Question: 1. (a) For each one of the statements below say whether the statement is true or false, explaining your answer. i. For two independent events A and B with P(A) > 0 and P(B) > 0, then: P(A[B) < P(A)+ P(B): ii. Suppose A and B are two events such that P(A) = 0:2, P(B) = 0:4 and: P(AjB)+ P(BjA) = 0:7: It holds that P(A\B) = 14=150.
07 Oct 2022,10:37 PM
SECTION A
Answer all five parts of question 1 (40 marks).
1. (a) For each one of the statements below say whether the statement is true or false, explaining your answer.
i. For two independent events A and B with P(A) > 0 and P(B) > 0, then:
P(A[B) < P(A)+ P(B):
ii. Suppose A and B are two events such that P(A) = 0:2, P(B) = 0:4 and:
P(AjB)+ P(BjA) = 0:7:
It holds that P(A\B) = 14=150.
iii. If B A, then the complementary events Ac and Bc are independent events.
iv. If T1 is an unbiased estimator of the parameter , and T2 is an unbiased estimator of the parameter , then T1T2 is an unbiased estimator of .
v. The signicance level of a test is the probability the null hypothesis is false.
(10 marks)
(b) A random variable X can take the values 0, 1 and 2, with:
P(X = 0) = 1 3; P(X = 1) = and P(X = 2) =
where 0 < < 8=3. One observation of X is taken at random, and we want to estimate the parameter . Consider the two estimators:
T1 = 2X and T2 = 4X(X 1):
i. Calculate the bias of each estimator.
(4 marks)
ii. Which of the two estimators would you prefer and why?
(5 marks)
(c) The random variable X is normally distributed with a mean of 1 and a variance
of 4. Calculate:
i. P X > 3:4X > 2:2
and ii. P X > 3:4jXj > 2:2 :
(5 marks)
(d) Two fair six-sided dice are thrown. If their total score is greater than or equal
to 8 they are thrown once more. even?
What is the probability the total score is
(5 marks)
(e) The random variable X has the probability density function given by:
(
kx 5 for x 1
0 otherwise:
i. Determine the value of k.
(3 marks) ii. Compute E(X) and Var(X).
(5 marks) iii. Derive the cumulative distribution function of X.
(3 marks)
SECTION B
Answer all three questions from this section.
2. (a) The proportion of pregnant women among all women who take a particular pregnancy test is 0.75. Two-thirds of pregnant women are at an early stage of pregnancy. If a woman who takes the test is pregnant at an early stage, the test will be positive (indicating that she is pregnant) with probability 0.6. If she is pregnant at a later stage, the test will be positive with probability 0.8. If she is not pregnant, the test will be positive with probability 0.2.
i. What is the probability the test will be positive?
(2 marks) ii. What is the probability the test will show a false result?
(3 marks)
iii. If 10 women independently each take a pregnancy test, what could you say about the distribution of the number of these tests that are positive? Your answer should include a sketch plot of the probability distribution.
(5 marks)
(b) Let fX1;X2;:::;Xng be a random sample from the continuous uniform distribution dened over the interval [0;3].
i. Derive the maximum likelihood estimator of .
(4 marks)
ii. Given that the method of moments estimator of is 2X=3, check whether the method of moments estimator is consistent. Justify your answer.
(6 marks)
Hint: You may use any results on the formula sheet at the end of the question paper and you may state expressions for E(X) and Var(X) without proof.
3. (a) Let fX1;X2;:::;X16g be a random sample of size n = 16 from N(;2), where 2 = 2:56 is known. A researcher decides to test:
H0 : = 12 vs. H1 : > 12
using a 5% signicance level.
i. Calculate the power of the test when = 12:4.
(6 marks)
ii. Briey explain in two dierent ways how you could increase the power of this test.
(4 marks)
(b) A company produces copper wire to a particular specication of breaking strength using four dierent types of machines (A, B, C and D). One machine of each type is selected at random and ve copper wire samples are measured from each machine. The measurements were:
Machine type
A B C D
Sample wire 1 Sample wire 2 Sample wire 3 Sample wire 4 Sample wire 5 11.1 11.5 11.7 13.5 11.4 11.7 14.1 11.7 12.1 12.2 11.2 11.5 12.9 11.5 11.0 13.8 13.3 10.7 13.3 12.0
Sample mean 11.84 12.36 11.62 12.62
You are given that:
XX
xij nx = 19:418
j=1 i=1
where the overall sample mean is x = 12:11.
i. Test the null hypothesis that the mean breaking strengths of the four types of machines are the same. Use a 5% signicance level.
(6 marks)
ii. Compute a 99% condence interval for the dierence of the mean breaking strengths between machine types B and C.
(4 marks)
4. Suppose X and Y are two independent random variables with the following probability distributions:
X = x 1 0 1
P(X = x) 0.30 0.40 0.30
and
Y = y 1 0 1
P(Y = y) 0.40 0.20 0.40
The random variables S and T are dened as:
S = X2 +Y2 and T = X +Y:
(a) Construct the table of the joint probability distribution of S and T.
(8 marks)
(b) Calculate the following quantities:
i. Var(T), given that E(T) = 0.
(2 marks)
ii. Cov(S;T).
(3 marks)
iii. E(S jT = 0).
(4 marks)
(c) Are S and T uncorrelated? Are S and T independent? Justify your answers. (3 marks)
Formulae for Statistics
Discrete distributions
Continuous distributions
UL21/0186 Page 7 of 31
Sample quantities
Sample variance
Sample covariance
Sample correlation
s2 = 1 X(xi x)2 = 1 Xx2 nx2 i=1 i=1
X X
(xi x)(yi y) = xiyi nxy
i=1 i=1
n
xiyi nxy
s n i=1 n x nx2 y ny2
i=1 i=1
Inference
Variance of sample mean
One-sample t statistic
Two-sample t statistic
2 n
X S= n
s
n+m 2 X Y 0
1=n +1=m (n 1)SX +(m 1)SY
STATISTICAL TABLES
Cumulative normal distribution
Critical values of the t distribution
Critical values of the F distribution
Critical values of the chi-squared distribution
New Cambridge Statistical Tables pages 17-29
.
STATISTICAL TABLES 1
TABLE A.1
Cumulative Standardized Normal Distribution
A(z)
A(z) is the integral of the standardized normal distribution from to z (in other words, the area under the curve to the left of z). It gives the probability of a normal random variable not being more than z standard deviations above its mean. Values of z of particular importance:
z A(z)
1.645 0.9500 Lower limit of right 5% tail 1.960 0.9750 Lower limit of right 2.5% tail 2.326 0.9900 Lower limit of right 1% tail 2.576 0.9950 Lower limit of right 0.5% tail 3.090 0.9990 Lower limit of right 0.1% tail 3.291 0.9995 Lower limit of right 0.05% tail
-4 -3 -2 -1 0 1 z 2 3 4
z 0.00 0.01
0.0 0.5000 0.5040 0.1 0.5398 0.5438 0.2 0.5793 0.5832 0.3 0.6179 0.6217 0.4 0.6554 0.6591 0.5 0.6915 0.6950 0.6 0.7257 0.7291 0.7 0.7580 0.7611 0.8 0.7881 0.7910 0.9 0.8159 0.8186 1.0 0.8413 0.8438 1.1 0.8643 0.8665 1.2 0.8849 0.8869 1.3 0.9032 0.9049 1.4 0.9192 0.9207 1.5 0.9332 0.9345 1.6 0.9452 0.9463 1.7 0.9554 0.9564 1.8 0.9641 0.9649 1.9 0.9713 0.9719 2.0 0.9772 0.9778 2.1 0.9821 0.9826 2.2 0.9861 0.9864 2.3 0.9893 0.9896 2.4 0.9918 0.9920 2.5 0.9938 0.9940 2.6 0.9953 0.9955 2.7 0.9965 0.9966 2.8 0.9974 0.9975 2.9 0.9981 0.9982 3.0 0.9987 0.9987 3.1 0.9990 0.9991 3.2 0.9993 0.9993 3.3 0.9995 0.9995 3.4 0.9997 0.9997 3.5 0.9998 0.9998 3.6 0.9998 0.9998
0.02 0.03
0.5080 0.5120 0.5478 0.5517 0.5871 0.5910 0.6255 0.6293 0.6628 0.6664 0.6985 0.7019 0.7324 0.7357 0.7642 0.7673 0.7939 0.7967 0.8212 0.8238 0.8461 0.8485 0.8686 0.8708 0.8888 0.8907 0.9066 0.9082 0.9222 0.9236 0.9357 0.9370 0.9474 0.9484 0.9573 0.9582 0.9656 0.9664 0.9726 0.9732 0.9783 0.9788 0.9830 0.9834 0.9868 0.9871 0.9898 0.9901 0.9922 0.9925 0.9941 0.9943 0.9956 0.9957 0.9967 0.9968 0.9976 0.9977 0.9982 0.9983 0.9987 0.9988 0.9991 0.9991 0.9994 0.9994 0.9995 0.9996 0.9997 0.9997 0.9998 0.9998 0.9999
0.04 0.05
0.5160 0.5199 0.5557 0.5596 0.5948 0.5987 0.6331 0.6368 0.6700 0.6736 0.7054 0.7088 0.7389 0.7422 0.7704 0.7734 0.7995 0.8023 0.8264 0.8289 0.8508 0.8531 0.8729 0.8749 0.8925 0.8944 0.9099 0.9115 0.9251 0.9265 0.9382 0.9394 0.9495 0.9505 0.9591 0.9599 0.9671 0.9678 0.9738 0.9744 0.9793 0.9798 0.9838 0.9842 0.9875 0.9878 0.9904 0.9906 0.9927 0.9929 0.9945 0.9946 0.9959 0.9960 0.9969 0.9970 0.9977 0.9978 0.9984 0.9984 0.9988 0.9989 0.9992 0.9992 0.9994 0.9994 0.9996 0.9996 0.9997 0.9997 0.9998 0.9998
0.06 0.07
0.5239 0.5279 0.5636 0.5675 0.6026 0.6064 0.6406 0.6443 0.6772 0.6808 0.7123 0.7157 0.7454 0.7486 0.7764 0.7794 0.8051 0.8078 0.8315 0.8340 0.8554 0.8577 0.8770 0.8790 0.8962 0.8980 0.9131 0.9147 0.9279 0.9292 0.9406 0.9418 0.9515 0.9525 0.9608 0.9616 0.9686 0.9693 0.9750 0.9756 0.9803 0.9808 0.9846 0.9850 0.9881 0.9884 0.9909 0.9911 0.9931 0.9932 0.9948 0.9949 0.9961 0.9962 0.9971 0.9972 0.9979 0.9979 0.9985 0.9985 0.9989 0.9989 0.9992 0.9992 0.9994 0.9995 0.9996 0.9996 0.9997 0.9997 0.9998 0.9998
0.08 0.09
0.5319 0.5359 0.5714 0.5753 0.6103 0.6141 0.6480 0.6517 0.6844 0.6879 0.7190 0.7224 0.7517 0.7549 0.7823 0.7852 0.8106 0.8133 0.8365 0.8389 0.8599 0.8621 0.8810 0.8830 0.8997 0.9015 0.9162 0.9177 0.9306 0.9319 0.9429 0.9441 0.9535 0.9545 0.9625 0.9633 0.9699 0.9706 0.9761 0.9767 0.9812 0.9817 0.9854 0.9857 0.9887 0.9890 0.9913 0.9916 0.9934 0.9936 0.9951 0.9952 0.9963 0.9964 0.9973 0.9974 0.9980 0.9981 0.9986 0.9986 0.9990 0.9990 0.9993 0.9993 0.9995 0.9995 0.9996 0.9997 0.9997 0.9998 0.9998 0.9998
.
STATISTICAL TABLES 2
TABLE A.2
t Distribution: Critical Values of t
Significance level
Degrees of freedom
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
26 27 28 29 30
32 34 36 38 40
42 44 46 48 50
60 70 80 90 100
120 150 200 300 400
500 600
Two-tailed test: One-tailed test:
10% 5% 2% 5% 2.5% 1%
6.314 12.706 31.821 2.920 4.303 6.965 2.353 3.182 4.541 2.132 2.776 3.747 2.015 2.571 3.365
1.943 2.447 3.143 1.894 2.365 2.998 1.860 2.306 2.896 1.833 2.262 2.821 1.812 2.228 2.764
1.796 2.201 2.718 1.782 2.179 2.681 1.771 2.160 2.650 1.761 2.145 2.624 1.753 2.131 2.602
1.746 2.120 2.583 1.740 2.110 2.567 1.734 2.101 2.552 1.729 2.093 2.539 1.725 2.086 2.528
1.721 2.080 2.518 1.717 2.074 2.508 1.714 2.069 2.500 1.711 2.064 2.492 1.708 2.060 2.485
1.706 2.056 2.479 1.703 2.052 2.473 1.701 2.048 2.467 1.699 2.045 2.462 1.697 2.042 2.457
1.694 2.037 2.449 1.691 2.032 2.441 1.688 2.028 2.434 1.686 2.024 2.429 1.684 2.021 2.423
1.682 2.018 2.418 1.680 2.015 2.414 1.679 2.013 2.410 1.677 2.011 2.407 1.676 2.009 2.403
1.671 2.000 2.390 1.667 1.994 2.381 1.664 1.990 2.374 1.662 1.987 2.368 1.660 1.984 2.364
1.658 1.980 2.358 1.655 1.976 2.351 1.653 1.972 2.345 1.650 1.968 2.339 1.649 1.966 2.336
1.648 1.965 2.334 1.647 1.964 2.333
1.645 1.960 2.326
1% 0.2% 0.5% 0.1%
63.657 318.309 9.925 22.327 5.841 10.215 4.604 7.173 4.032 5.893
3.707 5.208 3.499 4.785 3.355 4.501 3.250 4.297 3.169 4.144
3.106 4.025 3.055 3.930 3.012 3.852 2.977 3.787 2.947 3.733
2.921 3.686 2.898 3.646 2.878 3.610 2.861 3.579 2.845 3.552
2.831 3.527 2.819 3.505 2.807 3.485 2.797 3.467 2.787 3.450
2.779 3.435 2.771 3.421 2.763 3.408 2.756 3.396 2.750 3.385
2.738 3.365 2.728 3.348 2.719 3.333 2.712 3.319 2.704 3.307
2.698 3.296 2.692 3.286 2.687 3.277 2.682 3.269 2.678 3.261
2.660 3.232 2.648 3.211 2.639 3.195 2.632 3.183 2.626 3.174
2.617 3.160 2.609 3.145 2.601 3.131 2.592 3.118 2.588 3.111
2.586 3.107 2.584 3.104
2.576 3.090
0.1% 0.05%
636.619 31.599 12.924 8.610 6.869
5.959 5.408 5.041 4.781 4.587
4.437 4.318 4.221 4.140 4.073
4.015 3.965 3.922 3.883 3.850
3.819 3.792 3.768 3.745 3.725
3.707 3.690 3.674 3.659 3.646
3.622 3.601 3.582 3.566 3.551
3.538 3.526 3.515 3.505 3.496
3.460 3.435 3.416 3.402 3.390
3.373 3.357 3.340 3.323 3.315
3.310 3.307
3.291
.
