THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MATHEMATICS
FINAL EXAMINATION
MATH3811
STATISTICAL INFERENCE
(1) TIME ALLOWED – 2 Hours
(2) TOTAL NUMBER OF QUESTIONS – 4
(3) ANSWER ALL QUESTIONS
(4) THE QUESTIONS ARE NOT OF EQUAL VALUE
(5) TOTAL NUMBER OF MARKS – 100
(6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE
All answers must be written in ink. Except where they are expressly required pencils
may only be used for drawing, sketching or graphical work.
MATH3811 Page 2
1. [28 marks] Suppose a sample X = (X1, X2, . . . , Xn) from the density
f(x, θ) =
{
(1 + θ)xθ, x ∈ (0, 1),
0 elsewhere
is given where θ > −1 is the unknown parameter.
a) Calculate E(X1).
b) Find a minimal sufficient statistic T for θ. Give reasons for your answer.
c) Calculate the Fisher information about θ contained in the statistic T that
you found in b). Show that it is equal to n
(1+θ)2
.
d) Find the MLE of θ and also the MLE of h(θ) = 1
1+θ
.
e) Find the Cramer-Rao lower bound for the variance of an unbiased esti-
mator of h(θ) = 1
1+θ
.
f) Write down the score function. Find out if an UMVUE of h(θ) = 1
1+θ
exists and if it exists, write it down. Is there an unbiased estimator of
h(θ) = θ whose variance attains the Cramer-Rao bound? Explain your
answers.
g) State the asymptotic distribution of
√
n(hˆmle − h(θ)) for h(θ) = 11+θ .
Hint: The “delta method” implies that for any smooth function h(θ) :
√
n(h(θˆmle)− h(θ0))→d N(0, (∂h
∂θ
(θ0))
2I−1(θ0)),
I(θ) = E{ ∂
∂θ
[lnf(x, θ)]}2 = E{− ∂
2
∂θ2
[lnf(x, θ)]}.
2. [28 marks]
a) Let X = (X1, X2, . . . , Xn) be i.i.d. observations, each from a Poisson
distribution:
f(y|λ) = e(−λ) · λy/(y!), y = 0, 1, 2, . . . , λ > 0.
The prior on λ is believed to be Gamma(2,3).
(Note that, for any values of α > 0, β > 0, the density of the Gamma(α, β)
distribution is given by
τ(λ) =
{
1
βα·Γ(α) · λα−1 exp(−λ/β) , λ > 0;
0 else
and Γ(α) =
∫∞
0 e
−xxα−1dx is the Gamma function with the property
Γ(α+ 1) = αΓ(α).)
Please see over . . .
MATH3811 Page 3
i) Find the posterior density h(λ|X) of λ given X = (X1, X2, . . . , Xn).
Show that, like the prior, the posterior density is also a member of
the Gamma family.
ii) Find the Bayesian estimator of λ for quadratic error-loss with respect
to the prior τ(λ).
b) In a preliminary testing of a random number generator, the following ten
values were generated:
x1 = 0.621, x2 = 0.503, x3 = 0.203, x4 = 0.477, x5 = 0.710,
x6 = 0.581, x7 = 0.329, x8 = 0.480, x9 = 0.554, x10 = 0.382.
Carry out a Kolmogorov-Smirnov test to determine if the hypothesis of
a Uniform [0,1] distribution can be accepted. Use α = 0.05.
You can use the following extract of Table F for one-sample Kolmogorov
tests:
n 0.2 0.1 0.05 0.02 0.01
8 .358 .410 .454 .507 .542
9 .339 .387 .430 .480 .513
10 .323 .369 .409 .457 .489
(Each table entry is the value of the Kolmogorov statistic Dn for sample
of size n = 8, 9, 10 with the corresponding p-value given on the top row.)
