5.22
5.55
5.78
5.73
5.67
5.63
5.59
5.85
6.13
6.02
5.93
6.27
6.65
6.44
7.34
6.94
7.93
16.1
10.8
8.91
17
57.2
5.15
5.04
4.93
5.49
5.38
5.27
5.61
5.85
5.79
5.73
5.69
5.65
5.91
6.20
6.09
5.99
6.34
6.73
6.51
7.43
7.02
8.03
16.4
11.0
9.03
18
58.0
5.20
5.09
4.97
5.55
5.43
5.31
5.66
5.90
5.84
5.79
5.74
5.70
6.27
6.15
6.05
5.97
6.40
6.80
6.58
7.51
7.10
8.12
16.6
11.1
9.13
19
58.8
5.24
5.13
5.01
5.59
5.47
5.36
5.71
5.96
5.90
5.84
5.79
5.75
6.33
6.21
6.11
6.03
6.47
6.87
6.64
7.59
7.17
8.21
16.8
11.2
9.23
20
59.6
Appendix C: Statistical
Tables
3.40
3.36
3.31
3.49
3.44
3.58
3.53
3.67
3.65
3.63
3.61
3.59
3.70
3.77
3.73
4.33
4.26
4.53
4.41
5.67
5.30
5.06
5.22
10.9
7.50
6.29
4.90
4.68
9.80
6.82
5.76
5
37.1
Back Matter
3.88
3.82
4.04
3.95
4.16
4.60
4.34
8.33
5.91
5.04
4
32.8
r
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
2
3
4
3
27.0
Critical Values of the Studentized Range Distribution for 0.05
nr
2
1 18.0
TABLE 6
854
© The McGraw−Hill
Companies, 2009
5.24
4.95
4.74
4.60
4.48
4.39
4.32
4.26
4.21
4.17
4.13
4.10
4.07
4.05
4.02
3.96
3.89
3.82
3.76
3.70
3.64
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
24
30
40
60
120
4.60
4.50
4.40
4.70
5.02
4.91
4.80
5.09
5.05
5.25
5.19
5.14
5.40
5.32
4.82
4.71
4.60
4.93
5.29
5.17
5.05
5.38
5.33
5.56
5.49
5.43
5.73
5.63
6.14
5.97
5.84
7.01
6.63
6.35
7.56
8.42
9.96
24.7
13.3
5
186
4.99
4.87
4.76
5.11
5.51
5.37
5.24
5.60
5.55
5.13
5.01
4.88
5.27
5.69
5.54
5.40
5.79
5.73
5.99
5.92
5.85
6.32
6.19
6.08
6.10
5.98
5.88
5.80
5.72
5.66
6.67
6.48
7.68
7.24
6.91
8.32
9.32
11.1
28.2
15.0
7
216
6.43
6.25
7.37
6.96
6.66
7.97
8.91
10.6
26.6
14.2
6
202
5.25
5.12
4.99
5.39
5.84
5.69
5.54
5.94
5.89
6.16
6.08
6.01
6.51
6.37
6.26
6.87
6.67
7.94
7.47
7.13
8.61
9.67
11.5
29.5
15.6
8
227
5.36
5.21
5.08
5.50
5.45
5.30
5.16
5.60
6.09
5.92
5.76
6.27
6.20
6.14
6.15
6.08
6.02
5.97
5.81
5.65
6.44
6.35
6.81
6.67
6.54
7.21
6.99
8.37
7.87
7.49
9.10
10.2
12.3
31.7
16.7
10
246
6.31
6.22
6.67
6.53
6.41
7.05
6.84
8.17
7.68
7.32
8.87
9.97
11.9
30.7
16.2
9
237
5.53
5.38
5.23
5.69
6.19
6.02
5.85
6.38
6.31
6.25
6.55
6.46
6.94
6.79
6.66
7.36
7.13
8.55
8.03
7.65
9.30
10.5
32.6
17.1
12.6
11
253
5.60
5.44
5.29
5.77
6.29
6.11
5.93
6.48
6.41
6.34
6.66
6.56
7.06
6.90
6.77
7.48
7.25
8.71
8.18
7.78
9.49
10.7
33.4
17.5
12.8
12
260
5.73
5.56
5.40
6.08
5.90
6.01
5.84
5.67
5.51
5.35
6.45
6.26
6.66
6.58
6.51
6.84
6.74
7.26
7.10
6.96
7.71
7.46
9.00
8.44
8.03
9.81
11.1
34.8
18.2
13.3
14
272
6.37
6.19
6.57
6.50
6.43
6.76
6.66
7.17
7.01
6.87
7.60
7.36
8.86
8.31
7.91
9.65
10.9
34.1
17.9
13.1
13
266
5.79
5.61
5.45
6.14
5.96
6.52
6.33
6.73
6.65
6.58
6.93
6.82
7.36
7.19
7.05
7.81
7.56
9.12
8.55
8.13
9.95
11.2
35.4
18.5
13.6
15
277
5.84
5.66
5.49
6.20
6.02
6.59
6.39
6.80
6.72
6.65
7.00
6.90
7.44
7.27
7.12
7.91
7.65
9.24
8.66
8.23
10.1
11.4
36.0
18.8
13.7
16
282
5.89
5.71
5.54
6.26
6.07
6.65
6.45
6.87
6.79
6.72
7.07
6.97
7.52
7.34
7.20
7.99
7.73
9.35
8.76
8.32
10.2
11.6
36.5
19.1
13.9
17
286
5.93
5.75
5.57
6.31
6.12
6.71
6.51
6.94
6.85
6.78
7.14
7.03
7.59
7.42
7.27
8.07
7.81
9.46
8.85
8.41
10.3
11.7
37.0
19.3
14.1
18
290
5.98
5.79
5.61
6.36
6.17
6.76
6.56
7.00
6.91
6.84
7.20
7.09
7.66
7.48
7.33
8.15
7.88
9.55
8.94
8.49
10.4
11.8
37.5
19.5
14.2
19
294
6.02
5.83
5.65
6.41
6.21
6.82
6.61
7.05
6.96
6.89
7.26
7.15
7.73
7.55
7.39
8.22
7.95
9.03
8.57
9.65
10.5
11.9
37.9
19.8
14.4
20
298
Appendix C: Statistical
Tables
4.28
4.20
4.12
4.37
4.64
4.54
4.45
4.70
4.67
4.83
4.78
4.74
4.96
4.89
5.77
5.62
5.50
6.54
6.20
5.96
5.92
5.63
5.43
5.27
5.14
5.04
7.03
7.80
9.17
22.3
12.2
4
164
6.33
6.97
8.12
19.0
10.6
3
135
r
Critical Values of the Studentized Range Distribution for 0.01
Back Matter
Source: E. S. Pearson and H. O. Hartley, eds., Biometrika Tables for Statisticians, vol. 1, 3rd ed. (Cambridge University Press, 1966). Reprinted by permission by the Biometrika Trustees.