STATISTICAL TABLES 3
TABLE A.3
F Distribution: Critical Values of F (5% significance level)
v1 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 v2
1 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88 243.91 245.36 246.46 247.32 248.01 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.42 19.43 19.44 19.45 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.71 8.69 8.67 8.66 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.87 5.84 5.82 5.80 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.64 4.60 4.58 4.56
6 5.99 5.14 7 5.59 4.74 8 5.32 4.46 9 5.12 4.26 10 4.96 4.10
11 4.84 3.98 12 4.75 3.89 13 4.67 3.81 14 4.60 3.74 15 4.54 3.68
16 4.49 3.63 17 4.45 3.59 18 4.41 3.55 19 4.38 3.52 20 4.35 3.49
21 4.32 3.47 22 4.30 3.44 23 4.28 3.42 24 4.26 3.40 25 4.24 3.39
26 4.22 3.37 27 4.21 3.35 28 4.20 3.34 29 4.18 3.33 30 4.17 3.32
35 4.12 3.27 40 4.08 3.23 50 4.03 3.18 60 4.00 3.15 70 3.98 3.13
80 3.96 3.11 90 3.95 3.10 100 3.94 3.09 120 3.92 3.07 150 3.90 3.06
200 3.89 3.04 250 3.88 3.03 300 3.87 3.03 400 3.86 3.02 500 3.86 3.01
600 3.86 3.01 750 3.85 3.01 1000 3.85 3.00
4.76 4.53 4.35 4.12 4.07 3.84 3.86 3.63 3.71 3.48
3.59 3.36 3.49 3.26 3.41 3.18 3.34 3.11 3.29 3.06
3.24 3.01 3.20 2.96 3.16 2.93 3.13 2.90 3.10 2.87
3.07 2.84 3.05 2.82 3.03 2.80 3.01 2.78 2.99 2.76
2.98 2.74 2.96 2.73 2.95 2.71 2.93 2.70 2.92 2.69
2.87 2.64 2.84 2.61 2.79 2.56 2.76 2.53 2.74 2.50
2.72 2.49 2.71 2.47 2.70 2.46 2.68 2.45 2.66 2.43
2.65 2.42 2.64 2.41 2.63 2.40 2.63 2.39 2.62 2.39
2.62 2.39 2.62 2.38 2.61 2.38
4.39 4.28 3.97 3.87 3.69 3.58 3.48 3.37 3.33 3.22
3.20 3.09 3.11 3.00 3.03 2.92 2.96 2.85 2.90 2.79
2.85 2.74 2.81 2.70 2.77 2.66 2.74 2.63 2.71 2.60
2.68 2.57 2.66 2.55 2.64 2.53 2.62 2.51 2.60 2.49
2.59 2.47 2.57 2.46 2.56 2.45 2.55 2.43 2.53 2.42
2.49 2.37 2.45 2.34 2.40 2.29 2.37 2.25 2.35 2.23
2.33 2.21 2.32 2.20 2.31 2.19 2.29 2.18 2.27 2.16
2.26 2.14 2.25 2.13 2.24 2.13 2.24 2.12 2.23 2.12
2.23 2.11 2.23 2.11 2.22 2.11
4.21 4.15 3.79 3.73 3.50 3.44 3.29 3.23 3.14 3.07
3.01 2.95 2.91 2.85 2.83 2.77 2.76 2.70 2.71 2.64
2.66 2.59 2.61 2.55 2.58 2.51 2.54 2.48 2.51 2.45
2.49 2.42 2.46 2.40 2.44 2.37 2.42 2.36 2.40 2.34
2.39 2.32 2.37 2.31 2.36 2.29 2.35 2.28 2.33 2.27
2.29 2.22 2.25 2.18 2.20 2.13 2.17 2.10 2.14 2.07
2.13 2.06 2.11 2.04 2.10 2.03 2.09 2.02 2.07 2.00
2.06 1.98 2.05 1.98 2.04 1.97 2.03 1.96 2.03 1.96
2.02 1.95 2.02 1.95 2.02 1.95
4.10 4.06 3.68 3.64 3.39 3.35 3.18 3.14 3.02 2.98
2.90 2.85 2.80 2.75 2.71 2.67 2.65 2.60 2.59 2.54
2.54 2.49 2.49 2.45 2.46 2.41 2.42 2.38 2.39 2.35
2.37 2.32 2.34 2.30 2.32 2.27 2.30 2.25 2.28 2.24
2.27 2.22 2.25 2.20 2.24 2.19 2.22 2.18 2.21 2.16
2.16 2.11 2.12 2.08 2.07 2.03 2.04 1.99 2.02 1.97
2.00 1.95 1.99 1.94 1.97 1.93 1.96 1.91 1.94 1.89
1.93 1.88 1.92 1.87 1.91 1.86 1.90 1.85 1.90 1.85
1.90 1.85 1.89 1.84 1.89 1.84
4.00 3.96 3.57 3.53 3.28 3.24 3.07 3.03 2.91 2.86
2.79 2.74 2.69 2.64 2.60 2.55 2.53 2.48 2.48 2.42
2.42 2.37 2.38 2.33 2.34 2.29 2.31 2.26 2.28 2.22
2.25 2.20 2.23 2.17 2.20 2.15 2.18 2.13 2.16 2.11
2.15 2.09 2.13 2.08 2.12 2.06 2.10 2.05 2.09 2.04
2.04 1.99 2.00 1.95 1.95 1.89 1.92 1.86 1.89 1.84
1.88 1.82 1.86 1.80 1.85 1.79 1.83 1.78 1.82 1.76
1.80 1.74 1.79 1.73 1.78 1.72 1.78 1.72 1.77 1.71
1.77 1.71 1.77 1.70 1.76 1.70
3.92 3.90 3.87 3.49 3.47 3.44 3.20 3.17 3.15 2.99 2.96 2.94 2.83 2.80 2.77
2.70 2.67 2.65 2.60 2.57 2.54 2.51 2.48 2.46 2.44 2.41 2.39 2.38 2.35 2.33
2.33 2.30 2.28 2.29 2.26 2.23 2.25 2.22 2.19 2.21 2.18 2.16 2.18 2.15 2.12
2.16 2.12 2.10 2.13 2.10 2.07 2.11 2.08 2.05 2.09 2.05 2.03 2.07 2.04 2.01
2.05 2.02 1.99 2.04 2.00 1.97 2.02 1.99 1.96 2.01 1.97 1.94 1.99 1.96 1.93
1.94 1.91 1.88 1.90 1.87 1.84 1.85 1.81 1.78 1.82 1.78 1.75 1.79 1.75 1.72
1.77 1.73 1.70 1.76 1.72 1.69 1.75 1.71 1.68 1.73 1.69 1.66 1.71 1.67 1.64
1.69 1.66 1.62 1.68 1.65 1.61 1.68 1.64 1.61 1.67 1.63 1.60 1.66 1.62 1.59
1.66 1.62 1.59 1.66 1.62 1.58 1.65 1.61 1.58
.
STATISTICAL TABLES 4
TABLE A.3 (continued)
F Distribution: Critical Values of F (5% significance level)
v1 25 30 35 40 50 60 75 100 150 200 v2
1 249.26 250.10 250.69 251.14 251.77 252.20 252.62 253.04 253.46 253.68 2 19.46 19.46 19.47 19.47 19.48 19.48 19.48 19.49 19.49 19.49 3 8.63 8.62 8.60 8.59 8.58 8.57 8.56 8.55 8.54 8.54 4 5.77 5.75 5.73 5.72 5.70 5.69 5.68 5.66 5.65 5.65 5 4.52 4.50 4.48 4.46 4.44 4.43 4.42 4.41 4.39 4.39
6 3.83 3.81 7 3.40 3.38 8 3.11 3.08 9 2.89 2.86 10 2.73 2.70
11 2.60 2.57 12 2.50 2.47 13 2.41 2.38 14 2.34 2.31 15 2.28 2.25
16 2.23 2.19 17 2.18 2.15 18 2.14 2.11 19 2.11 2.07 20 2.07 2.04
21 2.05 2.01 22 2.02 1.98 23 2.00 1.96 24 1.97 1.94 25 1.96 1.92
26 1.94 1.90 27 1.92 1.88 28 1.91 1.87 29 1.89 1.85 30 1.88 1.84
35 1.82 1.79 40 1.78 1.74 50 1.73 1.69 60 1.69 1.65 70 1.66 1.62
80 1.64 1.60 90 1.63 1.59 100 1.62 1.57 120 1.60 1.55 150 1.58 1.54
200 1.56 1.52 250 1.55 1.50 300 1.54 1.50 400 1.53 1.49 500 1.53 1.48
600 1.52 1.48 750 1.52 1.47 1000 1.52 1.47
3.79 3.77 3.75 3.36 3.34 3.32 3.06 3.04 3.02 2.84 2.83 2.80 2.68 2.66 2.64
2.55 2.53 2.51 2.44 2.43 2.40 2.36 2.34 2.31 2.28 2.27 2.24 2.22 2.20 2.18
2.17 2.15 2.12 2.12 2.10 2.08 2.08 2.06 2.04 2.05 2.03 2.00 2.01 1.99 1.97
1.98 1.96 1.94 1.96 1.94 1.91 1.93 1.91 1.88 1.91 1.89 1.86 1.89 1.87 1.84
1.87 1.85 1.82 1.86 1.84 1.81 1.84 1.82 1.79 1.83 1.81 1.77 1.81 1.79 1.76
1.76 1.74 1.70 1.72 1.69 1.66 1.66 1.63 1.60 1.62 1.59 1.56 1.59 1.57 1.53
1.57 1.54 1.51 1.55 1.53 1.49 1.54 1.52 1.48 1.52 1.50 1.46 1.50 1.48 1.44
1.48 1.46 1.41 1.47 1.44 1.40 1.46 1.43 1.39 1.45 1.42 1.38 1.45 1.42 1.38
1.44 1.41 1.37 1.44 1.41 1.37 1.43 1.41 1.36
3.74 3.73 3.71 3.30 3.29 3.27 3.01 2.99 2.97 2.79 2.77 2.76 2.62 2.60 2.59
2.49 2.47 2.46 2.38 2.37 2.35 2.30 2.28 2.26 2.22 2.21 2.19 2.16 2.14 2.12
2.11 2.09 2.07 2.06 2.04 2.02 2.02 2.00 1.98 1.98 1.96 1.94 1.95 1.93 1.91
1.92 1.90 1.88 1.89 1.87 1.85 1.86 1.84 1.82 1.84 1.82 1.80 1.82 1.80 1.78
1.80 1.78 1.76 1.79 1.76 1.74 1.77 1.75 1.73 1.75 1.73 1.71 1.74 1.72 1.70
1.68 1.66 1.63 1.64 1.61 1.59 1.58 1.55 1.52 1.53 1.51 1.48 1.50 1.48 1.45
1.48 1.45 1.43 1.46 1.44 1.41 1.45 1.42 1.39 1.43 1.40 1.37 1.41 1.38 1.34
1.39 1.35 1.32 1.37 1.34 1.31 1.36 1.33 1.30 1.35 1.32 1.28 1.35 1.31 1.28
1.34 1.31 1.27 1.34 1.30 1.26 1.33 1.30 1.26
3.70 3.69 3.26 3.25 2.96 2.95 2.74 2.73 2.57 2.56
2.44 2.43 2.33 2.32 2.24 2.23 2.17 2.16 2.10 2.10
2.05 2.04 2.00 1.99 1.96 1.95 1.92 1.91 1.89 1.88
1.86 1.84 1.83 1.82 1.80 1.79 1.78 1.77 1.76 1.75
1.74 1.73 1.72 1.71 1.70 1.69 1.69 1.67 1.67 1.66
1.61 1.60 1.56 1.55 1.50 1.48 1.45 1.44 1.42 1.40
1.39 1.38 1.38 1.36 1.36 1.34 1.33 1.32 1.31 1.29
1.28 1.26 1.27 1.25 1.26 1.23 1.24 1.22 1.23 1.21
1.23 1.20 1.22 1.20 1.22 1.19
.
STATISTICAL TABLES 5
TABLE A.3 (continued)
F Distribution: Critical Values of F (1% significance level)
v1 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 v2
1 4052.18 4999.50 5403.35 5624.58 5763.65 5858.99 5928.36 5981.07 6022.47 6055.85 6106.32 6142.67 6170.10 6191.53 6208.73 2 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.42 99.43 99.44 99.44 99.45 3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23 27.05 26.92 26.83 26.75 26.69 4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.37 14.25 14.15 14.08 14.02 5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.89 9.77 9.68 9.61 9.55
6 13.75 10.92 7 12.25 9.55 8 11.26 8.65 9 10.56 8.02 10 10.04 7.56
11 9.65 7.21 12 9.33 6.93 13 9.07 6.70 14 8.86 6.51 15 8.68 6.36
16 8.53 6.23 17 8.40 6.11 18 8.29 6.01 19 8.18 5.93 20 8.10 5.85
21 8.02 5.78 22 7.95 5.72 23 7.88 5.66 24 7.82 5.61 25 7.77 5.57
26 7.72 5.53 27 7.68 5.49 28 7.64 5.45 29 7.60 5.42 30 7.56 5.39
35 7.42 5.27 40 7.31 5.18 50 7.17 5.06 60 7.08 4.98 70 7.01 4.92
80 6.96 4.88 90 6.93 4.85 100 6.90 4.82 120 6.85 4.79 150 6.81 4.75
200 6.76 4.71 250 6.74 4.69 300 6.72 4.68 400 6.70 4.66 500 6.69 4.65
600 6.68 4.64 750 6.67 4.63 1000 6.66 4.63
9.78 9.15 8.45 7.85 7.59 7.01 6.99 6.42 6.55 5.99
6.22 5.67 5.95 5.41 5.74 5.21 5.56 5.04 5.42 4.89
5.29 4.77 5.18 4.67 5.09 4.58 5.01 4.50 4.94 4.43
4.87 4.37 4.82 4.31 4.76 4.26 4.72 4.22 4.68 4.18
4.64 4.14 4.60 4.11 4.57 4.07 4.54 4.04 4.51 4.02
4.40 3.91 4.31 3.83 4.20 3.72 4.13 3.65 4.07 3.60
4.04 3.56 4.01 3.53 3.98 3.51 3.95 3.48 3.91 3.45
3.88 3.41 3.86 3.40 3.85 3.38 3.83 3.37 3.82 3.36
3.81 3.35 3.81 3.34 3.80 3.34
8.75 8.47 7.46 7.19 6.63 6.37 6.06 5.80 5.64 5.39
5.32 5.07 5.06 4.82 4.86 4.62 4.69 4.46 4.56 4.32
4.44 4.20 4.34 4.10 4.25 4.01 4.17 3.94 4.10 3.87
4.04 3.81 3.99 3.76 3.94 3.71 3.90 3.67 3.85 3.63
3.82 3.59 3.78 3.56 3.75 3.53 3.73 3.50 3.70 3.47
3.59 3.37 3.51 3.29 3.41 3.19 3.34 3.12 3.29 3.07
3.26 3.04 3.23 3.01 3.21 2.99 3.17 2.96 3.14 2.92
3.11 2.89 3.09 2.87 3.08 2.86 3.06 2.85 3.05 2.84
3.05 2.83 3.04 2.83 3.04 2.82
8.26 8.10 6.99 6.84 6.18 6.03 5.61 5.47 5.20 5.06
4.89 4.74 4.64 4.50 4.44 4.30 4.28 4.14 4.14 4.00
4.03 3.89 3.93 3.79 3.84 3.71 3.77 3.63 3.70 3.56
3.64 3.51 3.59 3.45 3.54 3.41 3.50 3.36 3.46 3.32
3.42 3.29 3.39 3.26 3.36 3.23 3.33 3.20 3.30 3.17
3.20 3.07 3.12 2.99 3.02 2.89 2.95 2.82 2.91 2.78
2.87 2.74 2.84 2.72 2.82 2.69 2.79 2.66 2.76 2.63
2.73 2.60 2.71 2.58 2.70 2.57 2.68 2.56 2.68 2.55
2.67 2.54 2.66 2.53 2.66 2.53
7.98 7.87 6.72 6.62 5.91 5.81 5.35 5.26 4.94 4.85
4.63 4.54 4.39 4.30 4.19 4.10 4.03 3.94 3.89 3.80
3.78 3.69 3.68 3.59 3.60 3.51 3.52 3.43 3.46 3.37
3.40 3.31 3.35 3.26 3.30 3.21 3.26 3.17 3.22 3.13
3.18 3.09 3.15 3.06 3.12 3.03 3.09 3.00 3.07 2.98
2.96 2.88 2.89 2.80 2.78 2.70 2.72 2.63 2.67 2.59
2.64 2.55 2.61 2.52 2.59 2.50 2.56 2.47 2.53 2.44
2.50 2.41 2.48 2.39 2.47 2.38 2.45 2.37 2.44 2.36
2.44 2.35 2.43 2.34 2.43 2.34
7.72 7.60 6.47 6.36 5.67 5.56 5.11 5.01 4.71 4.60
4.40 4.29 4.16 4.05 3.96 3.86 3.80 3.70 3.67 3.56
3.55 3.45 3.46 3.35 3.37 3.27 3.30 3.19 3.23 3.13
3.17 3.07 3.12 3.02 3.07 2.97 3.03 2.93 2.99 2.89
2.96 2.86 2.93 2.82 2.90 2.79 2.87 2.77 2.84 2.74
2.74 2.64 2.66 2.56 2.56 2.46 2.50 2.39 2.45 2.35
2.42 2.31 2.39 2.29 2.37 2.27 2.34 2.23 2.31 2.20
2.27 2.17 2.26 2.15 2.24 2.14 2.23 2.13 2.22 2.12
2.21 2.11 2.21 2.11 2.20 2.10
7.52 7.45 7.40 6.28 6.21 6.16 5.48 5.41 5.36 4.92 4.86 4.81 4.52 4.46 4.41
4.21 4.15 4.10 3.97 3.91 3.86 3.78 3.72 3.66 3.62 3.56 3.51 3.49 3.42 3.37
3.37 3.31 3.26 3.27 3.21 3.16 3.19 3.13 3.08 3.12 3.05 3.00 3.05 2.99 2.94
2.99 2.93 2.88 2.94 2.88 2.83 2.89 2.83 2.78 2.85 2.79 2.74 2.81 2.75 2.70
2.78 2.72 2.66 2.75 2.68 2.63 2.72 2.65 2.60 2.69 2.63 2.57 2.66 2.60 2.55
2.56 2.50 2.44 2.48 2.42 2.37 2.38 2.32 2.27 2.31 2.25 2.20 2.27 2.20 2.15
2.23 2.17 2.12 2.21 2.14 2.09 2.19 2.12 2.07 2.15 2.09 2.03 2.12 2.06 2.00
2.09 2.03 1.97 2.07 2.01 1.95 2.06 1.99 1.94 2.05 1.98 1.92 2.04 1.97 1.92
2.03 1.96 1.91 2.02 1.96 1.90 2.02 1.95 1.90
.