3. [22 marks] Let X = (X1,X2, . . . ,Xn) be i.i.d. random variables, each with
a density
f(x, θ) =
 1√2pixθe{−
1
2
[
ln(x)
θ
]2}, x > 0
0 elsewhere
where θ > 0 is a parameter. (This is called the log-normal density.)
a) Prove that the family L(X, θ) has a monotone likelihood ratio in T =∑n
i=1(lnXi)
2.
b) Argue that there is a uniformly most powerful (UMP) α−size test of the
hypothesis H0 : θ ≤ θ0 against H1 : θ > θ0 and exhibit its structure.
c) Using the density transformation formula (or otherwise) show that
Yi = lnXi
has a N(0, θ2) distribution.
d) Using c) (or otherwise), find the threshold constant in the test and hence
determine completely the uniformly most powerful α− size test ϕ∗ of
H0 : θ ≤ θ0 versus H1 : θ > θ0.
Calculate the power function Eθϕ
∗ and sketch a graph of it.
Please see over . . .
MATH3811 Page 4
4. [22 marks]
a) In an agricultural experiment, three blocks of similar size and location
have been selected. On each block, four varieties (A, B, C and D) of
potatoes have been grown on plots of equal size. The yields of potato
from the 12 plots have been registered in the table below:
Variety A Variety B Variety C Variety D
Block 1 68 67 71 77
Block 2 82 83 86 89
Block 3 56 59 64 60
Analyze these data using the Friedman test. At 10% level of significance,
decide if the hypothesis of equal yields for the four varieties can be ac-
cepted.
b) Two groups of students attempt a test with the following results:
Group 1: 91, 83, 76, 81
Group 2: 72, 61, 74, 63, 82
Using a (large sample) Wilcoxon test and α = 0.05, test the hypothesis
of no difference between the two groups.
Some useful formulae
1. Friedman’s Test:
F =
12l
K(K + 1)
K∑
i=1
(R¯i. − K + 1
2
)2 =
12
lK(K + 1)
K∑
i=1
R2i. − 3l(K + 1)
has χ2K−1 as a limiting distribution under the null hypothesis.
2. rth order statistic (r = 1, 2, . . . , n) of the sample of size n from a
distribution with a density fX(.) and a cdf FX(.) :
fX(r)(yr) =
n!
(r − 1)!(n− r)! [FX(yr)]
r−1[1− FX(yr)]n−rfX(yr)
Joint density of the couple (X(i), X(j)), 1 ≤ i < j ≤ n :
fX(i),X(j)(x, y) =
n!
(i− 1)!(j − 1− i)!(n− j)!fX(x)fX(y)[FX(x)]
i−1[FX(y)−FX(x)]j−1−i[1−FX(y)]n−j
for −∞ < x < y <∞.
3. Wilcoxon: Two independent samplesX1, X2, . . . , Xm and Y1, Y2, . . . Yn,
Wm+n =
∑m
i=1R(Xi). Then
Wm+n−m(m+n+1)/2√
mn(m+n+1)/12
→d N(0, 1).
4. Bayesian inference:
h(θ|X) = f(X|θ)τ(θ)
g(X)
, g(X) =
∫
Θ
f(X|θ)τ(θ)dθ.
5. Density transformation: Y = W (X) −→ fY (y) = fX(W−1(y))|d(W−1(y)dy |.
UNIVERSITY OF NEW SOUTH WALES • SCHOOL OF MATHEMATICS AND STATISTICS
Statistical Tables
t distribution critical values
Key: Table entry for p and C is the critical value t∗ with probability p lying to its right
and probability C lying between −t∗ and t∗.