6.51
5.70
4
5
14.0
8.26
2
3
nr 2
1 90.0
TABLE 6 (concluded)
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
© The McGraw−Hill
Companies, 2009
855
856
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
776
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 7
Critical Values of the Durbin-Watson Test Statistic for 0.05
k1
k2
k3
k4
k5
n
15
16
dL
1.08
1.10
dU
1.36
1.37
dL
0.95
0.98
dU
1.54
1.54
dL
0.82
0.86
dU
1.75
1.73
dL
0.69
0.74
dU
1.97
1.93
dL
0.56
0.62
dU
2.21
2.15
17
18
19
20
1.13
1.16
1.18
1.20
1.38
1.39
1.40
1.41
1.02
1.05
1.08
1.10
1.54
1.53
1.53
1.54
0.90
0.93
0.97
1.00
1.71
1.69
1.68
1.68
0.78
0.82
0.86
0.90
1.90
1.87
1.85
1.83
0.67
0.71
0.75
0.79
2.10
2.06
2.02
1.99
21
22
23
24
25
26
27
28
29
1.22
1.24
1.26
1.27
1.29
1.30
1.32
1.33
1.34
1.42
1.43
1.44
1.45
1.45
1.46
1.47
1.48
1.48
1.13
1.15
1.17
1.19
1.21
1.22
1.24
1.26
1.27
1.54
1.54
1.54
1.55
1.55
1.55
1.56
1.56
1.56
1.03
1.05
1.08
1.10
1.12
1.14
1.16
1.18
1.20
1.67
1.66
1.66
1.66
1.66
1.65
1.65
1.65
0.93
0.96
0.99
1.01
1.04
1.06
1.08
1.10
1.81
1.80
1.79
1.78
1.77
1.76
1.76
1.75
0.83
0.86
0.90
0.93
0.95
0.98
1.01
1.03
1.96
1.94
1.92
1.90
1.89
1.88
1.86
1.85
30
31
32
33
34
35
36
37
38
39
40
45
50
55
60
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
1.49
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
1.28
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
1.57
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
1.21
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
1.65
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
1.12
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
1.74
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
1.05
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
1.84
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
1.55
1.62
1.51
1.65
1.48
1.69
1.44
1.73
1.41
1.77
65
70
75
1.57
1.58
1.60
1.63
1.64
1.65
1.54
1.55
1.57
1.66
1.67
1.68
1.50
1.52
1.54
1.70
1.70
1.71
1.47
1.49
1.51
1.73
1.74
1.74
1.44
1.46
1.49
1.77
1.77
1.77
80
85
1.61
1.62
1.66
1.67
1.59
1.60
1.69
1.70
1.56
1.57
1.72
1.72
1.53
1.55
1.74
1.75
1.51
1.52
1.77
1.77
90
95
1.63
1.64
1.68
1.69
1.61
1.62
1.70
1.71
1.59
1.60
1.73
1.73
1.57
1.58
1.75
1.75
1.54
1.56
1.78
1.78
100
1.65
1.69
1.63
1.72
1.61
1.74
1.59
1.76
1.57
1.78
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 7 (concluded)
Critical Values of the Durbin-Watson Test Statistic for 0.01
k1
k2
k3
857
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
k4
k5
n
15
16
dL
0.81
0.84
dU
1.07
1.09
dL
0.70
0.74
dU
1.25
1.25
dL
0.59
0.63
dU
1.46
1.44
dL
0.49
0.53
dU
1.70
1.66
dL
0.39
0.44
dU
1.96
1.90
17
18
19
20
0.87
0.90
0.93
0.95
1.10
1.12
1.13
1.15
0.77
0.80
0.83
0.86
1.25
1.26
1.26
1.27
0.67
0.71
0.74
0.77
1.43
1.42
1.41
1.41
0.57
0.61
0.65
0.68
1.63
1.60
1.58
1.57
0.48
0.52
0.56
0.60
1.85
1.80
1.77
1.74
21
22
23
24
25
26
27
28
29
0.97
1.00
1.02
1.05
1.05
1.07
1.09
1.10
1.12
1.16
1.17
1.19
1.20
1.21
1.22
1.23
1.24
1.25
0.89
0.91
0.94
0.96
0.98
1.00
1.02
1.04
1.05
1.27
1.28
1.29
1.30
1.30
1.31
1.32
1.32
1.33
0.80
0.83
0.86
0.88
0.90
0.93
0.95
0.97
0.99
1.41
1.40
1.40
1.41
1.41
1.41
1.41
1.41
0.72
0.75
0.77
0.80
0.83
0.85
0.88
0.90
1.55
1.54
1.53
1.53
1.52
1.52
1.51
1.51
0.63
0.66
0.70
0.72
0.75
0.78
0.81
0.83
1.71
1.69
1.67
1.66
1.65
1.64
1.63
1.62
30
31
32
33
34
35
36
37
38
39
40
45
50
55
60
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
1.26
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
1.07
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
1.34
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
1.01
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
1.42
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
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
1.51
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
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
1.61
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
1.38
1.45
1.35
1.48
1.32
1.52
1.28
1.56
1.25
1.60
65
70
75
1.41
1.43
1.45
1.47
1.49
1.50
1.38
1.40
1.42
1.50
1.52
1.53
1.35
1.37
1.39
1.53
1.55
1.56
1.31
1.34
1.37
1.57
1.58
1.59
1.28
1.31
1.34
1.61
1.61
1.62
80
85
1.47
1.48
1.52
1.53
1.44
1.46
1.54
1.55
1.42
1.43
1.57
1.58
1.39
1.41
1.60
1.60
1.36
1.39
1.62
1.63
90
95
1.50
1.51
1.54
1.55
1.47
1.49
1.56
1.57
1.45
1.47
1.59
1.60
1.43
1.45
1.61
1.62
1.41
1.42
1.64
1.64
100
1.52
1.56
1.50
1.58
1.48
1.60
1.46
1.63
1.44
1.65
Source: J. Durbin and G. S. Watson, “Testing for Serial Correlation in Least Squares Regression, II,” Biometrika 38
(1951), pp. 159–78. Reproduced by permission of the Biometrika Trustees.