STATISTICAL TABLES 6
TABLE A.3 (continued)
F Distribution: Critical Values of F (1% significance level)
v1 25 30 35 40 50 60 75 100 150 200 v2
1 6239.83 6260.65 6275.57 6286.78 6302.52 6313.03 6323.56 6334.11 6344.68 6349.97 2 99.46 99.47 99.47 99.47 99.48 99.48 99.49 99.49 99.49 99.49 3 26.58 26.50 26.45 26.41 26.35 26.32 26.28 26.24 26.20 26.18 4 13.91 13.84 13.79 13.75 13.69 13.65 13.61 13.58 13.54 13.52 5 9.45 9.38 9.33 9.29 9.24 9.20 9.17 9.13 9.09 9.08
6 7.30 7.23 7 6.06 5.99 8 5.26 5.20 9 4.71 4.65 10 4.31 4.25
11 4.01 3.94 12 3.76 3.70 13 3.57 3.51 14 3.41 3.35 15 3.28 3.21
16 3.16 3.10 17 3.07 3.00 18 2.98 2.92 19 2.91 2.84 20 2.84 2.78
21 2.79 2.72 22 2.73 2.67 23 2.69 2.62 24 2.64 2.58 25 2.60 2.54
26 2.57 2.50 27 2.54 2.47 28 2.51 2.44 29 2.48 2.41 30 2.45 2.39
35 2.35 2.28 40 2.27 2.20 50 2.17 2.10 60 2.10 2.03 70 2.05 1.98
80 2.01 1.94 90 1.99 1.92 100 1.97 1.89 120 1.93 1.86 150 1.90 1.83
200 1.87 1.79 250 1.85 1.77 300 1.84 1.76 400 1.82 1.75 500 1.81 1.74
600 1.80 1.73 750 1.80 1.72 1000 1.79 1.72
7.18 7.14 5.94 5.91 5.15 5.12 4.60 4.57 4.20 4.17
3.89 3.86 3.65 3.62 3.46 3.43 3.30 3.27 3.17 3.13
3.05 3.02 2.96 2.92 2.87 2.84 2.80 2.76 2.73 2.69
2.67 2.64 2.62 2.58 2.57 2.54 2.53 2.49 2.49 2.45
2.45 2.42 2.42 2.38 2.39 2.35 2.36 2.33 2.34 2.30
2.23 2.19 2.15 2.11 2.05 2.01 1.98 1.94 1.93 1.89
1.89 1.85 1.86 1.82 1.84 1.80 1.81 1.76 1.77 1.73
1.74 1.69 1.72 1.67 1.70 1.66 1.69 1.64 1.68 1.63
1.67 1.63 1.66 1.62 1.66 1.61
7.09 7.06 5.86 5.82 5.07 5.03 4.52 4.48 4.12 4.08
3.81 3.78 3.57 3.54 3.38 3.34 3.22 3.18 3.08 3.05
2.97 2.93 2.87 2.83 2.78 2.75 2.71 2.67 2.64 2.61
2.58 2.55 2.53 2.50 2.48 2.45 2.44 2.40 2.40 2.36
2.36 2.33 2.33 2.29 2.30 2.26 2.27 2.23 2.25 2.21
2.14 2.10 2.06 2.02 1.95 1.91 1.88 1.84 1.83 1.78
1.79 1.75 1.76 1.72 1.74 1.69 1.70 1.66 1.66 1.62
1.63 1.58 1.61 1.56 1.59 1.55 1.58 1.53 1.57 1.52
1.56 1.51 1.55 1.50 1.54 1.50
7.02 6.99 5.79 5.75 5.00 4.96 4.45 4.41 4.05 4.01
3.74 3.71 3.50 3.47 3.31 3.27 3.15 3.11 3.01 2.98
2.90 2.86 2.80 2.76 2.71 2.68 2.64 2.60 2.57 2.54
2.51 2.48 2.46 2.42 2.41 2.37 2.37 2.33 2.33 2.29
2.29 2.25 2.26 2.22 2.23 2.19 2.20 2.16 2.17 2.13
2.06 2.02 1.98 1.94 1.87 1.82 1.79 1.75 1.74 1.70
1.70 1.65 1.67 1.62 1.65 1.60 1.61 1.56 1.57 1.52
1.53 1.48 1.51 1.46 1.50 1.44 1.48 1.42 1.47 1.41
1.46 1.40 1.45 1.39 1.44 1.38
6.95 6.93 5.72 5.70 4.93 4.91 4.38 4.36 3.98 3.96
3.67 3.66 3.43 3.41 3.24 3.22 3.08 3.06 2.94 2.92
2.83 2.81 2.73 2.71 2.64 2.62 2.57 2.55 2.50 2.48
2.44 2.42 2.38 2.36 2.34 2.32 2.29 2.27 2.25 2.23
2.21 2.19 2.18 2.16 2.15 2.13 2.12 2.10 2.09 2.07
1.98 1.96 1.90 1.87 1.78 1.76 1.70 1.68 1.65 1.62
1.61 1.58 1.57 1.55 1.55 1.52 1.51 1.48 1.46 1.43
1.42 1.39 1.40 1.36 1.38 1.35 1.36 1.32 1.34 1.31
1.34 1.30 1.33 1.29 1.32 1.28
.
STATISTICAL TABLES 7
TABLE A.3 (continued)
F Distribution: Critical Values of F (0.1% significance level)
v1 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 v2
1 4.05e05 5.00e05 5.40e05 5.62e05 5.76e05 5.86e05 5.93e05 5.98e05 6.02e05 6.06e05 6.11e05 6.14e05 6.17e05 6.19e05 6.21e05 2 998.50 999.00 999.17 999.25 999.30 999.33 999.36 999.37 999.39 999.40 999.42 999.43 999.44 999.44 999.45 3 167.03 148.50 141.11 137.10 134.58 132.85 131.58 130.62 129.86 129.25 128.32 127.64 127.14 126.74 126.42 4 74.14 61.25 56.18 53.44 51.71 50.53 49.66 49.00 48.47 48.05 47.41 46.95 46.60 46.32 46.10 5 47.18 37.12 33.20 31.09 29.75 28.83 28.16 27.65 27.24 26.92 26.42 26.06 25.78 25.57 25.39
6 35.51 27.00 7 29.25 21.69 8 25.41 18.49 9 22.86 16.39 10 21.04 14.91
11 19.69 13.81 12 18.64 12.97 13 17.82 12.31 14 17.14 11.78 15 16.59 11.34
16 16.12 10.97 17 15.72 10.66 18 15.38 10.39 19 15.08 10.16 20 14.82 9.95
21 14.59 9.77 22 14.38 9.61 23 14.20 9.47 24 14.03 9.34 25 13.88 9.22
26 13.74 9.12 27 13.61 9.02 28 13.50 8.93 29 13.39 8.85 30 13.29 8.77
35 12.90 8.47 40 12.61 8.25 50 12.22 7.96 60 11.97 7.77 70 11.80 7.64
80 11.67 7.54 90 11.57 7.47 100 11.50 7.41 120 11.38 7.32 150 11.27 7.24
200 11.15 7.15 250 11.09 7.10 300 11.04 7.07 400 10.99 7.03 500 10.96 7.00
600 10.94 6.99 750 10.91 6.97 1000 10.89 6.96
23.70 21.92 18.77 17.20 15.83 14.39 13.90 12.56 12.55 11.28
11.56 10.35 10.80 9.63 10.21 9.07 9.73 8.62 9.34 8.25
9.01 7.94 8.73 7.68 8.49 7.46 8.28 7.27 8.10 7.10
7.94 6.95 7.80 6.81 7.67 6.70 7.55 6.59 7.45 6.49
7.36 6.41 7.27 6.33 7.19 6.25 7.12 6.19 7.05 6.12
6.79 5.88 6.59 5.70 6.34 5.46 6.17 5.31 6.06 5.20
5.97 5.12 5.91 5.06 5.86 5.02 5.78 4.95 5.71 4.88
5.63 4.81 5.59 4.77 5.56 4.75 5.53 4.71 5.51 4.69
5.49 4.68 5.48 4.67 5.46 4.65
20.80 20.03 16.21 15.52 13.48 12.86 11.71 11.13 10.48 9.93
9.58 9.05 8.89 8.38 8.35 7.86 7.92 7.44 7.57 7.09
7.27 6.80 7.02 6.56 6.81 6.35 6.62 6.18 6.46 6.02
6.32 5.88 6.19 5.76 6.08 5.65 5.98 5.55 5.89 5.46
5.80 5.38 5.73 5.31 5.66 5.24 5.59 5.18 5.53 5.12
5.30 4.89 5.13 4.73 4.90 4.51 4.76 4.37 4.66 4.28
4.58 4.20 4.53 4.15 4.48 4.11 4.42 4.04 4.35 3.98
4.29 3.92 4.25 3.88 4.22 3.86 4.19 3.83 4.18 3.81
4.16 3.80 4.15 3.79 4.14 3.78
19.46 19.03 15.02 14.63 12.40 12.05 10.70 10.37 9.52 9.20
8.66 8.35 8.00 7.71 7.49 7.21 7.08 6.80 6.74 6.47
6.46 6.19 6.22 5.96 6.02 5.76 5.85 5.59 5.69 5.44
5.56 5.31 5.44 5.19 5.33 5.09 5.23 4.99 5.15 4.91
5.07 4.83 5.00 4.76 4.93 4.69 4.87 4.64 4.82 4.58
4.59 4.36 4.44 4.21 4.22 4.00 4.09 3.86 3.99 3.77
3.92 3.70 3.87 3.65 3.83 3.61 3.77 3.55 3.71 3.49
3.65 3.43 3.61 3.40 3.59 3.38 3.56 3.35 3.54 3.33
3.53 3.32 3.52 3.31 3.51 3.30
18.69 18.41 14.33 14.08 11.77 11.54 10.11 9.89 8.96 8.75
8.12 7.92 7.48 7.29 6.98 6.80 6.58 6.40 6.26 6.08
5.98 5.81 5.75 5.58 5.56 5.39 5.39 5.22 5.24 5.08
5.11 4.95 4.99 4.83 4.89 4.73 4.80 4.64 4.71 4.56
4.64 4.48 4.57 4.41 4.50 4.35 4.45 4.29 4.39 4.24
4.18 4.03 4.02 3.87 3.82 3.67 3.69 3.54 3.60 3.45
3.53 3.39 3.48 3.34 3.44 3.30 3.38 3.24 3.32 3.18
3.26 3.12 3.23 3.09 3.21 3.07 3.18 3.04 3.16 3.02
3.15 3.01 3.14 3.00 3.13 2.99
17.99 17.68 13.71 13.43 11.19 10.94 9.57 9.33 8.45 8.22
7.63 7.41 7.00 6.79 6.52 6.31 6.13 5.93 5.81 5.62
5.55 5.35 5.32 5.13 5.13 4.94 4.97 4.78 4.82 4.64
4.70 4.51 4.58 4.40 4.48 4.30 4.39 4.21 4.31 4.13
4.24 4.06 4.17 3.99 4.11 3.93 4.05 3.88 4.00 3.82
3.79 3.62 3.64 3.47 3.44 3.27 3.32 3.15 3.23 3.06
3.16 3.00 3.11 2.95 3.07 2.91 3.02 2.85 2.96 2.80
2.90 2.74 2.87 2.71 2.85 2.69 2.82 2.66 2.81 2.64
2.80 2.63 2.78 2.62 2.77 2.61
17.45 17.27 17.12 13.23 13.06 12.93 10.75 10.60 10.48 9.15 9.01 8.90 8.05 7.91 7.80
7.24 7.11 7.01 6.63 6.51 6.40 6.16 6.03 5.93 5.78 5.66 5.56 5.46 5.35 5.25
5.20 5.09 4.99 4.99 4.87 4.78 4.80 4.68 4.59 4.64 4.52 4.43 4.49 4.38 4.29
4.37 4.26 4.17 4.26 4.15 4.06 4.16 4.05 3.96 4.07 3.96 3.87 3.99 3.88 3.79
3.92 3.81 3.72 3.86 3.75 3.66 3.80 3.69 3.60 3.74 3.63 3.54 3.69 3.58 3.49
3.48 3.38 3.29 3.34 3.23 3.14 3.41 3.04 2.95 3.02 2.91 2.83 2.93 2.83 2.74
2.87 2.76 2.68 2.82 2.71 2.63 2.78 2.68 2.59 2.72 2.62 2.53 2.67 2.56 2.48
2.61 2.51 2.42 2.58 2.48 2.39 2.56 2.46 2.37 2.53 2.43 2.34 2.52 2.41 2.33
2.51 2.40 2.32 2.49 2.39 2.31 2.48 2.38 2.30
.