Upper tail probability p
df .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001 .0005
1 1.000 1.376 1.963 3.078 6.314 12.71 15.89 31.82 63.66 127.3 318.3 636.6
2 0.816 1.061 1.386 1.886 2.920 4.303 4.849 6.965 9.925 14.09 22.33 31.60
3 0.765 0.978 1.250 1.638 2.353 3.182 3.482 4.541 5.841 7.453 10.21 12.92
4 0.741 0.941 1.190 1.533 2.132 2.776 2.999 3.747 4.604 5.598 7.173 8.610
5 0.727 0.920 1.156 1.476 2.015 2.571 2.757 3.365 4.032 4.773 5.893 6.869
6 0.718 0.906 1.134 1.440 1.943 2.447 2.612 3.143 3.707 4.317 5.208 5.959
7 0.711 0.896 1.119 1.415 1.895 2.365 2.517 2.998 3.499 4.029 4.785 5.408
8 0.706 0.889 1.108 1.397 1.860 2.306 2.449 2.896 3.355 3.833 4.501 5.041
9 0.703 0.883 1.100 1.383 1.833 2.262 2.398 2.821 3.250 3.690 4.297 4.781
10 0.700 0.879 1.093 1.372 1.812 2.228 2.359 2.764 3.169 3.581 4.144 4.587
11 0.697 0.876 1.088 1.363 1.796 2.201 2.328 2.718 3.106 3.497 4.025 4.437
12 0.695 0.873 1.083 1.356 1.782 2.179 2.303 2.681 3.055 3.428 3.930 4.318
13 0.694 0.870 1.079 1.350 1.771 2.160 2.282 2.650 3.012 3.372 3.852 4.221
14 0.692 0.868 1.076 1.345 1.761 2.145 2.264 2.624 2.977 3.326 3.787 4.140
15 0.691 0.866 1.074 1.341 1.753 2.131 2.249 2.602 2.947 3.286 3.733 4.073
16 0.690 0.865 1.071 1.337 1.746 2.120 2.235 2.583 2.921 3.252 3.686 4.015
17 0.689 0.863 1.069 1.333 1.740 2.110 2.224 2.567 2.898 3.222 3.646 3.965
18 0.688 0.862 1.067 1.330 1.734 2.101 2.214 2.552 2.878 3.197 3.610 3.922
19 0.688 0.861 1.066 1.328 1.729 2.093 2.205 2.539 2.861 3.174 3.579 3.883
20 0.687 0.860 1.064 1.325 1.725 2.086 2.197 2.528 2.845 3.153 3.552 3.850
21 0.686 0.859 1.063 1.323 1.721 2.080 2.189 2.518 2.831 3.135 3.527 3.819
22 0.686 0.858 1.061 1.321 1.717 2.074 2.183 2.508 2.819 3.119 3.505 3.792
23 0.685 0.858 1.060 1.319 1.714 2.069 2.177 2.500 2.807 3.104 3.485 3.768
24 0.685 0.857 1.059 1.318 1.711 2.064 2.172 2.492 2.797 3.091 3.467 3.745
25 0.684 0.856 1.058 1.316 1.708 2.060 2.167 2.485 2.787 3.078 3.450 3.725
26 0.684 0.856 1.058 1.315 1.706 2.056 2.162 2.479 2.779 3.067 3.435 3.707
27 0.684 0.855 1.057 1.314 1.703 2.052 2.158 2.473 2.771 3.057 3.421 3.690
28 0.683 0.855 1.056 1.313 1.701 2.048 2.154 2.467 2.763 3.047 3.408 3.674
29 0.683 0.854 1.055 1.311 1.699 2.045 2.150 2.462 2.756 3.038 3.396 3.659
30 0.683 0.854 1.055 1.310 1.697 2.042 2.147 2.457 2.750 3.030 3.385 3.646
40 0.681 0.851 1.050 1.303 1.684 2.021 2.123 2.423 2.704 2.971 3.307 3.551
50 0.679 0.849 1.047 1.299 1.676 2.009 2.109 2.403 2.678 2.937 3.261 3.496
60 0.679 0.848 1.045 1.296 1.671 2.000 2.099 2.390 2.660 2.915 3.232 3.460
80 0.678 0.846 1.043 1.292 1.664 1.990 2.088 2.374 2.639 2.887 3.195 3.416
100 0.677 0.845 1.042 1.290 1.660 1.984 2.081 2.364 2.626 2.871 3.174 3.390
1000 0.675 0.842 1.037 1.282 1.646 1.962 2.056 2.330 2.581 2.813 3.098 3.300
.50 .60 .70 0.80 .90 .95 .96 .98 .99 .995 .998 .999
Probability C
Standard normal probabilities
Key: Table entry for z is the area under the standard normal curve to the left of z.
z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
−3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
−3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
−3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
−3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
−3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
−2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
−2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
−2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
−2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
−2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
−2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
−2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
−2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
−2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
−2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
−1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
−1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
−1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
−1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
−1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
−1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
−1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
−1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
−1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
−1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
−0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
−0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
−0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
−0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
−0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
−0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
−0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
−0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
−0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
−0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
χ2 distribution critical values
Key: Table entry for p is the critical value with probability p lying to its right.