777
858
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
778
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 8
Cumulative Distribution Function: F(r) for the Total Number of Runs R in Samples of
Sizes n1 and n2
Number of Runs, r
(n1, n2)
(2, 3)
2
0.200
3
0.500
4
0.900
5
1.000
6
7
8
9
10
(2, 4)
(2, 5)
(2, 6)
0.133
0.095
0.071
0.400
0.333
0.286
0.800
0.714
0.643
1.000
1.000
1.000
(2, 7)
(2, 8)
0.056
0.044
0.250
0.222
0.583
0.533
1.000
1.000
(2, 9)
(2, 10)
0.036
0.030
0.200
0.182
0.491
0.455
1.000
1.000
(3, 3)
(3, 4)
(3, 5)
0.100
0.057
0.036
0.300
0.200
0.143
0.700
0.543
0.429
0.900
0.800
1.000
0.971
1.000
(3, 6)
(3, 7)
(3, 8)
(3, 9)
(3, 10)
0.024
0.017
0.012
0.009
0.007
0.107
0.083
0.067
0.055
0.045
0.345
0.283
0.236
0.200
0.171
0.714
0.643
0.583
0.533
0.491
0.455
0.929
0.881
0.833
0.788
0.745
0.706
1.000
1.000
1.000
1.000
1.000
1.000
(4, 4)
(4, 5)
0.029
0.016
0.114
0.071
0.371
0.262
0.629
0.500
0.886
0.786
0.971
0.929
1.000
0.992
(4, 6)
(4, 7)
(4, 8)
(4, 9)
(4, 10)
0.010
0.006
0.004
0.003
0.002
0.048
0.033
0.024
0.018
0.014
0.190
0.142
0.109
0.085
0.068
0.405
0.333
0.279
0.236
0.203
0.690
0.606
0.533
0.471
0.419
0.881
0.833
0.788
0.745
0.706
0.976
0.954
0.929
0.902
0.874
1.000
1.000
1.000
1.000
1.000
1.000
(5, 5)
(5, 6)
(5, 7)
(5, 8)
(5, 9)
(5, 10)
0.008
0.004
0.003
0.002
0.001
0.001
0.040
0.024
0.015
0.010
0.007
0.005
0.167
0.110
0.076
0.054
0.039
0.029
0.357
0.262
0.197
0.152
0.119
0.095
0.643
0.522
0.424
0.347
0.287
0.239
0.833
0.738
0.652
0.576
0.510
0.455
0.960
0.911
0.854
0.793
0.734
0.678
0.992
0.976
0.955
0.929
0.902
0.874
(6, 6)
(6, 7)
(6, 8)
(6, 9)
0.002
0.013
0.067
0.175
0.392
0.608
0.825
0.933
0.987
0.001
0.001
0.000
0.008
0.005
0.003
0.043
0.028
0.019
0.121
0.086
0.063
0.296
0.226
0.175
0.500
0.413
0.343
0.733
0.646
0.566
0.879
0.821
0.762
0.966
0.937
0.902
(6, 10)
0.000
0.002
0.013
0.047
0.137
0.288
0.497
0.706
0.864
(7, 7)
0.001
0.004
0.025
(7, 8)
(7, 9)
0.000
0.000
0.002
0.001
0.015
0.010
0.078
0.051
0.035
0.209
0.149
0.108
0.383
0.296
0.231
0.617
0.514
0.427
0.791
0.704
0.622
0.922
0.867
0.806
(7, 10)
0.000
0.001
0.006
0.024
0.080
0.182
0.355
0.549
0.743
(8, 8)
0.000
0.001
0.009
0.032
0.100
0.214
0.405
0.595
0.786
(8, 9)
0.000
0.001
0.005
0.020
0.069
0.157
(8, 10)
0.000
0.000
0.003
0.013
0.048
0.117
0.319
0.251
0.500
0.419
0.702
0.621
(9, 9)
(9, 10)
0.000
0.000
0.000
0.000
0.003
0.002
0.012
0.008
0.044
0.029
0.109
0.077
0.238
0.179
0.399
0.319
0.601
0.510
(10, 10)
0.000
0.000
0.001
0.004
0.019
0.051
0.128
0.242
0.414
1.000
0.998
0.992
0.984
0.972
0.958
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 8 (concluded)
Cumulative Distribution Function: F(r) for the Total Number of Runs R in
Samples of Sizes n1 and n2
Number of Runs, r
(n1, n2)
(2, 3)
11
12
13
14
15
16
17
18
19
20
(2, 4)
(2, 5)
(2, 6)
(2, 7)
(2, 8)
(2, 9)
(2, 10)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(3, 7)
(3, 8)
(3, 9)
(3, 10)
(4, 4)
(4, 5)
(4, 6)
(4, 7)
(4, 8)
(4, 9)
(4, 10)
(5, 5)
(5, 6)
(5, 7)
(5, 8)
(5, 9)
(5, 10)
1.000
1.000
1.000
1.000
1.000
(6, 6)
(6, 7)
(6, 8)
0.998
0.992
0.984
1.000
0.999
0.998
1.000
1.000
(6, 9)
(6, 10)
0.972
0.958
0.994
0.990
1.000
1.000
(7, 7)
0.975
0.996
0.999
1.000
(7, 8)
(7, 9)
(7, 10)
0.949
0.916
0.879
0.988
0.975
0.957
0.998
0.994
0.990
1.000
0.999
0.998
1.000
1.000
1.000
859
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
(8, 8)
0.900
0.968
0.991
(8, 9)
(8, 10)
0.843
0.782
0.939
0.903
0.980
0.964
0.999
0.996
0.990
1.000
0.999
0.998
1.000
1.000
1.000
1.000
1.000
(9, 9)
(9, 10)
0.762
0.681
0.891
0.834
0.956
0.923
0.988
0.974
0.997
0.992
1.000
0.999
1.000
1.000
1.000
1.000
1.000
(10, 10)
0.586
0.758
0.872
0.949
0.981
0.996
0.999
1.000
1.000
1.000
Source: Reproduced from F. Swed and C. Eisenhart, “Tables for Testing Randomness of Grouping in a Sequence of
Alternatives,” Annals of Mathematical Statistics 14 (1943) by permission of the authors and of the Editor, Annals of
Mathematical Statistics.
779
860
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
780
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 9
Cumulative Distribution Function of the Mann-Whitney U Statistic: F(u) for n1 n2
and 3 n2 10
n2 3
n1
u
0
1
0.25
2
0.10
3
0.05
1
2
3
0.50
0.20
0.40
0.60
0.10
0.20
0.35
4
0.50
n2 4
n1
u
0
1
2
3
4
5
6
7
8
1
0.2000
0.4000
0.6000
2
0.0667
0.1333
0.2667
0.4000
0.6000
3
0.0286
0.0571
0.1143
0.2000
0.3143
0.4286
0.5714
4
0.0143
0.0286
0.0571
0.1000
0.1714
0.2429
0.3429
0.4429
0.5571
n2 5
n1
u
0
1
2
3
4
5
6
7
8
9
10
11
12
1
0.1667
0.3333
0.5000
2
0.0476
0.0952
0.1905
0.2857
0.4286
0.5714
3
0.0179
0.0357
0.0714
0.1250
0.1964
0.2857
0.3929
4
0.0079
0.0159
0.0317
0.0556
0.0952
0.1429
0.2063
5
0.0040
0.0079
0.0159
0.0278
0.0476
0.0754
0.1111
0.5000
0.2778
0.3651
0.4524
0.1548
0.2103
0.2738
0.5476
0.3452
0.4206
0.5000
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 9 (continued)
Cumulative Distribution Function of the Mann-Whitney U Statistic: F(u)
for n1 n2 and 3 n2 10
n2 6
n1
u
0
1
0.1429
2