STATISTICAL TABLES 8
TABLE A.3 (continued)
F Distribution: Critical Values of F (0.1% significance level)
v1 25 30 35 40 50 60 75 100 150 200 v2
1 6.24e05 6.26e05 6.28e05 6.29e05 6.30e05 6.31e05 6.32e05 6.33e05 6.35e05 6.35e05 2 999.46 999.47 999.47 999.47 999.48 999.48 999.49 999.49 999.49 999.49 3 125.84 125.45 125.17 124.96 124.66 124.47 124.27 124.07 123.87 123.77 4 45.70 45.43 45.23 45.09 44.88 44.75 44.61 44.47 44.33 44.26 5 25.08 24.87 24.72 24.60 24.44 24.33 24.22 24.12 24.01 23.95
6 16.85 16.67 7 12.69 12.53 8 10.26 10.11 9 8.69 8.55 10 7.60 7.47
11 6.81 6.68 12 6.22 6.09 13 5.75 5.63 14 5.38 5.25 15 5.07 4.95
16 4.82 4.70 17 4.60 4.48 18 4.42 4.30 19 4.26 4.14 20 4.12 4.00
21 4.00 3.88 22 3.89 3.78 23 3.79 3.68 24 3.71 3.59 25 3.63 3.52
26 3.56 3.44 27 3.49 3.38 28 3.43 3.32 29 3.38 3.27 30 3.33 3.22
35 3.13 3.02 40 2.98 2.87 50 2.79 2.68 60 2.67 2.55 70 2.58 2.47
80 2.52 2.41 90 2.47 2.36 100 2.43 2.32 120 2.37 2.26 150 2.32 2.21
200 2.26 2.15 250 2.23 2.12 300 2.21 2.10 400 2.18 2.07 500 2.17 2.05
600 2.16 2.04 750 2.15 2.03 1000 2.14 2.02
16.54 16.44 16.31 12.41 12.33 12.20 10.00 9.92 9.80 8.46 8.37 8.26 7.37 7.30 7.19
6.59 6.52 6.42 6.00 5.93 5.83 5.54 5.47 5.37 5.17 5.10 5.00 4.86 4.80 4.70
4.61 4.54 4.45 4.40 4.33 4.24 4.22 4.15 4.06 4.06 3.99 3.90 3.92 3.86 3.77
3.80 3.74 3.64 3.70 3.63 3.54 3.60 3.53 3.44 3.51 3.45 3.36 3.43 3.37 3.28
3.36 3.30 3.21 3.30 3.23 3.14 3.24 3.18 3.09 3.18 3.12 3.03 3.13 3.07 2.98
2.93 2.87 2.78 2.79 2.73 2.64 2.60 2.53 2.44 2.47 2.41 2.32 2.39 2.32 2.23
2.32 2.26 2.16 2.27 2.21 2.11 2.24 2.17 2.08 2.18 2.11 2.02 2.12 2.06 1.96
2.07 2.00 1.90 2.03 1.97 1.87 2.01 1.94 1.85 1.98 1.92 1.82 1.97 1.90 1.80
1.96 1.89 1.79 1.95 1.88 1.78 1.94 1.87 1.77
16.21 16.12 16.03 12.12 12.04 11.95 9.73 9.65 9.57 8.19 8.11 8.04 7.12 7.05 6.98
6.35 6.28 6.21 5.76 5.70 5.63 5.30 5.24 5.17 4.94 4.87 4.81 4.64 4.57 4.51
4.39 4.32 4.26 4.18 4.11 4.05 4.00 3.93 3.87 3.84 3.78 3.71 3.70 3.64 3.58
3.58 3.52 3.46 3.48 3.41 3.35 3.38 3.32 3.25 3.29 3.23 3.17 3.22 3.15 3.09
3.15 3.08 3.02 3.08 3.02 2.96 3.02 2.96 2.90 2.97 2.91 2.84 2.92 2.86 2.79
2.72 2.66 2.59 2.57 2.51 2.44 2.38 2.31 2.25 2.25 2.19 2.12 2.16 2.10 2.03
2.10 2.03 1.96 2.05 1.98 1.91 2.01 1.94 1.87 1.95 1.88 1.81 1.89 1.82 1.74
1.83 1.76 1.68 1.80 1.72 1.65 1.78 1.70 1.62 1.75 1.67 1.59 1.73 1.65 1.57
1.72 1.64 1.56 1.71 1.63 1.55 1.69 1.62 1.53
15.93 15.89 11.87 11.82 9.49 9.45 7.96 7.93 6.91 6.87
6.14 6.10 5.56 5.52 5.10 5.07 4.74 4.71 4.44 4.41
4.19 4.16 3.98 3.95 3.80 3.77 3.65 3.61 3.51 3.48
3.39 3.36 3.28 3.25 3.19 3.16 3.10 3.07 3.03 2.99
2.95 2.92 2.89 2.86 2.83 2.80 2.78 2.74 2.73 2.69
2.52 2.49 2.38 2.34 2.18 2.14 2.05 2.01 1.95 1.92
1.89 1.85 1.83 1.79 1.79 1.75 1.73 1.68 1.66 1.62
1.60 1.55 1.56 1.51 1.53 1.48 1.50 1.45 1.48 1.43
1.46 1.41 1.45 1.40 1.44 1.38
.
STATISTICAL TABLES 9
TABLE A.4
2 (Chi-Squared) Distribution: Critical Values of 2
Significance level
Degrees of 5% 1% 0.1% freedom
1 3.841 6.635 10.828 2 5.991 9.210 13.816 3 7.815 11.345 16.266 4 9.488 13.277 18.467 5 11.070 15.086 20.515 6 12.592 16.812 22.458 7 14.067 18.475 24.322 8 15.507 20.090 26.124 9 16.919 21.666 27.877 10 18.307 23.209 29.588
.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
ST104B ZB
BSc DEGREES AND GRADUATE DIPLOMAS IN ECONOMICS, MANAGEMENT, FINANCE AND THE SOCIAL SCIENCES, THE DIPLOMA IN ECONOMICS AND SOCIAL SCIENCES AND THE CERTIFICATE IN EDUCATION IN SOCIAL SCIENCES
Summer 2021 Online Assessment Instructions
ST104B Statistics 2
Tuesday, 18 May 2021: 06:00 – 10:00 (BST)
The assessment will be an open-book take-home online assessment within a 4-hour window. The requirements for this assessment remain the same as the closed-book exam, with an expected time/effort of 2 hours.
Candidates should answer FOUR of the following questions: Question 1 of Section A (40 Marks) and all questions from Section B (60 marks total). Candidates are strongly advised to divide their time accordingly.
You should complete this paper using pen and paper. Please use BLACK INK only.
Handwritten work then needs to be scanned, converted to PDF and then uploaded to the VLE as ONE individual file including the coversheet. Each scanned sheet should have your candidate number written clearly in the header. Please do not write your name anywhere on your submission.
You have until 10:00 (BST) on Tuesday, 18 May 2021 to upload your file into the VLE submission portal. However, you are advised not to leave your submission to the last minute.
Workings should be submitted for all questions requiring calculations. Any necessary assumptions introduced in answering a question are to be stated.
A list of formulae and extracts from statistical tables are provided after the final question of this paper.
You may use any calculator for any appropriate calculations, but you may not use any computer software to obtain solutions. Credit will only be given if all workings are shown.
© University of London 2021
Page 1 of 31 UL21/0187
If you think there is any information missing or any error in any question, then you should indicate this but proceed to answer the question stating any assumptions you have made.
The assessment has been designed with a duration of 4 hours to provide a more flexible window in which to complete the assessment and to appropriately test the course learning outcomes. As an open-book exam, the expected amount of effort required to complete all questions and upload your answers during this window is no more than 2 hours. Organise your time well.
You are assured that there will be no benefit in you going beyond the expected 2 hours of effort. Your assessment has been carefully designed to help you show what you have learned in the hours allocated.
This is an open book assessment and as such you may have access to additional materials including but not limited to subject guides and any recommended reading. But the work you submit is expected to be 100% your own. Therefore, unless instructed otherwise, you must not collaborate or confer with anyone during the assessment. The University of London will carry out checks to ensure the academic integrity of your work. Many students that break the University of London’s assessment regulations did not intend to cheat but did not properly understand the University of London’s regulations on referencing and plagiarism. The University of London considers all forms of plagiarism, whether deliberate or otherwise, a very serious matter and can apply severe penalties that might impact on your award.
The University of London 2020-21 Procedure for the consideration of Allegations of Assessment Offences is available online at:
Assessment Offence Procedures - University of London
Page 2 of 31 UL21/0187
SECTION A
Answer all five parts of question 1 (40 marks).
1. (a) For each one of the statements below say whether the statement is true or false, explaining your answer.
i. For two independent events A and B with P(A) > 0 and P(B) > 0, then:
P(A[B) < P(AjB)+ P(BjA):
ii. For two mutually exclusive events A and B such that P(A) > 0 and P(B) > 0, then:
P(A[B) > P(AjB)+ P(BjA):
iii. If B A and P(B) > 0, then the complementary events Ac and Bc cannot be independent events.
iv. If T1 and T2 are unbiased estimators of the parameter , then T1T2 is an unbiased estimator of 2.
v. The power of a test is the probability the alternative hypothesis is false. (10 marks)
(b) A random variable X can take the values 1, 0 and 1, where:
P(X = 1) = ; P(X = 0) = 1 3 and P(X = 1) = 2
for 0 < < 1=3. One observation of X is taken at random and we want to estimate the parameter . Consider the two estimators:
T1 = X and T2 = X2:
i. Calculate the bias of each estimator.
(4 marks)
ii. Which of the two estimators would you prefer and why?
(5 marks)
(c) The random variable X is normally distributed with a mean of 1 and a variance
of 4. Calculate:
i. P X < 2X < 4
and ii. P X < 2jXj < 4 :
(5 marks)
(d) Two fair six-sided dice are thrown. If their total score is strictly less than 4, they are thrown once more. What is the probability the total score is odd?
(5 marks)
UL21/0187 Page 3 of 31
(e) The random variable X has the probability density function given by:
( f(x) = kx 4
for x 1
otherwise:
i. Determine the value of k.
(3 marks)
ii. Compute E(X) and Var(X).
(5 marks)
iii. Derive the cumulative distribution function of X.
(3 marks)
SECTION B
Answer all three questions from this section.
2. (a) A chess player who plays with white pieces will win any particular game with probability 0.4, draw with probability 0.4, and lose with probability 0.2. The corresponding probabilities when she plays with black pieces are 0.2, 0.5 and 0.3, respectively. Before each game a fair coin is tossed to decide whether she will play with the black pieces or the white pieces. Hence the probability she plays with the white pieces is 0.5. Assume that the results of dierent games are independent.
i. Given that she won her last game, what is the probability she was playing with the white pieces?
(2 marks)
ii. Given that she did not lose either of her last two games, what is the probability she was playing with the same colour of pieces in both games?
(3 marks)
iii. Considering the next 10 games, what could you say about the distribution of the number of these games that she wins? Your answer should include a sketch plot of the probability distribution.
(5 marks)
(b) Let fX1;X2;:::;Xng be a random sample from the continuous uniform distribution dened over the interval [0;5].
i. Derive the maximum likelihood estimator of .
(4 marks)
ii. Given that the method of moments estimator of is 2X=5, check whether the method of moments estimator is consistent. Justify your answer.
(6 marks)
Hint: You may use any results on the formula sheet at the end of the question paper and you may state expressions for E(X) and Var(X) without proof.
UL21/0187 Page 4 of 31
3. (a) Let fX1;X2;:::;X25g be a random sample of size n = 25 from N(;2), where 2 = 3:24 is known. A researcher decides to test:
H0 : = 15 vs. H1 : < 15
using a 1% signicance level.
i. Calculate the power of the test when = 14:7.
(6 marks)
ii. Briey explain in two dierent ways how you could increase the power of this test.
(4 marks)
(b) A company produces copper wire to a particular specication of breaking strength using four dierent types of machines (A, B, C and D). One machine of each type is selected at random and ve copper wire samples are measured from each machine. The measurements were:
Machine type
A B C D
Sample wire 1 Sample wire 2 Sample wire 3 Sample wire 4 Sample wire 5 15.2 16.4 15.5 15.9 16.1 13.7 13.9 14.0 12.9 13.0 15.0 14.9 14.7 15.2 14.5 14.8 13.8 12.9 13.5 14.0
Sample mean 15.82 13.50 14.86 13.80
You are given that:
XX
xij nx = 21:0095
j=1 i=1
where the overall sample mean is x = 14:495.
i. Test the null hypothesis that the mean breaking strengths of the four types of machines are the same. Use a 1% signicance level.
(6 marks)
ii. Compute a 95% condence interval for the dierence of the mean breaking strengths between machine types A and B.
(4 marks)
UL21/0187 Page 5 of 31
4. Suppose X and Y are two independent random variables with the following probability distributions:
X = x 1 0 1
P(X = x) 0.20 0.60 0.20
and
Y = y 1 0 1
P(Y = y) 0.30 0.40 0.30
The random variables V and W are dened as:
V = X2 +Y2 and W = X +Y:
(a) Construct the table of the joint probability distribution of V and W.
(8 marks)
(b) Calculate the following quantities:
i. Var(W), given that E(W) = 0.
(2 marks)
ii. Cov(V;W).
(3 marks)
iii. E(V jW = 0).
(4 marks)
(c) Are V and W uncorrelated? Are V and W independent? Justify your answers. (3 marks)
[END OF PAPER]
UL21/0187 Page 6 of 31
Formulae for Statistics
Discrete distributions
Continuous distributions
UL21/0187 Page 7 of 31
Sample quantities
Sample variance
Sample covariance
Sample correlation
s2 = 1 X(xi x)2 = 1 Xx2 nx2 i=1 i=1
X X
(xi x)(yi y) = xiyi nxy
i=1 i=1
n
xiyi nxy
s n i=1 n x nx2 y ny2
i=1 i=1
Inference
Variance of sample mean
One-sample t statistic
Two-sample t statistic
UL21/0187
2 n
X S= n
s
n+m 2 X Y 0
1=n +1=m (n 1)SX +(m 1)SY
Page 8 of 31
STATISTICAL TABLES
Cumulative normal distribution
Critical values of the t distribution
Critical values of the F distribution
Critical values of the chi-squared distribution
New Cambridge Statistical Tables pages 17-29
© C. Dougherty 2001, 2002 (c.dougherty@lse.ac.uk). These tables have been computed to accompany the text C. Dougherty Introduction to Econometrics (second edition 2002, Oxford University Press, Oxford), They may be reproduced freely provided that this attribution is retained.
STATISTICAL TABLES 1
TABLE A.1
Cumulative Standardized Normal Distribution
A(z)
A(z) is the integral of the standardized normal distribution from to z (in other words, the area under the curve to the left of z). It gives the probability of a normal random variable not being more than z standard deviations above its mean. Values of z of particular importance:
z A(z)
1.645 0.9500 Lower limit of right 5% tail 1.960 0.9750 Lower limit of right 2.5% tail 2.326 0.9900 Lower limit of right 1% tail 2.576 0.9950 Lower limit of right 0.5% tail 3.090 0.9990 Lower limit of right 0.1% tail 3.291 0.9995 Lower limit of right 0.05% tail
-4 -3 -2 -1 0 1 z 2 3 4
z 0.00 0.01
0.0 0.5000 0.5040 0.1 0.5398 0.5438 0.2 0.5793 0.5832 0.3 0.6179 0.6217 0.4 0.6554 0.6591 0.5 0.6915 0.6950 0.6 0.7257 0.7291 0.7 0.7580 0.7611 0.8 0.7881 0.7910 0.9 0.8159 0.8186 1.0 0.8413 0.8438 1.1 0.8643 0.8665 1.2 0.8849 0.8869 1.3 0.9032 0.9049 1.4 0.9192 0.9207 1.5 0.9332 0.9345 1.6 0.9452 0.9463 1.7 0.9554 0.9564 1.8 0.9641 0.9649 1.9 0.9713 0.9719 2.0 0.9772 0.9778 2.1 0.9821 0.9826 2.2 0.9861 0.9864 2.3 0.9893 0.9896 2.4 0.9918 0.9920 2.5 0.9938 0.9940 2.6 0.9953 0.9955 2.7 0.9965 0.9966 2.8 0.9974 0.9975 2.9 0.9981 0.9982 3.0 0.9987 0.9987 3.1 0.9990 0.9991 3.2 0.9993 0.9993 3.3 0.9995 0.9995 3.4 0.9997 0.9997 3.5 0.9998 0.9998 3.6 0.9998 0.9998
0.02 0.03
0.5080 0.5120 0.5478 0.5517 0.5871 0.5910 0.6255 0.6293 0.6628 0.6664 0.6985 0.7019 0.7324 0.7357 0.7642 0.7673 0.7939 0.7967 0.8212 0.8238 0.8461 0.8485 0.8686 0.8708 0.8888 0.8907 0.9066 0.9082 0.9222 0.9236 0.9357 0.9370 0.9474 0.9484 0.9573 0.9582 0.9656 0.9664 0.9726 0.9732 0.9783 0.9788 0.9830 0.9834 0.9868 0.9871 0.9898 0.9901 0.9922 0.9925 0.9941 0.9943 0.9956 0.9957 0.9967 0.9968 0.9976 0.9977 0.9982 0.9983 0.9987 0.9988 0.9991 0.9991 0.9994 0.9994 0.9995 0.9996 0.9997 0.9997 0.9998 0.9998 0.9999
0.04 0.05
0.5160 0.5199 0.5557 0.5596 0.5948 0.5987 0.6331 0.6368 0.6700 0.6736 0.7054 0.7088 0.7389 0.7422 0.7704 0.7734 0.7995 0.8023 0.8264 0.8289 0.8508 0.8531 0.8729 0.8749 0.8925 0.8944 0.9099 0.9115 0.9251 0.9265 0.9382 0.9394 0.9495 0.9505 0.9591 0.9599 0.9671 0.9678 0.9738 0.9744 0.9793 0.9798 0.9838 0.9842 0.9875 0.9878 0.9904 0.9906 0.9927 0.9929 0.9945 0.9946 0.9959 0.9960 0.9969 0.9970 0.9977 0.9978 0.9984 0.9984 0.9988 0.9989 0.9992 0.9992 0.9994 0.9994 0.9996 0.9996 0.9997 0.9997 0.9998 0.9998
0.06 0.07
0.5239 0.5279 0.5636 0.5675 0.6026 0.6064 0.6406 0.6443 0.6772 0.6808 0.7123 0.7157 0.7454 0.7486 0.7764 0.7794 0.8051 0.8078 0.8315 0.8340 0.8554 0.8577 0.8770 0.8790 0.8962 0.8980 0.9131 0.9147 0.9279 0.9292 0.9406 0.9418 0.9515 0.9525 0.9608 0.9616 0.9686 0.9693 0.9750 0.9756 0.9803 0.9808 0.9846 0.9850 0.9881 0.9884 0.9909 0.9911 0.9931 0.9932 0.9948 0.9949 0.9961 0.9962 0.9971 0.9972 0.9979 0.9979 0.9985 0.9985 0.9989 0.9989 0.9992 0.9992 0.9994 0.9995 0.9996 0.9996 0.9997 0.9997 0.9998 0.9998
0.08 0.09
0.5319 0.5359 0.5714 0.5753 0.6103 0.6141 0.6480 0.6517 0.6844 0.6879 0.7190 0.7224 0.7517 0.7549 0.7823 0.7852 0.8106 0.8133 0.8365 0.8389 0.8599 0.8621 0.8810 0.8830 0.8997 0.9015 0.9162 0.9177 0.9306 0.9319 0.9429 0.9441 0.9535 0.9545 0.9625 0.9633 0.9699 0.9706 0.9761 0.9767 0.9812 0.9817 0.9854 0.9857 0.9887 0.9890 0.9913 0.9916 0.9934 0.9936 0.9951 0.9952 0.9963 0.9964 0.9973 0.9974 0.9980 0.9981 0.9986 0.9986 0.9990 0.9990 0.9993 0.9993 0.9995 0.9995 0.9996 0.9997 0.9997 0.9998 0.9998 0.9998
.