Upper tail probability p
df .995 .99 .975 .95 .90 .10 .05 .025 .01 .005
1 0.000039 0.00016 0.00098 0.0039 0.0158 2.71 3.84 5.02 6.63 7.88
2 0.0100 0.0201 0.0506 0.1026 0.2107 4.61 5.99 7.38 9.21 10.60
3 0.0717 0.115 0.216 0.352 0.584 6.25 7.81 9.35 11.34 12.84
4 0.207 0.297 0.484 0.711 1.064 7.78 9.49 11.14 13.28 14.86
5 0.412 0.554 0.831 1.15 1.61 9.24 11.07 12.83 15.09 16.75
6 0.676 0.872 1.24 1.64 2.20 10.64 12.59 14.45 16.81 18.55
7 0.989 1.24 1.69 2.17 2.83 12.02 14.07 16.01 18.48 20.28
8 1.34 1.65 2.18 2.73 3.49 13.36 15.51 17.53 20.09 21.95
9 1.73 2.09 2.70 3.33 4.17 14.68 16.92 19.02 21.67 23.59
10 2.16 2.56 3.25 3.94 4.87 15.99 18.31 20.48 23.21 25.19
11 2.60 3.05 3.82 4.57 5.58 17.28 19.68 21.92 24.72 26.76
12 3.07 3.57 4.40 5.23 6.30 18.55 21.03 23.34 26.22 28.30
13 3.57 4.11 5.01 5.89 7.04 19.81 22.36 24.74 27.69 29.82
14 4.07 4.66 5.63 6.57 7.79 21.06 23.68 26.12 29.14 31.32
15 4.60 5.23 6.26 7.26 8.55 22.31 25.00 27.49 30.58 32.80
16 5.14 5.81 6.91 7.96 9.31 23.54 26.30 28.85 32.00 34.27
17 5.70 6.41 7.56 8.67 10.09 24.77 27.59 30.19 33.41 35.72
18 6.26 7.01 8.23 9.39 10.86 25.99 28.87 31.53 34.81 37.16
19 6.84 7.63 8.91 10.12 11.65 27.20 30.14 32.85 36.19 38.58
20 7.43 8.26 9.59 10.85 12.44 28.41 31.41 34.17 37.57 40.00
21 8.03 8.90 10.28 11.59 13.24 29.62 32.67 35.48 38.93 41.40
22 8.64 9.54 10.98 12.34 14.04 30.81 33.92 36.78 40.29 42.80
23 9.26 10.20 11.69 13.09 14.85 32.01 35.17 38.08 41.64 44.18
24 9.89 10.86 12.40 13.85 15.66 33.20 36.42 39.36 42.98 45.56
25 10.52 11.52 13.12 14.61 16.47 34.38 37.65 40.65 44.31 46.93
26 11.16 12.20 13.84 15.38 17.29 35.56 38.89 41.92 45.64 48.29
27 11.81 12.88 14.57 16.15 18.11 36.74 40.11 43.19 46.96 49.64
28 12.46 13.56 15.31 16.93 18.94 37.92 41.34 44.46 48.28 50.99
29 13.12 14.26 16.05 17.71 19.77 39.09 42.56 45.72 49.59 52.34
30 13.79 14.95 16.79 18.49 20.60 40.26 43.77 46.98 50.89 53.67
40 20.71 22.16 24.43 26.51 29.05 51.81 55.76 59.34 63.69 66.77
50 27.99 29.71 32.36 34.76 37.69 63.17 67.50 71.42 76.15 79.49
60 35.53 37.48 40.48 43.19 46.46 74.40 79.08 83.30 88.38 91.95
80 51.17 53.54 57.15 60.39 64.28 96.58 101.88 106.63 112.33 116.32
100 67.33 70.06 74.22 77.93 82.36 118.50 124.34 129.56 135.81 140.17
F distribution critical values
Key: p=Upper tail probability p, dfn=degrees of freedom in numerator, dfd=degrees of freedom in
denominator, ∗Multiply by 10, †Multiply by 100.