0.0357
3
0.0119
4
0.0048
5
0.0022
6
0.0011
1
2
3
0.2857
0.4286
0.5714
0.0714
0.1429
0.2143
0.0238
0.0476
0.0833
0.0095
0.0190
0.0333
0.0043
0.0087
0.0152
0.0022
0.0043
0.0076
0.3214
0.4286
0.5714
0.1310
0.1905
0.2738
0.3571
0.4524
0.5476
0.0571
0.0857
0.1286
0.1762
0.2381
0.3048
0.3810
0.4571
0.5429
0.0260
0.0411
0.0628
0.0887
0.1234
0.1645
0.2143
0.2684
0.3312
0.3961
0.4654
0.5346
0.0130
0.0206
0.0325
0.0465
0.0660
0.0898
0.1201
0.1548
0.1970
0.2424
0.2944
0.3496
0.4091
0.4686
0.5314
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
n2 7
n1
u
0
1
2
3
4
5
1
0.1250
0.2500
0.3750
0.5000
2
0.0278
0.0556
0.1111
0.1667
0.2500
0.3333
3
0.0083
0.0167
0.0333
0.0583
0.0917
0.1333
4
0.0030
0.0061
0.0121
0.0212
0.0364
0.0545
5
0.0013
0.0025
0.0051
0.0088
0.0152
0.0240
6
0.0006
0.0012
0.0023
0.0041
0.0070
0.0111
7
0.0003
0.0006
0.0012
0.0020
0.0035
0.0055
6
0.4444
0.1917
0.0818
0.0366
0.0175
0.0087
7
0.5556
0.2583
0.3333
0.1152
0.1576
0.0530
0.0745
0.0256
0.0367
0.0131
0.0189
0.4167
0.5000
0.2061
0.2636
0.3242
0.1010
0.1338
0.1717
0.0507
0.0688
0.0903
0.0265
0.0364
0.0487
0.3939
0.4636
0.5364
0.2159
0.2652
0.3194
0.1171
0.1474
0.1830
0.0641
0.0825
0.1043
0.3775
0.4381
0.5000
0.2226
0.2669
0.3141
0.1297
0.1588
0.1914
18
19
20
0.3654
0.4178
0.4726
0.2279
0.2675
0.3100
21
22
0.5274
0.3552
0.4024
8
9
10
11
12
13
14
15
16
17
23
24
861
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
0.4508
0.5000
781
862
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
782
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 9 (continued)
Cumulative Distribution Function of the Mann-Whitney U Statistic: F(u)
for n1 n2 and 3 n2 10
n2 8
n1
u
0
1
0.1111
2
0.0222
3
0.0061
4
0.0020
5
0.0008
6
0.0003
7
0.0002
8
0.0001
1
2
3
0.2222
0.3333
0.4444
0.0444
0.0889
0.1333
0.0121
0.0242
0.0424
0.0040
0.0081
0.0141
0.0016
0.0031
0.0054
0.0007
0.0013
0.0023
0.0003
0.0006
0.0011
0.0002
0.0003
0.0005
4
5
6
7
8
9
10
11
12
13
0.5556
0.2000
0.2667
0.3556
0.4444
0.5556
0.0667
0.0970
0.1394
0.1879
0.2485
0.3152
0.3879
0.4606
0.5394
0.0242
0.0364
0.0545
0.0768
0.1071
0.1414
0.1838
0.2303
0.2848
0.3414
0.0093
0.0148
0.0225
0.0326
0.0466
0.0637
0.0855
0.1111
0.1422
0.1772
0.0040
0.0063
0.0100
0.0147
0.0213
0.0296
0.0406
0.0539
0.0709
0.0019
0.0030
0.0047
0.0070
0.0103
0.0145
0.0200
0.0270
0.0361
0.0009
0.0015
0.0023
0.0035
0.0052
0.0074
0.0103
0.0141
0.0190
0.4040
0.4667
0.5333
0.2176
0.2618
0.3108
0.3621
0.4165
0.4716
0.5284
0.0906
0.1142
0.1412
0.1725
0.2068
0.2454
0.2864
0.3310
0.3773
0.4259
0.4749
0.5251
0.0469
0.0603
0.0760
0.0946
0.1159
0.1405
0.1678
0.1984
0.2317
0.2679
0.3063
0.3472
0.3894
0.4333
0.4775
0.5225
0.0249
0.0325
0.0415
0.0524
0.0652
0.0803
0.0974
0.1172
0.1393
0.1641
0.1911
0.2209
0.2527
0.2869
0.3227
0.3605
0.3992
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
0.4392
0.4796
0.5204
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 9 (continued)
Cumulative Distribution Function of the Mann-Whitney U Statistic: F(u)
for n1 n2 and 3 n2 10
n2 9
n1
u
0
1
0.1000
2
0.0182
3
0.0045
4
0.0014
5
0.0005
6
0.0002
7
0.0001
8
0.0000
9
0.0000
1
2
3
0.2000
0.3000
0.4000
0.0364
0.0727
0.1091
0.0091
0.0182
0.0318
0.0028
0.0056
0.0098
0.0010
0.0020
0.0035
0.0004
0.0008
0.0014
0.0002
0.0003
0.0006
0.0001
0.0002
0.0003
0.0000
0.0001
0.0001
4
5
6
7
8
9
10
11
0.5000
0.1636
0.2182
0.2909
0.3636
0.4545
0.5455
0.0500
0.0727
0.1045
0.1409
0.1864
0.2409
0.3000
0.3636
0.0168
0.0252
0.0378
0.0531
0.0741
0.0993
0.1301
0.1650
0.0060
0.0095
0.0145
0.0210
0.0300
0.0415
0.0599
0.0734
0.0024
0.0038
0.0060
0.0088
0.0128
0.0180
0.0248
0.0010
0.0017
0.0026
0.0039
0.0058
0.0082
0.0115
0.0005
0.0008
0.0012
0.0019
0.0028
0.0039
0.0056
0.0002
0.0004
0.0006
0.0009
0.0014
0.0020
0.0028
0.4318
0.5000
0.2070
0.2517
0.3021
0.3552
0.4126
0.4699
0.5301
0.0949
0.1199
0.1489
0.1818
0.2188
0.2592
0.3032
0.3497
0.3986
0.4491
0.5000
0.0332
0.0440
0.0567
0.0723
0.0905
0.1119
0.1361
0.1638
0.1924
0.2280
0.2643
0.3035
0.3445
0.3878
0.4320
0.4773
0.5227
0.0156
0.0209
0.0274
0.0356
0.0454
0.0571
0.0708
0.0869
0.1052
0.1261
0.1496
0.1755
0.2039
0.2349
0.2680
0.3032
0.3403
0.3788
0.4185
0.0076
0.0103
0.0137
0.0180
0.0232
0.0296
0.0372
0.0464
0.0570
0.0694
0.0836
0.0998
0.1179
0.1383
0.1606
0.1852
0.2117
0.2404
0.2707
0.0039
0.0053
0.0071
0.0094
0.0122
0.0157
0.0200
0.0252
0.0313
0.0385
0.0470
0.0567
0.0680
0.0807
0.0951
0.1112
0.1290
0.1487
0.1701
0.4591
0.5000
0.3029
0.3365
0.3715
0.1933
0.2181
0.2447
33
34
35
0.4074
0.4442
0.4813
0.2729
0.3024
0.3332
36
0.5187
0.3652
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
863
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
37
38
39
0.3981
0.4317
0.4657
40
0.5000
783
864
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
784
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 9 (concluded)
Cumulative Distribution Function of the Mann-Whitney U Statistic: F(u)
for n1 n2 and 3 n2 10
n2 10
n1
u
1
0 0.0909
1 0.1818
2 0.2727
2
0.0152
0.0303
0.0606
3
0.0035
0.0070