STATISTICAL TABLES 2
TABLE A.2
t Distribution: Critical Values of t
Significance level
Degrees of freedom
1 2 3 4 5
6 7 8 9 10
11 12 13 14 15
16 17 18 19 20
21 22 23 24 25
26 27 28 29 30
32 34 36 38 40
42 44 46 48 50
60 70 80 90 100
120 150 200 300 400
500 600
Two-tailed test: One-tailed test:
10% 5% 2% 5% 2.5% 1%
6.314 12.706 31.821 2.920 4.303 6.965 2.353 3.182 4.541 2.132 2.776 3.747 2.015 2.571 3.365
1.943 2.447 3.143 1.894 2.365 2.998 1.860 2.306 2.896 1.833 2.262 2.821 1.812 2.228 2.764
1.796 2.201 2.718 1.782 2.179 2.681 1.771 2.160 2.650 1.761 2.145 2.624 1.753 2.131 2.602
1.746 2.120 2.583 1.740 2.110 2.567 1.734 2.101 2.552 1.729 2.093 2.539 1.725 2.086 2.528
1.721 2.080 2.518 1.717 2.074 2.508 1.714 2.069 2.500 1.711 2.064 2.492 1.708 2.060 2.485
1.706 2.056 2.479 1.703 2.052 2.473 1.701 2.048 2.467 1.699 2.045 2.462 1.697 2.042 2.457
1.694 2.037 2.449 1.691 2.032 2.441 1.688 2.028 2.434 1.686 2.024 2.429 1.684 2.021 2.423
1.682 2.018 2.418 1.680 2.015 2.414 1.679 2.013 2.410 1.677 2.011 2.407 1.676 2.009 2.403
1.671 2.000 2.390 1.667 1.994 2.381 1.664 1.990 2.374 1.662 1.987 2.368 1.660 1.984 2.364
1.658 1.980 2.358 1.655 1.976 2.351 1.653 1.972 2.345 1.650 1.968 2.339 1.649 1.966 2.336
1.648 1.965 2.334 1.647 1.964 2.333
1.645 1.960 2.326
1% 0.2% 0.5% 0.1%
63.657 318.309 9.925 22.327 5.841 10.215 4.604 7.173 4.032 5.893
3.707 5.208 3.499 4.785 3.355 4.501 3.250 4.297 3.169 4.144
3.106 4.025 3.055 3.930 3.012 3.852 2.977 3.787 2.947 3.733
2.921 3.686 2.898 3.646 2.878 3.610 2.861 3.579 2.845 3.552
2.831 3.527 2.819 3.505 2.807 3.485 2.797 3.467 2.787 3.450
2.779 3.435 2.771 3.421 2.763 3.408 2.756 3.396 2.750 3.385
2.738 3.365 2.728 3.348 2.719 3.333 2.712 3.319 2.704 3.307
2.698 3.296 2.692 3.286 2.687 3.277 2.682 3.269 2.678 3.261
2.660 3.232 2.648 3.211 2.639 3.195 2.632 3.183 2.626 3.174
2.617 3.160 2.609 3.145 2.601 3.131 2.592 3.118 2.588 3.111
2.586 3.107 2.584 3.104
2.576 3.090
0.1% 0.05%
636.619 31.599 12.924 8.610 6.869
5.959 5.408 5.041 4.781 4.587
4.437 4.318 4.221 4.140 4.073
4.015 3.965 3.922 3.883 3.850
3.819 3.792 3.768 3.745 3.725
3.707 3.690 3.674 3.659 3.646
3.622 3.601 3.582 3.566 3.551
3.538 3.526 3.515 3.505 3.496
3.460 3.435 3.416 3.402 3.390
3.373 3.357 3.340 3.323 3.315
3.310 3.307
3.291
.
STATISTICAL TABLES 3
TABLE A.3
F Distribution: Critical Values of F (5% significance level)
v1 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 v2
1 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88 243.91 245.36 246.46 247.32 248.01 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40 19.41 19.42 19.43 19.44 19.45 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 8.74 8.71 8.69 8.67 8.66 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 5.91 5.87 5.84 5.82 5.80 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 4.68 4.64 4.60 4.58 4.56
6 5.99 5.14 7 5.59 4.74 8 5.32 4.46 9 5.12 4.26 10 4.96 4.10
11 4.84 3.98 12 4.75 3.89 13 4.67 3.81 14 4.60 3.74 15 4.54 3.68
16 4.49 3.63 17 4.45 3.59 18 4.41 3.55 19 4.38 3.52 20 4.35 3.49
21 4.32 3.47 22 4.30 3.44 23 4.28 3.42 24 4.26 3.40 25 4.24 3.39
26 4.22 3.37 27 4.21 3.35 28 4.20 3.34 29 4.18 3.33 30 4.17 3.32
35 4.12 3.27 40 4.08 3.23 50 4.03 3.18 60 4.00 3.15 70 3.98 3.13
80 3.96 3.11 90 3.95 3.10 100 3.94 3.09 120 3.92 3.07 150 3.90 3.06
200 3.89 3.04 250 3.88 3.03 300 3.87 3.03 400 3.86 3.02 500 3.86 3.01
600 3.86 3.01 750 3.85 3.01 1000 3.85 3.00
4.76 4.53 4.35 4.12 4.07 3.84 3.86 3.63 3.71 3.48
3.59 3.36 3.49 3.26 3.41 3.18 3.34 3.11 3.29 3.06
3.24 3.01 3.20 2.96 3.16 2.93 3.13 2.90 3.10 2.87
3.07 2.84 3.05 2.82 3.03 2.80 3.01 2.78 2.99 2.76
2.98 2.74 2.96 2.73 2.95 2.71 2.93 2.70 2.92 2.69
2.87 2.64 2.84 2.61 2.79 2.56 2.76 2.53 2.74 2.50
2.72 2.49 2.71 2.47 2.70 2.46 2.68 2.45 2.66 2.43
2.65 2.42 2.64 2.41 2.63 2.40 2.63 2.39 2.62 2.39
2.62 2.39 2.62 2.38 2.61 2.38
4.39 4.28 3.97 3.87 3.69 3.58 3.48 3.37 3.33 3.22
3.20 3.09 3.11 3.00 3.03 2.92 2.96 2.85 2.90 2.79
2.85 2.74 2.81 2.70 2.77 2.66 2.74 2.63 2.71 2.60
2.68 2.57 2.66 2.55 2.64 2.53 2.62 2.51 2.60 2.49
2.59 2.47 2.57 2.46 2.56 2.45 2.55 2.43 2.53 2.42
2.49 2.37 2.45 2.34 2.40 2.29 2.37 2.25 2.35 2.23
2.33 2.21 2.32 2.20 2.31 2.19 2.29 2.18 2.27 2.16
2.26 2.14 2.25 2.13 2.24 2.13 2.24 2.12 2.23 2.12
2.23 2.11 2.23 2.11 2.22 2.11
4.21 4.15 3.79 3.73 3.50 3.44 3.29 3.23 3.14 3.07
3.01 2.95 2.91 2.85 2.83 2.77 2.76 2.70 2.71 2.64
2.66 2.59 2.61 2.55 2.58 2.51 2.54 2.48 2.51 2.45
2.49 2.42 2.46 2.40 2.44 2.37 2.42 2.36 2.40 2.34
2.39 2.32 2.37 2.31 2.36 2.29 2.35 2.28 2.33 2.27
2.29 2.22 2.25 2.18 2.20 2.13 2.17 2.10 2.14 2.07
2.13 2.06 2.11 2.04 2.10 2.03 2.09 2.02 2.07 2.00
2.06 1.98 2.05 1.98 2.04 1.97 2.03 1.96 2.03 1.96
2.02 1.95 2.02 1.95 2.02 1.95
4.10 4.06 3.68 3.64 3.39 3.35 3.18 3.14 3.02 2.98
2.90 2.85 2.80 2.75 2.71 2.67 2.65 2.60 2.59 2.54
2.54 2.49 2.49 2.45 2.46 2.41 2.42 2.38 2.39 2.35
2.37 2.32 2.34 2.30 2.32 2.27 2.30 2.25 2.28 2.24
2.27 2.22 2.25 2.20 2.24 2.19 2.22 2.18 2.21 2.16
2.16 2.11 2.12 2.08 2.07 2.03 2.04 1.99 2.02 1.97
2.00 1.95 1.99 1.94 1.97 1.93 1.96 1.91 1.94 1.89
1.93 1.88 1.92 1.87 1.91 1.86 1.90 1.85 1.90 1.85
1.90 1.85 1.89 1.84 1.89 1.84
4.00 3.96 3.57 3.53 3.28 3.24 3.07 3.03 2.91 2.86
2.79 2.74 2.69 2.64 2.60 2.55 2.53 2.48 2.48 2.42
2.42 2.37 2.38 2.33 2.34 2.29 2.31 2.26 2.28 2.22
2.25 2.20 2.23 2.17 2.20 2.15 2.18 2.13 2.16 2.11
2.15 2.09 2.13 2.08 2.12 2.06 2.10 2.05 2.09 2.04
2.04 1.99 2.00 1.95 1.95 1.89 1.92 1.86 1.89 1.84
1.88 1.82 1.86 1.80 1.85 1.79 1.83 1.78 1.82 1.76
1.80 1.74 1.79 1.73 1.78 1.72 1.78 1.72 1.77 1.71
1.77 1.71 1.77 1.70 1.76 1.70
3.92 3.90 3.87 3.49 3.47 3.44 3.20 3.17 3.15 2.99 2.96 2.94 2.83 2.80 2.77
2.70 2.67 2.65 2.60 2.57 2.54 2.51 2.48 2.46 2.44 2.41 2.39 2.38 2.35 2.33
2.33 2.30 2.28 2.29 2.26 2.23 2.25 2.22 2.19 2.21 2.18 2.16 2.18 2.15 2.12
2.16 2.12 2.10 2.13 2.10 2.07 2.11 2.08 2.05 2.09 2.05 2.03 2.07 2.04 2.01
2.05 2.02 1.99 2.04 2.00 1.97 2.02 1.99 1.96 2.01 1.97 1.94 1.99 1.96 1.93
1.94 1.91 1.88 1.90 1.87 1.84 1.85 1.81 1.78 1.82 1.78 1.75 1.79 1.75 1.72
1.77 1.73 1.70 1.76 1.72 1.69 1.75 1.71 1.68 1.73 1.69 1.66 1.71 1.67 1.64
1.69 1.66 1.62 1.68 1.65 1.61 1.68 1.64 1.61 1.67 1.63 1.60 1.66 1.62 1.59
1.66 1.62 1.59 1.66 1.62 1.58 1.65 1.61 1.58
.
STATISTICAL TABLES 4
TABLE A.3 (continued)
F Distribution: Critical Values of F (5% significance level)
v1 25 30 35 40 50 60 75 100 150 200 v2
1 249.26 250.10 250.69 251.14 251.77 252.20 252.62 253.04 253.46 253.68 2 19.46 19.46 19.47 19.47 19.48 19.48 19.48 19.49 19.49 19.49 3 8.63 8.62 8.60 8.59 8.58 8.57 8.56 8.55 8.54 8.54 4 5.77 5.75 5.73 5.72 5.70 5.69 5.68 5.66 5.65 5.65 5 4.52 4.50 4.48 4.46 4.44 4.43 4.42 4.41 4.39 4.39
6 3.83 3.81 7 3.40 3.38 8 3.11 3.08 9 2.89 2.86 10 2.73 2.70
11 2.60 2.57 12 2.50 2.47 13 2.41 2.38 14 2.34 2.31 15 2.28 2.25
16 2.23 2.19 17 2.18 2.15 18 2.14 2.11 19 2.11 2.07 20 2.07 2.04
21 2.05 2.01 22 2.02 1.98 23 2.00 1.96 24 1.97 1.94 25 1.96 1.92
26 1.94 1.90 27 1.92 1.88 28 1.91 1.87 29 1.89 1.85 30 1.88 1.84
35 1.82 1.79 40 1.78 1.74 50 1.73 1.69 60 1.69 1.65 70 1.66 1.62
80 1.64 1.60 90 1.63 1.59 100 1.62 1.57 120 1.60 1.55 150 1.58 1.54
200 1.56 1.52 250 1.55 1.50 300 1.54 1.50 400 1.53 1.49 500 1.53 1.48
600 1.52 1.48 750 1.52 1.47 1000 1.52 1.47
3.79 3.77 3.75 3.36 3.34 3.32 3.06 3.04 3.02 2.84 2.83 2.80 2.68 2.66 2.64
2.55 2.53 2.51 2.44 2.43 2.40 2.36 2.34 2.31 2.28 2.27 2.24 2.22 2.20 2.18
2.17 2.15 2.12 2.12 2.10 2.08 2.08 2.06 2.04 2.05 2.03 2.00 2.01 1.99 1.97
1.98 1.96 1.94 1.96 1.94 1.91 1.93 1.91 1.88 1.91 1.89 1.86 1.89 1.87 1.84
1.87 1.85 1.82 1.86 1.84 1.81 1.84 1.82 1.79 1.83 1.81 1.77 1.81 1.79 1.76
1.76 1.74 1.70 1.72 1.69 1.66 1.66 1.63 1.60 1.62 1.59 1.56 1.59 1.57 1.53
1.57 1.54 1.51 1.55 1.53 1.49 1.54 1.52 1.48 1.52 1.50 1.46 1.50 1.48 1.44
1.48 1.46 1.41 1.47 1.44 1.40 1.46 1.43 1.39 1.45 1.42 1.38 1.45 1.42 1.38
1.44 1.41 1.37 1.44 1.41 1.37 1.43 1.41 1.36
3.74 3.73 3.71 3.30 3.29 3.27 3.01 2.99 2.97 2.79 2.77 2.76 2.62 2.60 2.59
2.49 2.47 2.46 2.38 2.37 2.35 2.30 2.28 2.26 2.22 2.21 2.19 2.16 2.14 2.12
2.11 2.09 2.07 2.06 2.04 2.02 2.02 2.00 1.98 1.98 1.96 1.94 1.95 1.93 1.91
1.92 1.90 1.88 1.89 1.87 1.85 1.86 1.84 1.82 1.84 1.82 1.80 1.82 1.80 1.78
1.80 1.78 1.76 1.79 1.76 1.74 1.77 1.75 1.73 1.75 1.73 1.71 1.74 1.72 1.70
1.68 1.66 1.63 1.64 1.61 1.59 1.58 1.55 1.52 1.53 1.51 1.48 1.50 1.48 1.45
1.48 1.45 1.43 1.46 1.44 1.41 1.45 1.42 1.39 1.43 1.40 1.37 1.41 1.38 1.34
1.39 1.35 1.32 1.37 1.34 1.31 1.36 1.33 1.30 1.35 1.32 1.28 1.35 1.31 1.28
1.34 1.31 1.27 1.34 1.30 1.26 1.33 1.30 1.26
3.70 3.69 3.26 3.25 2.96 2.95 2.74 2.73 2.57 2.56
2.44 2.43 2.33 2.32 2.24 2.23 2.17 2.16 2.10 2.10
2.05 2.04 2.00 1.99 1.96 1.95 1.92 1.91 1.89 1.88
1.86 1.84 1.83 1.82 1.80 1.79 1.78 1.77 1.76 1.75
1.74 1.73 1.72 1.71 1.70 1.69 1.69 1.67 1.67 1.66
1.61 1.60 1.56 1.55 1.50 1.48 1.45 1.44 1.42 1.40
1.39 1.38 1.38 1.36 1.36 1.34 1.33 1.32 1.31 1.29
1.28 1.26 1.27 1.25 1.26 1.23 1.24 1.22 1.23 1.21
1.23 1.20 1.22 1.20 1.22 1.19
.