dfn 1 2 3 4 5 6 7 8 9 10 12 15 20 24 30 40 60 120 ∞
dfd p
1 .05 161 200 216 225 230 234 237 239 241 242 244 246 248 249 250 251 252 253 254
.025 648 800 864 900 922 937 948 957 963 969 977 986 993 997 1001 1006 1010 1014 1018
.01 405∗ 500∗ 540∗ 563∗ 576∗ 586∗ 593∗ 598∗ 602∗ 606∗ 611∗ 616∗ 621∗ 624∗ 626∗ 629∗ 631∗ 634∗ 637∗
.005 162† 200† 216† 225† 231† 234† 237† 239† 241† 242† 244† 246† 248† 249† 250† 251† 253† 254† 255†
2 .05 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.50
.025 38.51 39.00 39.17 39.25 39.30 39.33 39.36 39.37 39.39 39.40 39.41 39.43 39.45 39.46 39.46 39.47 39.48 39.49 39.50
.01 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.50
.005 199 199 199 199 199 199 199 199 199 199 199 199 199 200 200 200 200 200 200
3 .05 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.53
.025 17.44 16.04 15.44 15.10 14.88 14.73 14.62 14.54 14.47 14.42 14.34 14.25 14.17 14.12 14.08 14.04 13.99 13.95 13.90
.01 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23 27.05 26.87 26.69 26.60 26.50 26.41 26.32 26.22 26.13
.005 55.55 49.80 47.47 46.19 45.39 44.84 44.43 44.13 43.88 43.69 43.39 43.08 42.78 42.62 42.47 42.31 42.15 41.99 41.83
4 .05 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63
.025 12.22 10.65 9.98 9.60 9.36 9.20 9.07 8.98 8.90 8.84 8.75 8.66 8.56 8.51 8.46 8.41 8.36 8.31 8.26
.01 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.37 14.20 14.02 13.93 13.84 13.75 13.65 13.56 13.46
.005 31.33 26.28 24.26 23.15 22.46 21.97 21.62 21.35 21.14 20.97 20.70 20.44 20.17 20.03 19.89 19.75 19.61 19.47 19.32
5 .05 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.36
.025 10.01 8.43 7.76 7.39 7.15 6.98 6.85 6.76 6.68 6.62 6.52 6.43 6.33 6.28 6.23 6.18 6.12 6.07 6.02
.01 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02
.005 22.78 18.31 16.53 15.56 14.94 14.51 14.20 13.96 13.77 13.62 13.38 13.15 12.90 12.78 12.66 12.53 12.40 12.27 12.14
6 .05 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67
.025 8.81 7.26 6.60 6.23 5.99 5.82 5.70 5.60 5.52 5.46 5.37 5.27 5.17 5.12 5.07 5.01 4.96 4.90 4.85
.01 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88
.005 18.63 14.54 12.92 12.03 11.46 11.07 10.79 10.57 10.39 10.25 10.03 9.81 9.59 9.47 9.36 9.24 9.12 9.00 8.88
7 .05 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23
.025 8.07 6.54 5.89 5.52 5.29 5.12 4.99 4.90 4.82 4.76 4.67 4.57 4.47 4.41 4.36 4.31 4.25 4.20 4.14
.01 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65
.005 16.24 12.40 10.88 10.05 9.52 9.16 8.89 8.68 8.51 8.38 8.18 7.97 7.75 7.64 7.53 7.42 7.31 7.19 7.08
8 .05 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93
.025 7.57 6.06 5.42 5.05 4.82 4.65 4.53 4.43 4.36 4.30 4.20 4.10 4.00 3.95 3.89 3.84 3.78 3.73 3.67
.01 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86
.005 14.69 11.04 9.60 8.81 8.30 7.95 7.69 7.50 7.34 7.21 7.01 6.81 6.61 6.50 6.40 6.29 6.18 6.06 5.95
9 .05 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71
.025 7.21 5.71 5.08 4.72 4.48 4.32 4.20 4.10 4.03 3.96 3.87 3.77 3.67 3.61 3.56 3.51 3.45 3.39 3.33
.01 10.56 8.02 6.99 6.42 6.06 5.80 5.61 5.47 5.35 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.31
.005 13.61 10.11 8.72 7.96 7.47 7.13 6.88 6.69 6.54 6.42 6.23 6.03 5.83 5.73 5.62 5.52 5.41 5.30 5.19