0.0140
4
0.0010
0.0020
0.0040
5
0.0003
0.0007
0.0013
6
0.0001
0.0002
0.0005
7
0.0001
0.0001
0.0002
8
0.0000
0.0000
0.0001
3 0.3636
4 0.4545
0.0909
0.1364
0.0245
0.0385
0.0070
0.0120
0.0023
0.0040
0.0009
0.0015
0.0004
0.0006
0.0002
0.0003
5 0.5455
0.1818
0.0559
0.0180
0.0063
0.0024
0.0010
0.2424
0.3030
0.3788
0.4545
0.5455
0.0804
0.1084
0.1434
0.1853
0.2343
0.2867
0.3462
0.4056
0.4685
0.5315
0.0270
0.0380
0.0529
0.0709
0.0939
0.1199
0.1518
0.1868
0.2268
0.2697
0.3177
0.3666
0.4196
0.4725
0.5275
0.0097
0.0140
0.0200
0.0276
0.0376
0.0496
0.0646
0.0823
0.1032
0.1272
0.1548
0.1855
0.2198
0.2567
0.2970
0.3393
0.3839
0.4296
0.4765
0.5235
0.0037
0.0055
0.0080
0.0112
0.0156
0.0210
0.0280
0.0363
0.0467
0.0589
0.0736
0.0903
0.1099
0.1317
0.1566
0.1838
0.2139
0.2461
0.2811
0.3177
0.3564
0.3962
0.4374
0.4789
0.5211
0.0015
0.0023
0.0034
0.0048
0.0068
0.0093
0.0125
0.0165
0.0215
0.0277
0.0351
0.0439
0.0544
0.0665
0.0806
0.0966
0.1148
0.1349
0.1574
0.1819
0.2087
0.2374
0.2681
0.3004
0.3345
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
9
0.0000
0.0000
0.0000
10
0.0000
0.0000
0.0000
0.0001
0.0000
0.0004
0.0001
0.0002
0.0001
0.0001
0.0007
0.0010
0.0015
0.0022
0.0031
0.0043
0.0058
0.0078
0.0103
0.0133
0.0171
0.0217
0.0273
0.0338
0.0416
0.0506
0.0610
0.0729
0.0864
0.1015
0.1185
0.1371
0.1577
0.1800
0.2041
0.0003
0.0005
0.0007
0.0011
0.0015
0.0021
0.0028
0.0038
0.0051
0.0066
0.0086
0.0110
0.0140
0.0175
0.0217
0.0267
0.0326
0.0394
0.0474
0.0564
0.0667
0.0782
0.0912
0.1055
0.1214
0.0002
0.0002
0.0004
0.0005
0.0008
0.0010
0.0014
0.0019
0.0026
0.0034
0.0045
0.0057
0.0073
0.0093
0.0116
0.0144
0.0177
0.0216
0.0262
0.0315
0.0376
0.0446
0.0526
0.0615
0.0716
31
0.3698
0.2299
32
0.4063
0.2574
0.1388
0.1577
0.0827
0.0952
33
34
0.4434
0.4811
0.2863
0.3167
0.1781
0.2001
0.1088
0.1237
35
36
37
0.5189
0.3482
0.3809
0.4143
0.2235
0.2483
0.2745
0.1399
0.1575
0.1763
0.4484
0.4827
0.5173
0.3019
0.3304
0.3598
0.1965
0.2179
0.2406
41
42
43
0.3901
0.4211
0.4524
0.2644
0.2894
0.3153
44
45
0.4841
0.5159
0.3421
0.3697
38
39
40
46
47
48
0.3980
0.4267
0.4559
49
0.4853
50
0.5147
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 10
One-Tailed
P 0.05
P 0.025
Critical Values of the Wilcoxon T Statistic
Two-Tailed
P 0.10
P 0.05
n5
1
n6
2
1
P 0.01
P 0.02
P 0.005
P 0.01
One-Tailed
Two-Tailed
n 11
n 12
P 0.05
P 0.025
P 0.10
P 0.05
14
11
17
14
n7
4
2
0
n8
6
4
865
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
n9
8
6
n 10
11
8
2
3
5
0
2
3
n 13
n 14
n 15
n 16
21
17
26
21
30
25
36
30
P 0.01
P 0.02
7
10
13
16
20
24
P 0.005
P 0.01
5
7
10
13
16
19
One-Tailed
P 0.05
P 0.025
P 0.01
Two-Tailed
P 0.10
P 0.05
P 0.02
n 17
41
35
28
n 18
47
40
33
n 19
54
46
38
n 20
60
52
43
n 21
68
59
49
n 22
75
66
56
P 0.005
P 0.01
23
28
32
37
43
49
One-Tailed
P 0.05
P 0.025
P 0.01
Two-Tailed
P 0.10
P 0.05
P 0.02
n 23
83
73
62
n 24
92
81
69
n 25
101
90
77
n 26
110
98
85
n 27
120
107
93
n 28
130
117
102
P 0.005
P 0.01
55
68
68
76
84
92
One-Tailed
P 0.05
P 0.025
P 0.01
Two-Tailed
P 0.10
P 0.05
P 0.02
n 29
141
127
111
n 30
152
137
120
n 31
163
148
130
n 32
175
159
141
n 33
188
171
151
n 34
201
183
162
149
P 0.005
P 0.01
100
109
118
128
138
One-Tailed
P 0.05
P 0.025
P 0.01
Two-Tailed
P 0.10
P 0.05
P 0.02
n 35
214
195
174
n 36
228
208
186
n 37
242
222
198
n 38
256
235
211
n 39
271
250
224
P 0.005
P 0.01
160
171
183
195
208
One-Tailed
P 0.05
Two-Tailed
P 0.10
n 40
287
n 41
303
n 42
319
n 43
336
n 44
353
n 45
371
P 0.025
P 0.01
P 0.05
P 0.02
264
238
279
252
295
267
311
281
327
297
344
313
P 0.005
P 0.01
221
234
248
262
277
292
One-Tailed
P 0.05
P 0.025
Two-Tailed
P 0.10
P 0.05
n 46
389
361
n 47
408
379
n 48
427
397
n 49
446
415
n 50
466
434
P 0.01
P 0.02
329
345
362
380
398
P 0.005
P 0.01
307
323
339
356
373
Source: Reproduced from F. Wilcoxon and R. A. Wilcox, Some Rapid Approximate Statistical Procedures (1964), p. 28,
with the permission of American Cyanamid Company.
785
866
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
786
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 11
Critical Values of Spearman’s Rank Correlation Coefficient
n
5
6
0.05
0.900
0.829
0.025
—
0.886
0.01
—
0.943
0.005
—
—
7
8
0.714
0.643
0.786
0.738
0.893
0.833
—
0.881
9
10
11
0.600
0.564
0.523
0.683
0.648
0.623
0.783
0.745
0.736
0.833
0.794
0.818
12
13
14
15
16
0.497
0.475
0.457
0.441
0.591
0.566
0.545
0.525
0.703
0.673
0.646
0.623
0.780
0.745
0.716
0.689
0.425
0.412
0.399
0.388
0.377
0.368
0.359
0.351
0.343
0.336
0.329
0.507
0.490
0.476
0.462
0.450
0.438
0.428
0.418
0.409
0.400
0.392
0.601
0.582
0.564
0.549
0.534
0.521
0.508
0.496
0.485
0.475
0.465
0.666
0.645
0.625
0.608
0.591
0.576
0.562
0.549
0.537
0.526
0.515
0.323
0.317
0.311
0.385
0.377
0.370
0.456
0.448
0.440
0.505
0.496
0.487
0.305
0.364
0.432
0.478
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Source: Reproduced by permission from E. G. Olds, “Distribution of Sums of Squares of Rank Differences for Small
Samples,” Annals of Mathematical Statistics 9 (1938).