STATISTICAL TABLES 5
TABLE A.3 (continued)
F Distribution: Critical Values of F (1% significance level)
v1 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 v2
1 4052.18 4999.50 5403.35 5624.58 5763.65 5858.99 5928.36 5981.07 6022.47 6055.85 6106.32 6142.67 6170.10 6191.53 6208.73 2 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39 99.40 99.42 99.43 99.44 99.44 99.45 3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35 27.23 27.05 26.92 26.83 26.75 26.69 4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66 14.55 14.37 14.25 14.15 14.08 14.02 5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16 10.05 9.89 9.77 9.68 9.61 9.55
6 13.75 10.92 7 12.25 9.55 8 11.26 8.65 9 10.56 8.02 10 10.04 7.56
11 9.65 7.21 12 9.33 6.93 13 9.07 6.70 14 8.86 6.51 15 8.68 6.36
16 8.53 6.23 17 8.40 6.11 18 8.29 6.01 19 8.18 5.93 20 8.10 5.85
21 8.02 5.78 22 7.95 5.72 23 7.88 5.66 24 7.82 5.61 25 7.77 5.57
26 7.72 5.53 27 7.68 5.49 28 7.64 5.45 29 7.60 5.42 30 7.56 5.39
35 7.42 5.27 40 7.31 5.18 50 7.17 5.06 60 7.08 4.98 70 7.01 4.92
80 6.96 4.88 90 6.93 4.85 100 6.90 4.82 120 6.85 4.79 150 6.81 4.75
200 6.76 4.71 250 6.74 4.69 300 6.72 4.68 400 6.70 4.66 500 6.69 4.65
600 6.68 4.64 750 6.67 4.63 1000 6.66 4.63
9.78 9.15 8.45 7.85 7.59 7.01 6.99 6.42 6.55 5.99
6.22 5.67 5.95 5.41 5.74 5.21 5.56 5.04 5.42 4.89
5.29 4.77 5.18 4.67 5.09 4.58 5.01 4.50 4.94 4.43
4.87 4.37 4.82 4.31 4.76 4.26 4.72 4.22 4.68 4.18
4.64 4.14 4.60 4.11 4.57 4.07 4.54 4.04 4.51 4.02
4.40 3.91 4.31 3.83 4.20 3.72 4.13 3.65 4.07 3.60
4.04 3.56 4.01 3.53 3.98 3.51 3.95 3.48 3.91 3.45
3.88 3.41 3.86 3.40 3.85 3.38 3.83 3.37 3.82 3.36
3.81 3.35 3.81 3.34 3.80 3.34
8.75 8.47 7.46 7.19 6.63 6.37 6.06 5.80 5.64 5.39
5.32 5.07 5.06 4.82 4.86 4.62 4.69 4.46 4.56 4.32
4.44 4.20 4.34 4.10 4.25 4.01 4.17 3.94 4.10 3.87
4.04 3.81 3.99 3.76 3.94 3.71 3.90 3.67 3.85 3.63
3.82 3.59 3.78 3.56 3.75 3.53 3.73 3.50 3.70 3.47
3.59 3.37 3.51 3.29 3.41 3.19 3.34 3.12 3.29 3.07
3.26 3.04 3.23 3.01 3.21 2.99 3.17 2.96 3.14 2.92
3.11 2.89 3.09 2.87 3.08 2.86 3.06 2.85 3.05 2.84
3.05 2.83 3.04 2.83 3.04 2.82
8.26 8.10 6.99 6.84 6.18 6.03 5.61 5.47 5.20 5.06
4.89 4.74 4.64 4.50 4.44 4.30 4.28 4.14 4.14 4.00
4.03 3.89 3.93 3.79 3.84 3.71 3.77 3.63 3.70 3.56
3.64 3.51 3.59 3.45 3.54 3.41 3.50 3.36 3.46 3.32
3.42 3.29 3.39 3.26 3.36 3.23 3.33 3.20 3.30 3.17
3.20 3.07 3.12 2.99 3.02 2.89 2.95 2.82 2.91 2.78
2.87 2.74 2.84 2.72 2.82 2.69 2.79 2.66 2.76 2.63
2.73 2.60 2.71 2.58 2.70 2.57 2.68 2.56 2.68 2.55
2.67 2.54 2.66 2.53 2.66 2.53
7.98 7.87 6.72 6.62 5.91 5.81 5.35 5.26 4.94 4.85
4.63 4.54 4.39 4.30 4.19 4.10 4.03 3.94 3.89 3.80
3.78 3.69 3.68 3.59 3.60 3.51 3.52 3.43 3.46 3.37
3.40 3.31 3.35 3.26 3.30 3.21 3.26 3.17 3.22 3.13
3.18 3.09 3.15 3.06 3.12 3.03 3.09 3.00 3.07 2.98
2.96 2.88 2.89 2.80 2.78 2.70 2.72 2.63 2.67 2.59
2.64 2.55 2.61 2.52 2.59 2.50 2.56 2.47 2.53 2.44
2.50 2.41 2.48 2.39 2.47 2.38 2.45 2.37 2.44 2.36
2.44 2.35 2.43 2.34 2.43 2.34
7.72 7.60 6.47 6.36 5.67 5.56 5.11 5.01 4.71 4.60
4.40 4.29 4.16 4.05 3.96 3.86 3.80 3.70 3.67 3.56
3.55 3.45 3.46 3.35 3.37 3.27 3.30 3.19 3.23 3.13
3.17 3.07 3.12 3.02 3.07 2.97 3.03 2.93 2.99 2.89
2.96 2.86 2.93 2.82 2.90 2.79 2.87 2.77 2.84 2.74
2.74 2.64 2.66 2.56 2.56 2.46 2.50 2.39 2.45 2.35
2.42 2.31 2.39 2.29 2.37 2.27 2.34 2.23 2.31 2.20
2.27 2.17 2.26 2.15 2.24 2.14 2.23 2.13 2.22 2.12
2.21 2.11 2.21 2.11 2.20 2.10
7.52 7.45 7.40 6.28 6.21 6.16 5.48 5.41 5.36 4.92 4.86 4.81 4.52 4.46 4.41
4.21 4.15 4.10 3.97 3.91 3.86 3.78 3.72 3.66 3.62 3.56 3.51 3.49 3.42 3.37
3.37 3.31 3.26 3.27 3.21 3.16 3.19 3.13 3.08 3.12 3.05 3.00 3.05 2.99 2.94
2.99 2.93 2.88 2.94 2.88 2.83 2.89 2.83 2.78 2.85 2.79 2.74 2.81 2.75 2.70
2.78 2.72 2.66 2.75 2.68 2.63 2.72 2.65 2.60 2.69 2.63 2.57 2.66 2.60 2.55
2.56 2.50 2.44 2.48 2.42 2.37 2.38 2.32 2.27 2.31 2.25 2.20 2.27 2.20 2.15
2.23 2.17 2.12 2.21 2.14 2.09 2.19 2.12 2.07 2.15 2.09 2.03 2.12 2.06 2.00
2.09 2.03 1.97 2.07 2.01 1.95 2.06 1.99 1.94 2.05 1.98 1.92 2.04 1.97 1.92
2.03 1.96 1.91 2.02 1.96 1.90 2.02 1.95 1.90
.
STATISTICAL TABLES 6
TABLE A.3 (continued)
F Distribution: Critical Values of F (1% significance level)
v1 25 30 35 40 50 60 75 100 150 200 v2
1 6239.83 6260.65 6275.57 6286.78 6302.52 6313.03 6323.56 6334.11 6344.68 6349.97 2 99.46 99.47 99.47 99.47 99.48 99.48 99.49 99.49 99.49 99.49 3 26.58 26.50 26.45 26.41 26.35 26.32 26.28 26.24 26.20 26.18 4 13.91 13.84 13.79 13.75 13.69 13.65 13.61 13.58 13.54 13.52 5 9.45 9.38 9.33 9.29 9.24 9.20 9.17 9.13 9.09 9.08
6 7.30 7.23 7 6.06 5.99 8 5.26 5.20 9 4.71 4.65 10 4.31 4.25
11 4.01 3.94 12 3.76 3.70 13 3.57 3.51 14 3.41 3.35 15 3.28 3.21
16 3.16 3.10 17 3.07 3.00 18 2.98 2.92 19 2.91 2.84 20 2.84 2.78
21 2.79 2.72 22 2.73 2.67 23 2.69 2.62 24 2.64 2.58 25 2.60 2.54
26 2.57 2.50 27 2.54 2.47 28 2.51 2.44 29 2.48 2.41 30 2.45 2.39
35 2.35 2.28 40 2.27 2.20 50 2.17 2.10 60 2.10 2.03 70 2.05 1.98
80 2.01 1.94 90 1.99 1.92 100 1.97 1.89 120 1.93 1.86 150 1.90 1.83
200 1.87 1.79 250 1.85 1.77 300 1.84 1.76 400 1.82 1.75 500 1.81 1.74
600 1.80 1.73 750 1.80 1.72 1000 1.79 1.72
7.18 7.14 5.94 5.91 5.15 5.12 4.60 4.57 4.20 4.17
3.89 3.86 3.65 3.62 3.46 3.43 3.30 3.27 3.17 3.13
3.05 3.02 2.96 2.92 2.87 2.84 2.80 2.76 2.73 2.69
2.67 2.64 2.62 2.58 2.57 2.54 2.53 2.49 2.49 2.45
2.45 2.42 2.42 2.38 2.39 2.35 2.36 2.33 2.34 2.30
2.23 2.19 2.15 2.11 2.05 2.01 1.98 1.94 1.93 1.89
1.89 1.85 1.86 1.82 1.84 1.80 1.81 1.76 1.77 1.73
1.74 1.69 1.72 1.67 1.70 1.66 1.69 1.64 1.68 1.63
1.67 1.63 1.66 1.62 1.66 1.61
7.09 7.06 5.86 5.82 5.07 5.03 4.52 4.48 4.12 4.08
3.81 3.78 3.57 3.54 3.38 3.34 3.22 3.18 3.08 3.05
2.97 2.93 2.87 2.83 2.78 2.75 2.71 2.67 2.64 2.61
2.58 2.55 2.53 2.50 2.48 2.45 2.44 2.40 2.40 2.36
2.36 2.33 2.33 2.29 2.30 2.26 2.27 2.23 2.25 2.21
2.14 2.10 2.06 2.02 1.95 1.91 1.88 1.84 1.83 1.78
1.79 1.75 1.76 1.72 1.74 1.69 1.70 1.66 1.66 1.62
1.63 1.58 1.61 1.56 1.59 1.55 1.58 1.53 1.57 1.52
1.56 1.51 1.55 1.50 1.54 1.50
7.02 6.99 5.79 5.75 5.00 4.96 4.45 4.41 4.05 4.01
3.74 3.71 3.50 3.47 3.31 3.27 3.15 3.11 3.01 2.98
2.90 2.86 2.80 2.76 2.71 2.68 2.64 2.60 2.57 2.54
2.51 2.48 2.46 2.42 2.41 2.37 2.37 2.33 2.33 2.29
2.29 2.25 2.26 2.22 2.23 2.19 2.20 2.16 2.17 2.13
2.06 2.02 1.98 1.94 1.87 1.82 1.79 1.75 1.74 1.70
1.70 1.65 1.67 1.62 1.65 1.60 1.61 1.56 1.57 1.52
1.53 1.48 1.51 1.46 1.50 1.44 1.48 1.42 1.47 1.41
1.46 1.40 1.45 1.39 1.44 1.38
6.95 6.93 5.72 5.70 4.93 4.91 4.38 4.36 3.98 3.96
3.67 3.66 3.43 3.41 3.24 3.22 3.08 3.06 2.94 2.92
2.83 2.81 2.73 2.71 2.64 2.62 2.57 2.55 2.50 2.48
2.44 2.42 2.38 2.36 2.34 2.32 2.29 2.27 2.25 2.23
2.21 2.19 2.18 2.16 2.15 2.13 2.12 2.10 2.09 2.07
1.98 1.96 1.90 1.87 1.78 1.76 1.70 1.68 1.65 1.62
1.61 1.58 1.57 1.55 1.55 1.52 1.51 1.48 1.46 1.43
1.42 1.39 1.40 1.36 1.38 1.35 1.36 1.32 1.34 1.31
1.34 1.30 1.33 1.29 1.32 1.28
.