10 .05 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54
.025 6.94 5.46 4.83 4.47 4.24 4.07 3.95 3.85 3.78 3.72 3.62 3.52 3.42 3.37 3.31 3.26 3.20 3.14 3.08
.01 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.91
.005 12.83 9.43 8.08 7.34 6.87 6.54 6.30 6.12 5.97 5.85 5.66 5.47 5.27 5.17 5.07 4.97 4.86 4.75 4.64
12 .05 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30
.025 6.55 5.10 4.47 4.12 3.89 3.73 3.61 3.51 3.44 3.37 3.28 3.18 3.07 3.02 2.96 2.91 2.85 2.79 2.72
.01 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.36
.005 11.75 8.51 7.23 6.52 6.07 5.76 5.52 5.35 5.20 5.09 4.91 4.72 4.53 4.43 4.33 4.23 4.12 4.01 3.90
15 .05 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07
.025 6.20 4.77 4.15 3.80 3.58 3.41 3.29 3.20 3.12 3.06 2.96 2.86 2.76 2.70 2.64 2.59 2.52 2.46 2.40
.01 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87
.005 10.80 7.70 6.48 5.80 5.37 5.07 4.85 4.67 4.54 4.42 4.25 4.07 3.88 3.79 3.69 3.58 3.48 3.37 3.26
20 .05 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84
.025 5.87 4.46 3.86 3.51 3.29 3.13 3.01 2.91 2.84 2.77 2.68 2.57 2.46 2.41 2.35 2.29 2.22 2.16 2.09
.01 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42
.005 9.94 6.99 5.82 5.17 4.76 4.47 4.26 4.09 3.96 3.85 3.68 3.50 3.32 3.22 3.12 3.02 2.92 2.81 2.69
24 .05 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73
.025 5.72 4.32 3.72 3.38 3.15 2.99 2.87 2.78 2.70 2.64 2.54 2.44 2.33 2.27 2.21 2.15 2.08 2.01 1.94
.01 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21
.005 9.55 6.66 5.52 4.89 4.49 4.20 3.99 3.83 3.69 3.59 3.42 3.25 3.06 2.97 2.87 2.77 2.66 2.55 2.43
30 .05 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62
.025 5.57 4.18 3.59 3.25 3.03 2.87 2.75 2.65 2.57 2.51 2.41 2.31 2.20 2.14 2.07 2.01 1.94 1.87 1.79
.01 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.01
.005 9.18 6.35 5.24 4.62 4.23 3.95 3.74 3.58 3.45 3.34 3.18 3.01 2.82 2.73 2.63 2.52 2.42 2.30 2.18
40 .05 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51
.025 5.42 4.05 3.46 3.13 2.90 2.74 2.62 2.53 2.45 2.39 2.29 2.18 2.07 2.01 1.94 1.88 1.80 1.72 1.64
.01 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.80
.005 8.83 6.07 4.98 4.37 3.99 3.71 3.51 3.35 3.22 3.12 2.95 2.78 2.60 2.50 2.40 2.30 2.18 2.06 1.93
60 .05 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39
.025 5.29 3.93 3.34 3.01 2.79 2.63 2.51 2.41 2.33 2.27 2.17 2.06 1.94 1.88 1.82 1.74 1.67 1.58 1.48
.01 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60
.005 8.49 5.79 4.73 4.14 3.76 3.49 3.29 3.13 3.01 2.90 2.74 2.57 2.39 2.29 2.19 2.08 1.96 1.83 1.69
120 .05 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25
.025 5.15 3.80 3.23 2.89 2.67 2.52 2.39 2.30 2.22 2.16 2.05 1.94 1.82 1.76 1.69 1.61 1.53 1.43 1.31
.01 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38
.005 8.18 5.54 4.50 3.92 3.55 3.28 3.09 2.93 2.81 2.71 2.54 2.37 2.19 2.09 1.98 1.87 1.75 1.61 1.43
∞ .05 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00
.025 5.02 3.69 3.12 2.79 2.57 2.41 2.29 2.19 2.11 2.05 1.94 1.83 1.71 1.64 1.57 1.48 1.39 1.27 1.00
.01 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00
.005 7.88 5.30 4.28 3.72 3.35 3.09 2.90 2.74 2.62 2.52 2.36 2.19 2.00 1.90 1.79 1.67 1.53 1.36 1.00