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 12
Poisson Probability Distribution
This table gives values of
P(x) 867
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
m xe - m
x!
x
0
1
.005
.9950
.0050
.01
.9900
.0099
.02
.9802
.0192
.03
.9704
.0291
.04
.9608
.0384
.05
.9512
.0476
.06
.9418
.0565
.07
.9324
.0653
.08
.9231
.0738
.09
.9139
.0823
2
3
.0000
.0000
.0000
.0000
.0002
.0000
.0004
.0000
.0008
.0000
.0012
.0000
.0017
.0000
.0023
.0001
.0030
.0001
.0037
.0001
x
0
1
2
3
4
0.1
.9048
.0905
.0045
.0002
.0000
0.2
.8187
.1637
.0164
.0011
.0001
0.3
.7408
.2222
.0333
.0033
.0002
0.4
.6703
.2681
.0536
.0072
.0007
0.5
.6065
.3033
.0758
.0126
.0016
0.6
.5488
.3293
.0988
.0198
.0030
0.7
.4966
.3476
.1217
.0284
.0050
0.8
.4493
.3595
.1438
.0383
.0077
0.9
.4066
.3659
.1647
.0494
.0111
1.0
.3679
.3679
.1839
.0613
.0153
5
6
7
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0001
.0000
.0000
.0002
.0000
.0000
.0004
.0000
.0000
.0007
.0001
.0000
.0012
.0002
.0000
.0020
.0003
.0000
.0031
.0005
.0001
x
0
1
2
3
4
1.1
.3329
.3662
.2014
.0738
.0203
1.2
.3012
.3614
.2169
.0867
.0260
1.3
.2725
.3543
.2303
.0998
.0324
1.4
.2466
.3452
.2417
.1128
.0395
1.5
.2231
.3347
.2510
.1255
.0471
1.6
.2019
.3230
.2584
.1378
.0551
1.7
.1827
.3106
.2640
.1496
.0636
1.8
.1653
.2975
.2678
.1607
.0723
1.9
.1496
.2842
.2700
.1710
.0812
2.0
.1353
.2707
.2707
.1804
.0902
5
6
7
.0045
.0008
.0001
.0062
.0012
.0002
.0084
.0018
.0003
.0111
.0026
.0005
.0141
.0035
.0008
.0176
.0047
.0011
.0216
.0061
.0015
.0260
.0078
.0020
.0309
.0098
.0027
.0361
.0120
.0034
8
.0000
.0000
.0001
.0001
.0001
.0002
.0003
.0005
.0006
.0009
9
.0000
.0000
.0000
.0000
.0000
.0000
.0001
.0001
.0001
.0002
x
0
1
2.1
.1225
.2572
2.2
.1108
.2438
2.3
.1003
.2306
2.4
.0907
.2177
2.5
.0821
.2052
2.6
.0743
.1931
2.7
.0672
.1815
2.8
.0608
.1703
2.9
.0550
.1596
3.0
.0498
.1494
2
3
4
.2700
.1890
.0992
.2681
.1966
.1082
.2652
.2033
.1169
.2613
.2090
.1254
.2565
.2138
.1336
.2510
.2176
.1414
.2450
.2205
.1488
.2384
.2225
.1557
.2314
.2237
.1622
.2240
.2240
.1680
5
6
7
.0417
.0146
.0044
.0476
.0174
.0055
.0538
.0206
.0068
.0602
.0241
.0083
.0668
.0278
.0099
.0735
.0319
.0118
.0804
.0362
.0139
.0872
.0407
.0163
.0940
.0455
.0188
.1008
.0504
.0216
8
9
.0011
.0003
.0015
.0004
.0019
.0005
.0025
.0007
.0031
.0009
.0038
.0011
.0047
.0014
.0057
.0018
.0068
.0022
.0081
.0027
10
.0001
.0001
.0001
.0002
.0002
.0003
.0004
.0005
.0006
.0008
11
12
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0001
.0000
.0001
.0000
.0001
.0000
.0002
.0000
.0002
.0001
787
868
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
788
Back Matter
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 12 (continued)
Poisson Probability Distribution
x
0
1
2
3
4
3.1
.0450
.1397
.2165
.2237
.1734
3.2
.0408
.1304
.2087
.2226
.1781
3.3
.0369
.1217
.2008
.2209
.1823
3.4
.0334
.1135
.1929
.2186
.1858
3.5
.0302
.1057
.1850
.2158
.1888
3.6
.0273
.0984
.1771
.2125
.1912
3.7
.0247
.0915
.1692
.2087
.1931
3.8
.0224
.0850
.1615
.2046
.1944
3.9
.0202
.0789
.1539
.2001
.1951
4.0
.0183
.0733
.1465
.1954
.1954
5
6
7
8
9
.1075
.0555
.0246
.0095
.0033
.1140
.0608
.0278
.0111
.0040
.1203
.0662
.0312
.0129
.0047
.1264
.0716
.0348
.0148
.0056
.1322
.0771
.0385
.0169
.0066
.1377
.0826
.0425
.0191
.0076
.1429
.0881
.0466
.0215
.0089
.1477
.0936
.0508
.0241
.0102
.1522
.0989
.0551
.0269
.0116
.1563
.1042
.0595
.0298
.0132
10
11
12
13
14
.0010
.0003
.0001
.0000
.0000
.0013
.0004
.0001
.0000
.0000
.0016
.0005
.0001
.0000
.0000
.0019
.0006
.0002
.0000
.0000
.0023
.0007
.0002
.0001
.0000
.0028
.0009
.0003
.0001
.0000
.0033
.0011
.0003
.0001
.0000
.0039
.0013
.0004
.0001
.0000
.0045
.0016
.0005
.0002
.0000
.0053
.0019
.0006
.0002
.0001
x
0
1
2
3
4
4.1
.0166
.0679
.1393
.1904
.1951
4.2
.0150
.0630
.1323
.1852
.1944
4.3
.0136
.0583
.1254
.1798
.1933
4.4
.0123
.0540
.1188
.1743
.1917
4.5
.0111
.0500
.1125
.1687
.1898
4.6
.0101
.0462
.1063
.1631
.1875
4.7
.0091
.0427
.1005
.1574
.1849
4.8
.0082
.0395
.0948
.1517
.1820
4.9
.0074
.0365
.0894
.1460
.1789
5.0
.0067
.0337
.0842
.1404
.1755
5
6
7
8
9
.1600
.1093
.0640
.0328
.0150
.1633
.1143
.0686
.0360
.0168
.1662
.1191
.0732
.0393
.0188
.1687
.1237
.0778
.0428
.0209
.1708
.1281
.0824
.0463
.0232
.1725
.1323
.0869
.0500
.0255
.1738
.1362
.0914
.0537
.0280
.1747
.1398
.0959
.0575
.0307
.1753
.1432
.1002
.0614
.0334
.1755
.1462
.1044
.0653
.0363
10
11
12
13
14
15
.0061
.0023
.0008
.0002
.0001
.0000
.0071
.0027
.0009
.0003
.0001
.0000
.0081
.0032
.0011
.0004
.0001
.0000
.0092
.0037
.0014
.0005