STATISTICAL TABLES 7
TABLE A.3 (continued)
F Distribution: Critical Values of F (0.1% significance level)
v1 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 v2
1 4.05e05 5.00e05 5.40e05 5.62e05 5.76e05 5.86e05 5.93e05 5.98e05 6.02e05 6.06e05 6.11e05 6.14e05 6.17e05 6.19e05 6.21e05 2 998.50 999.00 999.17 999.25 999.30 999.33 999.36 999.37 999.39 999.40 999.42 999.43 999.44 999.44 999.45 3 167.03 148.50 141.11 137.10 134.58 132.85 131.58 130.62 129.86 129.25 128.32 127.64 127.14 126.74 126.42 4 74.14 61.25 56.18 53.44 51.71 50.53 49.66 49.00 48.47 48.05 47.41 46.95 46.60 46.32 46.10 5 47.18 37.12 33.20 31.09 29.75 28.83 28.16 27.65 27.24 26.92 26.42 26.06 25.78 25.57 25.39
6 35.51 27.00 7 29.25 21.69 8 25.41 18.49 9 22.86 16.39 10 21.04 14.91
11 19.69 13.81 12 18.64 12.97 13 17.82 12.31 14 17.14 11.78 15 16.59 11.34
16 16.12 10.97 17 15.72 10.66 18 15.38 10.39 19 15.08 10.16 20 14.82 9.95
21 14.59 9.77 22 14.38 9.61 23 14.20 9.47 24 14.03 9.34 25 13.88 9.22
26 13.74 9.12 27 13.61 9.02 28 13.50 8.93 29 13.39 8.85 30 13.29 8.77
35 12.90 8.47 40 12.61 8.25 50 12.22 7.96 60 11.97 7.77 70 11.80 7.64
80 11.67 7.54 90 11.57 7.47 100 11.50 7.41 120 11.38 7.32 150 11.27 7.24
200 11.15 7.15 250 11.09 7.10 300 11.04 7.07 400 10.99 7.03 500 10.96 7.00
600 10.94 6.99 750 10.91 6.97 1000 10.89 6.96
23.70 21.92 18.77 17.20 15.83 14.39 13.90 12.56 12.55 11.28
11.56 10.35 10.80 9.63 10.21 9.07 9.73 8.62 9.34 8.25
9.01 7.94 8.73 7.68 8.49 7.46 8.28 7.27 8.10 7.10
7.94 6.95 7.80 6.81 7.67 6.70 7.55 6.59 7.45 6.49
7.36 6.41 7.27 6.33 7.19 6.25 7.12 6.19 7.05 6.12
6.79 5.88 6.59 5.70 6.34 5.46 6.17 5.31 6.06 5.20
5.97 5.12 5.91 5.06 5.86 5.02 5.78 4.95 5.71 4.88
5.63 4.81 5.59 4.77 5.56 4.75 5.53 4.71 5.51 4.69
5.49 4.68 5.48 4.67 5.46 4.65
20.80 20.03 16.21 15.52 13.48 12.86 11.71 11.13 10.48 9.93
9.58 9.05 8.89 8.38 8.35 7.86 7.92 7.44 7.57 7.09
7.27 6.80 7.02 6.56 6.81 6.35 6.62 6.18 6.46 6.02
6.32 5.88 6.19 5.76 6.08 5.65 5.98 5.55 5.89 5.46
5.80 5.38 5.73 5.31 5.66 5.24 5.59 5.18 5.53 5.12
5.30 4.89 5.13 4.73 4.90 4.51 4.76 4.37 4.66 4.28
4.58 4.20 4.53 4.15 4.48 4.11 4.42 4.04 4.35 3.98
4.29 3.92 4.25 3.88 4.22 3.86 4.19 3.83 4.18 3.81
4.16 3.80 4.15 3.79 4.14 3.78
19.46 19.03 15.02 14.63 12.40 12.05 10.70 10.37 9.52 9.20
8.66 8.35 8.00 7.71 7.49 7.21 7.08 6.80 6.74 6.47
6.46 6.19 6.22 5.96 6.02 5.76 5.85 5.59 5.69 5.44
5.56 5.31 5.44 5.19 5.33 5.09 5.23 4.99 5.15 4.91
5.07 4.83 5.00 4.76 4.93 4.69 4.87 4.64 4.82 4.58
4.59 4.36 4.44 4.21 4.22 4.00 4.09 3.86 3.99 3.77
3.92 3.70 3.87 3.65 3.83 3.61 3.77 3.55 3.71 3.49
3.65 3.43 3.61 3.40 3.59 3.38 3.56 3.35 3.54 3.33
3.53 3.32 3.52 3.31 3.51 3.30
18.69 18.41 14.33 14.08 11.77 11.54 10.11 9.89 8.96 8.75
8.12 7.92 7.48 7.29 6.98 6.80 6.58 6.40 6.26 6.08
5.98 5.81 5.75 5.58 5.56 5.39 5.39 5.22 5.24 5.08
5.11 4.95 4.99 4.83 4.89 4.73 4.80 4.64 4.71 4.56
4.64 4.48 4.57 4.41 4.50 4.35 4.45 4.29 4.39 4.24
4.18 4.03 4.02 3.87 3.82 3.67 3.69 3.54 3.60 3.45
3.53 3.39 3.48 3.34 3.44 3.30 3.38 3.24 3.32 3.18
3.26 3.12 3.23 3.09 3.21 3.07 3.18 3.04 3.16 3.02
3.15 3.01 3.14 3.00 3.13 2.99
17.99 17.68 13.71 13.43 11.19 10.94 9.57 9.33 8.45 8.22
7.63 7.41 7.00 6.79 6.52 6.31 6.13 5.93 5.81 5.62
5.55 5.35 5.32 5.13 5.13 4.94 4.97 4.78 4.82 4.64
4.70 4.51 4.58 4.40 4.48 4.30 4.39 4.21 4.31 4.13
4.24 4.06 4.17 3.99 4.11 3.93 4.05 3.88 4.00 3.82
3.79 3.62 3.64 3.47 3.44 3.27 3.32 3.15 3.23 3.06
3.16 3.00 3.11 2.95 3.07 2.91 3.02 2.85 2.96 2.80
2.90 2.74 2.87 2.71 2.85 2.69 2.82 2.66 2.81 2.64
2.80 2.63 2.78 2.62 2.77 2.61
17.45 17.27 17.12 13.23 13.06 12.93 10.75 10.60 10.48 9.15 9.01 8.90 8.05 7.91 7.80
7.24 7.11 7.01 6.63 6.51 6.40 6.16 6.03 5.93 5.78 5.66 5.56 5.46 5.35 5.25
5.20 5.09 4.99 4.99 4.87 4.78 4.80 4.68 4.59 4.64 4.52 4.43 4.49 4.38 4.29
4.37 4.26 4.17 4.26 4.15 4.06 4.16 4.05 3.96 4.07 3.96 3.87 3.99 3.88 3.79
3.92 3.81 3.72 3.86 3.75 3.66 3.80 3.69 3.60 3.74 3.63 3.54 3.69 3.58 3.49
3.48 3.38 3.29 3.34 3.23 3.14 3.41 3.04 2.95 3.02 2.91 2.83 2.93 2.83 2.74
2.87 2.76 2.68 2.82 2.71 2.63 2.78 2.68 2.59 2.72 2.62 2.53 2.67 2.56 2.48
2.61 2.51 2.42 2.58 2.48 2.39 2.56 2.46 2.37 2.53 2.43 2.34 2.52 2.41 2.33
2.51 2.40 2.32 2.49 2.39 2.31 2.48 2.38 2.30
.
STATISTICAL TABLES 8
TABLE A.3 (continued)
F Distribution: Critical Values of F (0.1% significance level)
v1 25 30 35 40 50 60 75 100 150 200 v2
1 6.24e05 6.26e05 6.28e05 6.29e05 6.30e05 6.31e05 6.32e05 6.33e05 6.35e05 6.35e05 2 999.46 999.47 999.47 999.47 999.48 999.48 999.49 999.49 999.49 999.49 3 125.84 125.45 125.17 124.96 124.66 124.47 124.27 124.07 123.87 123.77 4 45.70 45.43 45.23 45.09 44.88 44.75 44.61 44.47 44.33 44.26 5 25.08 24.87 24.72 24.60 24.44 24.33 24.22 24.12 24.01 23.95
6 16.85 16.67 7 12.69 12.53 8 10.26 10.11 9 8.69 8.55 10 7.60 7.47
11 6.81 6.68 12 6.22 6.09 13 5.75 5.63 14 5.38 5.25 15 5.07 4.95
16 4.82 4.70 17 4.60 4.48 18 4.42 4.30 19 4.26 4.14 20 4.12 4.00
21 4.00 3.88 22 3.89 3.78 23 3.79 3.68 24 3.71 3.59 25 3.63 3.52
26 3.56 3.44 27 3.49 3.38 28 3.43 3.32 29 3.38 3.27 30 3.33 3.22
35 3.13 3.02 40 2.98 2.87 50 2.79 2.68 60 2.67 2.55 70 2.58 2.47
80 2.52 2.41 90 2.47 2.36 100 2.43 2.32 120 2.37 2.26 150 2.32 2.21
200 2.26 2.15 250 2.23 2.12 300 2.21 2.10 400 2.18 2.07 500 2.17 2.05
600 2.16 2.04 750 2.15 2.03 1000 2.14 2.02
16.54 16.44 16.31 12.41 12.33 12.20 10.00 9.92 9.80 8.46 8.37 8.26 7.37 7.30 7.19
6.59 6.52 6.42 6.00 5.93 5.83 5.54 5.47 5.37 5.17 5.10 5.00 4.86 4.80 4.70
4.61 4.54 4.45 4.40 4.33 4.24 4.22 4.15 4.06 4.06 3.99 3.90 3.92 3.86 3.77
3.80 3.74 3.64 3.70 3.63 3.54 3.60 3.53 3.44 3.51 3.45 3.36 3.43 3.37 3.28
3.36 3.30 3.21 3.30 3.23 3.14 3.24 3.18 3.09 3.18 3.12 3.03 3.13 3.07 2.98
2.93 2.87 2.78 2.79 2.73 2.64 2.60 2.53 2.44 2.47 2.41 2.32 2.39 2.32 2.23
2.32 2.26 2.16 2.27 2.21 2.11 2.24 2.17 2.08 2.18 2.11 2.02 2.12 2.06 1.96
2.07 2.00 1.90 2.03 1.97 1.87 2.01 1.94 1.85 1.98 1.92 1.82 1.97 1.90 1.80
1.96 1.89 1.79 1.95 1.88 1.78 1.94 1.87 1.77
16.21 16.12 16.03 12.12 12.04 11.95 9.73 9.65 9.57 8.19 8.11 8.04 7.12 7.05 6.98
6.35 6.28 6.21 5.76 5.70 5.63 5.30 5.24 5.17 4.94 4.87 4.81 4.64 4.57 4.51
4.39 4.32 4.26 4.18 4.11 4.05 4.00 3.93 3.87 3.84 3.78 3.71 3.70 3.64 3.58
3.58 3.52 3.46 3.48 3.41 3.35 3.38 3.32 3.25 3.29 3.23 3.17 3.22 3.15 3.09
3.15 3.08 3.02 3.08 3.02 2.96 3.02 2.96 2.90 2.97 2.91 2.84 2.92 2.86 2.79
2.72 2.66 2.59 2.57 2.51 2.44 2.38 2.31 2.25 2.25 2.19 2.12 2.16 2.10 2.03
2.10 2.03 1.96 2.05 1.98 1.91 2.01 1.94 1.87 1.95 1.88 1.81 1.89 1.82 1.74
1.83 1.76 1.68 1.80 1.72 1.65 1.78 1.70 1.62 1.75 1.67 1.59 1.73 1.65 1.57
1.72 1.64 1.56 1.71 1.63 1.55 1.69 1.62 1.53
15.93 15.89 11.87 11.82 9.49 9.45 7.96 7.93 6.91 6.87
6.14 6.10 5.56 5.52 5.10 5.07 4.74 4.71 4.44 4.41
4.19 4.16 3.98 3.95 3.80 3.77 3.65 3.61 3.51 3.48
3.39 3.36 3.28 3.25 3.19 3.16 3.10 3.07 3.03 2.99
2.95 2.92 2.89 2.86 2.83 2.80 2.78 2.74 2.73 2.69
2.52 2.49 2.38 2.34 2.18 2.14 2.05 2.01 1.95 1.92
1.89 1.85 1.83 1.79 1.79 1.75 1.73 1.68 1.66 1.62
1.60 1.55 1.56 1.51 1.53 1.48 1.50 1.45 1.48 1.43
1.46 1.41 1.45 1.40 1.44 1.38
.
STATISTICAL TABLES 9
TABLE A.4
2 (Chi-Squared) Distribution: Critical Values of 2
Significance level
Degrees of 5% 1% 0.1% freedom
1 3.841 6.635 10.828 2 5.991 9.210 13.816 3 7.815 11.345 16.266 4 9.488 13.277 18.467 5 11.070 15.086 20.515 6 12.592 16.812 22.458 7 14.067 18.475 24.322 8 15.507 20.090 26.124 9 16.919 21.666 27.877 10 18.307 23.209 29.588
.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.
Examiners’ commentaries 2021
Examiners’ commentaries 2021
ST104b Statistics 2
Important note
This commentary reects the examination and assessment arrangements for this course in the academic year 2020{21. The format and structure of the examination may change in future years, and any such changes will be publicised on the virtual learning environment (VLE).
Information about the subject guide and the Essential reading references
Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2019). You should always attempt to use the most recent edition of any Essential reading textbook, even if the commentary and/or online reading list and/or subject guide refer to an earlier edition. If dierent editions of Essential reading are listed, please check the VLE for reading supplements { if none are available, please use the contents list and index of the new edition to nd the relevant section.
General remarks
Learning outcomes
At the end of this half course, and having completed the Essential reading and activities, you should be able to:
apply and be competent users of standard statistical operators and be able to recall a variety of well-known distributions and their respective moments
explain the fundamentals of statistical inference and apply these principles to justify the use of an appropriate model and perform tests in a number of dierent settings
demonstrate understanding that statistical techniques are based on assumptions and the plausibility of such assumptions must be investigated when analysing real problems.
Format of the examination
The examination is two hours long and you must answer all four questions.
Question 1, for 40% of the marks, is a compulsory question with several parts. It is designed to test general knowledge and understanding of the whole syllabus. Here candidates are expected to give reasoned answers, with some explanation, avoiding one-word responses which will never be given any marks. More emphasis is given to understanding than to knowledge. Candidates should answer the rst part of this question (true or false statements) either by proving that the statement is true or false or, in the case of a false statement, providing a counterexample. It is not sucient to just provide the correct answer (no credit is given for this); an explanation is required. Furthermore, when trying to show that a certain statement is true, it is not sucient to show that the statement
1
ST104b Statistics 2
is true in a very specic case { this does not prove the statement is true, you must show the statement holds in all circumstances.
The other three questions are also compulsory and account for 20% of the total marks each. They are meant to test a greater depth of knowledge on parts of the syllabus. They are also longer and examine the ability to apply general knowledge and concepts to specic problems.
How to prepare for the examination
It is hard to overemphasise that memorising answers to past examination questions is not the best way to study for this paper. It is important for candidates to understand the material they write down, and to be able to develop it all from scratch as they write it. Often, there are several ways to obtain good marks for a question. If you cannot solve a certain section of a question you might still get full marks for subsequent sections as long as your reasoning is correct.
A very good mathematics background is extremely important. The course can be divided into a probability and distribution theory part and a statistics part. A mathematics background is important for both but especially for the probability and distribution theory part.
You should ensure you have an understanding of all parts of the half course. Specialising is a bad strategy as all questions are compulsory and there is no choice.
The paper is light on computations, as questions are answered with the use of a basic calculator only.
Key steps to improvement
You should understand all parts of the half course without exception. Remember that all questions in the paper are compulsory. Another reason why you should understand all parts of the course is that you should not expect to get very similar questions compared to previous years’ papers.
You should be able to write down or discuss denitions or models used in the syllabus.
It is important that you have the necessary mathematical skills. This means that you should understand your mathematics courses well too.
Routine computations are less important and you should spend more time understanding the concepts. Understanding concepts means being able to apply them, sometimes even in combined situations.
As stated earlier, the ST104b Statistics 2 examinations are not heavy in calculations.
Probability is the most important part of the half course as everything else depends on it. You must have a thorough understanding of all concepts. You should avoid making elementary mistakes which demonstrate a lack of understanding and are hence heavily penalised by the examiners. These include the following.
Calculating probabilities outside the range 0 to 1. Should that happen because of a calculation mistake, candidates must state that they think a mistake was made.
Calculating negative variances or sums of squares. Should that happen because of a calculation mistake, candidates must state that they think a mistake was made.
Finding a correlation outside the range 1 to 1.
Confusing independent and mutually exclusive events. You should spend a sucient amount of time studying the distinction between these.
You should get familiar with the logical thinking needed to answer the rst part of Question 1.
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Examiners’ commentaries 2021
When there is no evidence to reject a null hypothesis it does not mean that there is evidence to accept it.
You should have a clear understanding of conditional probabilities.
Examination revision strategy
Many candidates are disappointed to nd that their examination performance is poorer than they expected. This may be due to a number of reasons, but one particular failing is ‘question spotting’, that is, conning your examination preparation to a few questions and/or topics which have come up in past papers for the course. This can have serious consequences.
We recognise that candidates might not cover all topics in the syllabus in the same depth, but you need to be aware that examiners are free to set questions on any aspect of the syllabus. This means that you need to study enough of the syllabus to enable you to answer the required number of examination questions.
The syllabus can be found in the Course information sheet available on the VLE. You should read the syllabus carefully and ensure that you cover sucient material in preparation for the examination. Examiners will vary the topics and questions from year to year and may well set questions that have not appeared in past papers. Examination papers may legitimately include questions on any topic in the syllabus. So, although past papers can be helpful during your revision, you cannot assume that topics or specic questions that have come up in past examinations will occur again.
If you rely on a question-spotting strategy, it is likely you will nd yourself in diculties when you sit the examination. We strongly advise you not to adopt this strategy.
3
ST104b Statistics 2
Examiners’ commentaries 2021
ST104b Statistics 2
Important note
This commentary reects the examination and assessment arrangements for this course in the academic year 2020{21. The format and structure of the examination may change in future years, and any such changes will be publicised on the virtual learning environment (VLE).
Information about the subject guide and the Essential reading references
Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2019). You should always attempt to use the most recent edition of any Essential reading textbook, even if the commentary and/or online reading list and/or subject guide refer to an earlier edition. If dierent editions of Essential reading are listed, please check the VLE for reading supplements { if none are available, please use the contents list and index of the new edition to nd the relevant section.
Comments on specic questions { Zone A
Candidates should answer FOUR of the following questions: Question 1 of Section A (40 marks) and all questions from Section B (60 marks in total). Candidates are strongly advised to divide their time accordingly.
Section A
Answer all ve parts of question 1 (40 marks).
Question 1
(a) For each one of the statements below say whether the statement is true or false, explaining your answer.
i. For two independent events A and B with P(A) > 0 and P(B) > 0, then:
P(A [ B) < P(A) + P(B):
ii. Suppose A and B are two events such that P(A) = 0:2, P(B) = 0:4 and:
P(AjB) + P(B jA) = 0:7:
It holds that P(A \ B) = 14=150.
iii. If B A, then the complementary events Ac and Bc are independent events.
iv. If T1 is an unbiased estimator of the parameter , and T2 is an unbiased estimator of the parameter , then T1T2 is an unbiased estimator of .
v. The signicance level of a test is the probability the null hypothesis is false. (10 marks)
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Examiners’ commentaries 2021
Reading for this question
Chapter 2 of the subject guide covers the relevant aspects of set theory and probability theory required for parts i. to iii. In particular, statistical independence of two events is dened in Section 2.7, along with conditional probability. Section 7.4 provides the denition of an unbiased estimator for part iv. Finally, Section 9.5 denes the signicance level for part v.