.0001
.0000
.0104
.0043
.0016
.0006
.0002
.0001
.0118
.0049
.0019
.0007
.0002
.0001
.0132
.0056
.0022
.0008
.0003
.0001
.0147
.0064
.0026
.0009
.0003
.0001
.0164
.0073
.0030
.0011
.0004
.0001
.0181
.0082
.0034
.0013
.0005
.0002
x
0
1
2
3
4
5.1
.0061
.0311
.0793
.1348
.1719
5.2
.0055
.0287
.0746
.1293
.1681
5.3
.0050
.0265
.0701
.1239
.1641
5.4
.0045
.0244
.0659
.1185
.1600
5.5
.0041
.0225
.0618
.1133
.1558
5.6
.0037
.0207
.0580
.1082
.1515
5.7
.0033
.0191
.0544
.1033
.1472
5.8
.0030
.0176
.0509
.0985
.1428
5.9
.0027
.0162
.0477
.0938
.1383
6.0
.0025
.0149
.0446
.0892
.1339
5
6
7
8
9
.1753
.1490
.1086
.0692
.0392
.1748
.1515
.1125
.0731
.0423
.1740
.1537
.1163
.0771
.0454
.1728
.1555
.1200
.0810
.0486
.1714
.1571
.1234
.0849
.0519
.1697
.1584
.1267
.0887
.0552
.1678
.1594
.1298
.0925
.0586
.1656
.1601
.1326
.0962
.0620
.1632
.1605
.1353
.0998
.0654
.1606
.1606
.1377
.1033
.0688
10
11
12
13
14
.0200
.0093
.0039
.0015
.0006
.0220
.0104
.0045
.0018
.0007
.0241
.0116
.0051
.0021
.0008
.0262
.0129
.0058
.0024
.0009
.0285
.0143
.0065
.0028
.0011
.0309
.0157
.0073
.0032
.0013
.0334
.0173
.0082
.0036
.0015
.0359
.0190
.0092
.0041
.0017
.0386
.0207
.0102
.0046
.0019
.0413
.0225
.0113
.0052
.0022
15
16
17
.0002
.0001
.0000
.0002
.0001
.0000
.0003
.0001
.0000
.0003
.0001
.0000
.0004
.0001
.0000
.0005
.0002
.0001
.0006
.0002
.0001
.0007
.0002
.0001
.0008
.0003
.0001
.0009
.0003
.0001
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 12 (concluded)
869
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Poisson Probability Distribution
x
0
1
6.1
.0022
.0137
6.2
.0020
.0126
6.3
.0019
.0116
6.4
.0017
.0106
6.5
.0015
.0098
6.6
.0014
.0090
6.7
.0012
.0082
6.8
.0011
.0076
6.9
.0010
.0070
7.0
.0009
.0064
2
3
.0417
.0848
.0390
.0806
.0364
.0765
.0340
.0726
.0318
.0688
.0296
.0652
.0276
.0617
.0258
.0584
.0240
.0552
.0223
.0521
4
.1294
.1249
.1205
.1162
.1118
.1076
.1034
.0992
.0952
.0912
5
6
7
8
9
.1579
.1605
.1399
.1066
.0723
.1549
.1601
.1418
.1099
.0757
.1519
.1595
.1435
.1130
.0791
.1487
.1586
.1450
.1160
.0825
.1454
.1575
.1462
.1188
.0858
.1420
.1562
.1472
.1215
.0891
.1385
.1546
.1480
.1240
.0923
.1349
.1529
.1486
.1263
.0954
.1314
.1511
.1489
.1284
.0985
.1277
.1490
.1490
.1304
.1014
10
11
12
13
14
.0441
.0245
.0124
.0058
.0025
.0469
.0265
.0137
.0065
.0029
.0498
.0285
.0150
.0073
.0033
.0528
.0307
.0164
.0081
.0037
.0558
.0330
.0179
.0089
.0041
.0588
.0353
.0194
.0098
.0046
.0618
.0377
.0210
.0108
.0052
.0649
.0401
.0227
.0119
.0058
.0679
.0426
.0245
.0130
.0064
.0710
.0452
.0264
.0142
.0071
15
16
17
18
19
.0010
.0004
.0001
.0000
.0000
.0012
.0005
.0002
.0001
.0000
.0014
.0005
.0002
.0001
.0000
.0016
.0006
.0002
.0001
.0000
.0018
.0007
.0003
.0001
.0000
.0020
.0008
.0003
.0001
.0000
.0023
.0010
.0004
.0001
.0000
.0026
.0011
.0004
.0002
.0001
.0029
.0013
.0005
.0002
.0001
.0033
.0014
.0006
.0002
.0001
x
0
1
2
3
4
7.1
.0008
.0059
.0208
.0492
.0874
7.2
.0007
.0054
.0194
.0464
.0836
7.3
.0007
.0049
.0180
.0438
.0799
7.4
.0006
.0045
.0167
.0413
.0764
7.5
.0006
.0041
.0156
.0389
.0729
7.6
.0005
.0038
.0145
.0366
.0696
7.7
.0005
.0035
.0134
.0345
.0663
7.8
.0004
.0032
.0125
.0324
.0632
7.9
.0004
.0029
.0116
.0305
.0602
8.0
.0003
.0027
.0107
.0286
.0573
5
6
7
.1241
.1468
.1489
.1204
.1445
.1486
.1167
.1420
.1481
.1130
.1394
.1474
.1094
.1367
.1465
.1057
.1339
.1454
.1021
.1311
.1442
.0986
.1282
.1428
.0951
.1252
.1413
.0916
.1221
.1396
8
.1321
.1337
.1351
.1363
.1373
.1382
.1388
.1392
.1395
.1396
9
.1042
.1070
.1096
.1121
.1144
.1167
.1187
.1207
.1224
.1241
10
11
.0740
.0478
.0770
.0504
.0800
.0531
.0829
.0558
.0858
.0585
.0887
.0613
.0914
.0640
.0941
.0667
.0967
.0695
.0993
.0722
12
13
14
.0283
.0154
.0078
.0303
.0168
.0086
.0323
.0181
.0095
.0344
.0196
.0104
.0366
.0211
.0113
.0388
.0227
.0123
.0411
.0243
.0134
.0434
.0260
.0145
.0457
.0278
.0157
.0481
.0296
.0169
15
16
17
.0037
.0016
.0007
.0041
.0019
.0008
.0046
.0021
.0009
.0051
.0024
.0010
.0057
.0026
.0012
.0062
.0030
.0013
.0069
.0033
.0015
.0075
.0037
.0017
.0083
.0041
.0019
.0090
.0045
.0021
18
19
.0003
.0001
.0003
.0001
.0004
.0001
.0004
.0002
.0005
.0002
.0006
.0002
.0006
.0003
.0007
.0003
.0008
.0003
.0009
.0004
20
.0000
.0000
.0001
.0001
.0001
.0001
.0001
.0001
.0001
.0002
21
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0001
.0001
789
870
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
790
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
Appendix C
TABLE 13
Control Chart Constants
For Estimating
Sigma
For
X Chart
For X Chart
(Standard
Given)
For R Chart
D3
For R Chart
(Standard Given)
D4
D1
For s Chart
(Standard Given)
n
c4
d2
A2
A3
A
2
3
0.7979
0.8862
1.128
1.693
1.880
1.023
2.659
1.954
2.121
1.732
0
0
3.267
2.575
0
0
3.686
4.358
D2
0
0
B3
3.267
2.568
B4
0
0
B5
2.606
2.276
B6
4
5
6
0.9213
0.9400
0.9515
2.059
2.326
2.534
0.729
0.577
0.483
1.628
1.427
1.287
1.500
1.342
1.225
0
0
0
2.282
2.115
2.004
0
0
0
4.698
4.918
5.078
0
0
0.030
2.266
2.089
1.970
0
0
0.029
2.088
1.964
1.874
7
8
9
10
15
20
0.9594
0.9650
0.9693
0.9727
0.9823
0.9869
2.704
2.847
2.970
3.078
3.472
3.735
0.419
0.373
0.337
0.308
0.223
0.180
1.182
1.099
1.032
0.975
0.789
0.680
1.134
1.061
1.000
0.949
0.775
0.671
0.076
0.136
0.184
0.223
0.348
0.414
1.924
1.864
1.816
1.777
1.652
1.586
0.205
0.387
0.546
0.687
1.207
1.548
5.203
5.307
5.394
5.469
5.737
5.922
0.118
0.185
0.239
0.284
0.428
0.510
1.882
1.815
1.761
1.716
1.572
1.490
0.113
0.179
0.232
0.276
0.421
0.504
1.806
1.751
1.707
1.669
1.544
1.470
25
0.9896
3.931
0.153
0.606
0.600
0.459
1.541
1.804
6.058
0.565
1.435
0.559
1.420
Source: T. P. Ryan, Statistical Methods for Quality Improvement © 1989 New York: John Wiley & Sons. This material is used by permission of John Wiley &
Sons, Inc.
Aczel−Sounderpandian:
Complete Business
Statistics, Seventh Edition
Back Matter
Statistical Tables
TABLE 14
1559
5550
4735
5333
8495
1947
4785
9972
0472
4727
3658
6906
3793
3376
6126
0466
9908
7594
5715
7932
6311
0476
5317
7474
7460
1002
5449
9453
0471
5469
2782
3129
7092
9566
5863
5881
6416
9568
0452
8762
0194
3306
7198
8350
7449
6126
1851
7698
0810
6647
3867
1172
6701
8244
8009
1947
9562
0729
6904
Random Numbers
9068
6245
6214
1313
2956
3353
6325
9163
4629
6994
3226
9758
6916
5966
0224
7566
9832
3636
9301
4739
2025
1624
3903
8876
6800
1494
6891
0809
2725
2668
9603
7217
9885
0501
7000
9151
9939
3012
9538
5920
0270
5478
3079
4846
4279
5399
7940
4218
0118
7149
7111
7278
1668
0620
4031
8315
4821
9026
2569
9290
7313
8037
3063
1121
1197
1868
5833
2007
1175
5981
0244
0132
1614
7169
1320
8185
1224
5847
4567
5250
3470
6098
1918
1987
9972
9047
7151
7588
1996
1877
5020
3714
8352
1714
2321
9569
6316
5730
8989
7601
2797
2171
1309
4224
0852
9908
2726
4979
1409
5549
7527
5067
8736
7884
9755
8050
9631
3251
8303
0117
1385
1134
8484
7363
5020
0100
4464
5622
9025
0259
8873
4025
3596
8777
8835
6808
3524
6797
6099
1600
9438
9828
2758
3877
6297
9982
6573
2249
4159
3788
8557
1062
9276
3147
0439
9065
1893
4777
0342
1605
6972
0612
1018
5491
3860
5130
0458
6809
9439
2492
0413
2649
2215
7187
0106
8096
0079
8508
7652
1882
8676
2920
9003
9100
5758
3312
2341
1080
4609
8987
0721
1593
8470
0384
1184
0077
4540
6718
0675
3482
2061
0737
6104
1075
0411
0546
3857
9809
0853
7804
0634
7218
6755
1705
0710
1186
2169
3897
4996
0928
4584
2496
6557
1536
3132
1059
3313
3427
6211
7961
1429
2382
4074
2782
8906
8838
8954
5069
0828
6241
7934
9313
0823
3696
8728
8562
1437
1269
4975
1537
5097
5448
3699
3404
6674
8488
7539
3261
5505
6664
6880
4006
7762
1120
0110
6637
2570
9465
9524
2401
6922
2510
4680
2158
9245
7073
4133
0023
6599
4988
2091
4946
8011
1722
5739
0082
9793
9457
7825
6253
1931
4743
4665
5713
8738
1051
6354
2957
9960
0670
3434
7379
6496
1193
5192
6721
5957
4814
6695
7286
9575
1272
6752
8061
3639
9681
7749
5167
0391
1500
0477
8091
6129
6132
8010
4544
2186
6228
0379
1032
5759
9881
1639
6558
7082
7650
9780
9328
4642
9750
9918
4314
8592
7919
9024
3071
5120
9261
4181
1252
6669
9436
3212
8503
871
© The McGraw−Hill
Companies, 2009
Appendix C: Statistical
Tables
6677
7668
0530
5304
5263
4261
7391
6297
2497
1471
7331
7556
2098
6090
2686
4669
1181
4391
5438
9777
3204
4195
9993
9170
5763
0669
7153
5090
1224
1701
0544
3945
7774
1697
4838
6947
7071
9149
9562
9495
9228
9429
0597
4430
6086
1541
7269
9662
4557
7477
6651
4903
8572
8120
8088
6060
4973
8854
6333
6415
1096
9210
1582
0968
0066
1250
5653
4219
7206
0792
1975
6683
8083
1796
1402
8627
2016
6508
1621
9637
2660
8191
2776
2061
8557
8881
2053
3124
3141
2660
1696
6674
7153
1954
7102
9596
4848
8534
1594
5558
3937
5948
9481
1955
7894
7047
4795
0245
7320
4267
1023
0634
6500
2490
2319
4879
3435
0952
0342
5780
0177
6198
0069
2714
5501
7782
5339
2027
5383
7898
0901
5450
1150
3905
1968
6167
9673
7244
1091
2150
8488
4025
9373
0513
3367
7570
6563
6147
6737
2286
2775
6208
1680
0097
8758
8634
9321
8600
3597
7573
5753
9048
9860
1843
0382
7718
4861
5822
8099
5745
1140
8127
9122
0635
8900
4206
1641
Source: T. P. Ryan, Statistical Methods for Quality Improvement © 1989 New York: John Wiley & Sons. This material is
used by permission of John Wiley & Sons, Inc.
791