Approaching the question
i. True. Since, due to independence:
P(A[B) = P(A)+ P(B) P(A\B)
= P(A)+ P(B) P(A)P(B)
< P(A)+ P(B)
and both P(A) and P(B) are strictly positive.
ii. True. Since P(A) = 0:2, P(B) = 0:4 and:
P(AjB)+ P(BjA) = 0:7
it follows that (5+5=2)P(A\B) = 0:7 implying that P(A\B) = 14=150.
iii. False. When B A it holds that:
P(Ac \Bc) = P(Ac)
which in general is not equal to P(Ac)P(Bc).
iv. False. E(T1T2) = E(T1)E(T2) when T1 and T2 are correlated.
v. False. The signicance level is the probability of rejecting a null hypothesis given that it is true.
(b) A random variable X can take the values 0, 1 and 2, with:
P(X = 0) = 1 3; P(X = 1) = and P(X = 2) =
where 0 < < 8=3. One observation of X is taken at random, and we want to estimate the parameter . Consider the two estimators:
T1 = 2X and T2 = 4X(X 1):
i. Calculate the bias of each estimator.
(4 marks)
ii. Which of the two estimators would you prefer and why?
(5 marks)
Reading for this question
Section 7.4 of the subject guide covers the relevant material on estimation criteria, i.e. bias, variance and mean squared error (MSE).
Approaching the question
i. We nd:
and:
E(T1) = 2 4 +4 8 =
E(T2) = 8 =
hence for both of these estimators of the bias is zero.
5
ST104b Statistics 2
ii. Since both are unbiased estimators we prefer the one with the smaller MSE, i.e. with the smaller variance in this case, or equivalently the smaller second moment. We have:
E(T2) = E(4X2) = 4 +16 = 3
and: E(T2) = E(16X2(X 1)2) = 64 = 8:
Since 3 < 8, we prefer T1 because the variance (and hence mean squared error) is smaller.
(c) The random variable X is normally distributed with a mean of 1 and a variance
of 4. Calculate:
i. P X > 3:4X > 2:2 and ii. P X > 3:4jXj > 2:2 :
(5 marks)
Reading for this question
Section 2.7.3 of the subject guide denes conditional probability, and Section 4.5.3 introduces the normal distribution, including the standardisation transformation.
Approaching the question
i. We standardise and nd:
P(X > 3:4) P(Z > 1:2) 0:1151 P(X > 2:2) P(Z > 0:6) 0:2743
ii. Similarly, we nd:
P X > 3:4jXj > 2:2 = P(fX < 2:2g[fX > 2:2g)
P(Z > 1:2)
P(Z < 1:6)+ P(Z > 0:6)
0:1151 0:0548+0:2743
= 0:3497:
(d) Two fair six-sided dice are thrown. If their total score is greater than or equal to 8 they are thrown once more. What is the probability the total score is even?
(5 marks)
Reading for this question
This probability problem makes use of classical probability, which can be found in Section 2.6 of the subject guide.
Approaching the question
The total score is even if the total score of the rst throw is 2, 4 or 6, or the total score of the rst throw is greater than or equal to 8 and the total score of the second throw is even. We hence nd (for example, by a quick table) that the required probability is given by:
36 + 3 + 5 + 5+4+3+2+1 1 = 11 = 0:4583
since the probability that the sum of the outcomes of two dice being even is equal to 1/2 as this corresponds to both dice getting an even outcome, or both dice getting an outcome that is odd.
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Examiners’ commentaries 2021
(e) The random variable X has the probability density function given by:
( f(x) = kx 5
for x 1
otherwise:
i. Determine the value of k.
(3 marks)
ii. Compute E(X) and Var(X).
(5 marks)
iii. Derive the cumulative distribution function of X.
(3 marks)
Reading for this question
Section 3.5 of the subject guide covers all required aspects in this question related to continuous random variables.
Approaching the question
i. The probability density function should integrate to 1 and hence:
so k = 4.
ii. We compute:
and:
hence:
Z 1 1
1 = k 1 x 5 dx = k 4x4 1 = 4
E(X) = 4Z 1 x 4 dx = 4 1
Z 1
E(X2) = 4 x 3 dx = 2 1
2 Var(X) = E(X2) (E(X))2 = 2 3 = 9:
iii. For the cumulative distribution function, note that for x < 1 we have P(X x) = 0. For x 1 we get: Z x
P(X x) = f(t)dt = 1 x 4: 1
Hence in full: (
F(x) = 1 x 4
for x < 1
for x 1:
Section B
Answer all three questions from this section.
Question 2
(a) The proportion of pregnant women among all women who take a particular pregnancy test is 0.75. Two-thirds of pregnant women are at an early stage of pregnancy. If a woman who takes the test is pregnant at an early stage, the test will be positive (indicating that she is pregnant) with probability 0.6. If she is pregnant at a later stage, the test will be positive with probability 0.8. If she is not pregnant, the test will be positive with probability 0.2.
7
ST104b Statistics 2
i. What is the probability the test will be positive?
(2 marks) ii. What is the probability the test will show a false result?
(3 marks)
iii. If 10 women independently each take a pregnancy test, what could you say about the distribution of the number of these tests that are positive? Your answer should include a sketch plot of the probability distribution.
(5 marks)
Reading for this question
This question makes use of the total probability formula, covered in Section 2.7.5 of the subject guide, and recognition of the applicability of the binomial distribution, covered in Section 4.4.3.
Approaching the question
i. A positive test comes from a woman who is not pregnant, at early stage of pregnancy, or at a later stage of pregnancy. Adding up the respective probabilities, we nd that the probability that the test is positive is given by:
0:250:2+0:50:6+0:250:8 = 0:55:
ii. A false result occurs when a pregnant woman tests negative, or when a woman who is not pregnant tests positive. We hence have that the probability of this occurring is given by:
0:250:2+0:50:4+0:250:2 = 0:3:
iii. If 10 women independently each take a pregnancy test then the number of positive tests follows a binomial distribution with n = 10 and = 0:55. Possible outcomes are f0;1;2;:::;10g and the respective probabilities are given by n x(1 )n x. The sketch plot should indicate a slight asymmetry.
(b) Let fX1;X2;:::;Xng be a random sample from the continuous uniform distribution dened over the interval [0;3].
i. Derive the maximum likelihood estimator of .
(4 marks)
ii. Given that the method of moments estimator of is 2X=3, check whether the method of moments estimator is consistent. Justify your answer.
(6 marks)
Hint: You may use any results on the formula sheet at the end of the question paper and you may state expressions for E(X) and Var(X) without proof.
Reading for this question
Method of moments and maximum likelihood estimation are covered in Sections 7.5 and 7.7 of the subject guide, respectively. Details of the continuous uniform distribution can be found in Section 4.5.1.
Approaching the question
i. Due to independence, the likelihood function is:
n
L() = (3) 1 = 3 n n
i=1
which is decreasing in , for X(n), and L() = 0 for < X(n), where X(n) is the maximum value in the random sample. Hence the maximum likelihood estimator of 3 is X(n), so by the invariance principle the maximum likelihood estimator of is:
= X(n) :
8
Examiners’ commentaries 2021
ii. Using the formula sheet (with a = 0 and b = 3), we have:
E(X) = 0+3 = 2 :
Since: E() = E3 X = 2 E(X) = 2 E(X) = 2 3 =
then is an unbiased estimator of . We have, from the formula sheet (with a = 0 and b = 3):
Var(X) = (3120)2 = 12 = 342 :
Therefore:
Var() = Var2X = 4 Var(X) = 4 Var(X) = 4 32 = n:
Since is an unbiased estimator:
2 MSE() = Var() = 3n ! 0
as n ! 1, hence is a consistent estimator of .
Question 3
(a) Let fX1;X2;:::;X16g be a random sample of size n = 16 from N(;2), where 2 = 2:56 is known. A researcher decides to test:
H0 : = 12 vs. H1 : > 12
using a 5% signicance level.
i. Calculate the power of the test when = 12:4.
(6 marks)
ii. Briey explain in two dierent ways how you could increase the power of this test.
(4 marks)
Reading for this question
Section 9.8 of the subject guide denes a power function, from which the power of a test can be calculated. Section 9.5.4 explains the hypothesis testing procedure for one-sided tests for normal means.
Approaching the question
i. The test statistic under H0 is:
2:56=16 N(0;1)
and the critical value at a 5% signicance level is z0:05 = 1:645. Hence we reject H0 if:
p2:56=16 > 1:645 equivalently x > 12+1:6450:4 = 12:658:
When = 12:4, the power of the test is:
P(X > 12:658j = 12:4) = P Z > 12:658: 12:4 = P(Z > 0:645) 0:2595:
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ST104b Statistics 2
ii. The power of the test could be increased either by increasing the sample size or by using a larger signicance level, such as 10%. Both ways would increase the probability of rejecting the null hypothesis. (Note although 2 aects the power, this is not under the control of the researcher.)
(b) A company produces copper wire to a particular specication of breaking strength using four dierent types of machines (A, B, C and D). One machine of each type is selected at random and ve copper wire samples are measured from each machine. The measurements were:
Machine type
A B C D
Sample wire 1 Sample wire 2 Sample wire 3 Sample wire 4 Sample wire 5 11.1 11.5 11.7 13.5 11.4 11.7 14.1 11.7 12.1 12.2 11.2 11.5 12.9 11.5 11.0 13.8 13.3 10.7 13.3 12.0
Sample mean 11.84 12.36 11.62 12.62
You are given that:
X Xx2 nx2 = 19:418
j=1 i=1
where the overall sample mean is x = 12:11.
i. Test the null hypothesis that the mean breaking strengths of the four types of machines are the same. Use a 5% signicance level.
(6 marks)
ii. Compute a 99% condence interval for the dierence of the mean breaking strengths between machine types B and C.
(4 marks)
Reading for this question
Section 10.5 of the subject guide covers all aspects concerning one-way analysis of variance necessary to perform the required F test and to construct a condence interval for the dierence of means.
Approaching the question
i. We test:
vs.
H0 : There is no dierence in mean breaking strengths
H1 : There is a dierence.
The sum of squares between machines is:
5((11:84 12:11)2 +(12:36 12:11)2 +(11:62 12:11)2
+(12:62 12:11)2)
= 3:178
and so the error sum of squares is:
19:418 3:178 = 16:24:
The test statistic value is: 3:178=(4 1) 16:24=(20 4)
The 5% critical value is F0:05;3;16 = 3:24. Since 1:0437 < 3:24 we do not reject H0. There is no evidence of a dierence in means.
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Examiners’ commentaries 2021
ii. A 99% condence interval for B C is computed using:
s
xB xC t0:005;16 b2 nB + nC :
Therefore, we have:
12:36 11:622:921s 16:24 5 + 5
which gives ( 1:121;2:601).
Question 4
Suppose X and Y are two independent random variables with the following probability distributions:
X = x 1 0 1
P(X = x) 0.30 0.40 0.30
and
Y = y 1 0 1
P(Y = y) 0.40 0.20 0.40
The random variables S and T are dened as:
S = X2 + Y 2 and T = X + Y:
(a) Construct the table of the joint probability distribution of S and T.
(8 marks)
(b) Calculate the following quantities:
i. Var(T), given that E(T) = 0.
(2 marks)
ii. Cov(S;T).
(3 marks)
iii. E(S jT = 0).
(4 marks)
(c) Are S and T uncorrelated? Are S and T independent? Justify your answers.
(3 marks)
Reading for this question
Sections 5.4 to 5.7 of the subject guide are relevant for this question, which span joint, marginal and conditional distributions, as well as covariance and correlation.
Approaching the question
(a) The joint probability distribution of S and T is:
S
0 1 2
2
1 T 0
1 2 0 0
0.08 0 0 0 0.22 0 0.22 0 0.12 0 0.24 0 0.12
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ST104b Statistics 2
(b) i. Since E(T) = 0, we have:
Var(T) = E(T2)
2
= t2p(t)
t= 2
= ( 2)2 0:12+( 1)2 0:22+02 0:32+12 0:22+22 0:12
= 1:4:
ii. We have that:
2 2
E(ST) = stp(s;t) = ( 40:12)+( 10:22)+1 0:22+4 0:12 = 0: s=0 t= 2
Since E(T) = 0, then:
Cov(S;T) = E(ST) E(S)E(T) = E(ST) = 0:
iii. We have:
2
E(S jT = 0) = s=0 spSjT=0(sjt = 0) = 0 0:32 +2 0:32 = 1:5:
(c) The random variables S and T are uncorrelated, since Cov(S;T) = 0. However:
P(T = 2) = 0:12 and P(S = 0) = 0:08 ) P(T = 2)P(S = 0) = 0:0096
but:
P(fT = 2g\fS = 0g) = 0 = P(T = 2)P(S = 0)
which is sucient to show that S and T are not independent.
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Examiners’ commentaries 2021
Examiners’ commentaries 2021
ST104b Statistics 2
Important note
This commentary reects the examination and assessment arrangements for this course in the academic year 2020{21. The format and structure of the examination may change in future years, and any such changes will be publicised on the virtual learning environment (VLE).
Information about the subject guide and the Essential reading references
Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2019). You should always attempt to use the most recent edition of any Essential reading textbook, even if the commentary and/or online reading list and/or subject guide refer to an earlier edition. If dierent editions of Essential reading are listed, please check the VLE for reading supplements { if none are available, please use the contents list and index of the new edition to nd the relevant section.
Comments on specic questions { Zone B
Candidates should answer FOUR of the following questions: Question 1 of Section A (40 marks) and all questions from Section B (60 marks in total). Candidates are strongly advised to divide their time accordingly.
Section A
Answer all ve parts of question 1 (40 marks).
Question 1
(a) For each one of the statements below say whether the statement is true or false, explaining your answer.
i. For two independent events A and B with P(A) > 0 and P(B) > 0, then:
P(A [ B) < P(AjB) + P(B jA):
ii. For two mutually exclusive events A and B such that P(A) > 0 and P(B) > 0, then:
P(A [ B) > P(AjB) + P(B jA):
iii. If B A and P(B) > 0, then the complementary events Ac and Bc cannot be independent events.
iv. If T1 and T2 are unbiased estimators of the parameter , then T1T2 is an unbiased estimator of 2.
v. The power of a test is the probability the alternative hypothesis is false.
(10 marks)
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ST104b Statistics 2
Reading for this question
Chapter 2 of the subject guide covers the relevant aspects of set theory and probability theory required for parts i. to iii. In particular, statistical independence of two events is dened in Section 2.7, along with conditional probability. Section 7.4 provides the denition of an unbiased estimator for part iv. Finally, Section 9.8 denes the power of a test for part v.
Approaching the question
i. True. Note that:
P(A[B) = P(A)+ P(B) P(A\B)
= P(A)+ P(B) P(A)P(B)
< P(A)+ P(B)
= P(AjB)+ P(BjA)
as both P(A) and P(B) are strictly positive.
ii. True. As A and B are mutually exclusive events we have that:
P(AjB)+ P(BjA) = 0 < P(A)+ P(B) = P(A[B):
iii. False. When A is the sample space, S, and hence P(A) = 1, then for any event B we have B S and:
P(Bc \Ac) = 0 = P(Bc)P(Ac):
iv. False. E(T1T2) = 2 when T1 and T2 are correlated.
v. False. The power of a test is the probability of rejecting a null hypothesis when it is false.
(b) A random variable X can take the values 1, 0 and 1, where:
P(X = 1) = ; P(X = 0) = 1 3 and P(X = 1) = 2
for 0 < < 1=3. One observation of X is taken at random and we want to estimate the parameter . Consider the two estimators:
2 T1 = X and T2 = 3 :
i. Calculate the bias of each estimator.
(4 marks)
ii. Which of the two estimators would you prefer and why?
(5 marks)
Reading for this question
Section 7.4 of the subject guide covers the relevant material on estimation criteria, i.e. bias, variance and mean squared error (MSE).
Approaching the question
i. We nd:
and:
E(T1) = +2 =
E(T2) = 3 + 2 =
hence for both of these estimators of the bias is zero.
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Examiners’ commentaries 2021
ii. Since both are unbiased estimators we prefer the one with smaller MSE, i.e. with the smaller variance in this case, or equivalently the smaller second moment. We have:
E(T2) = E(X2) = +2 = 3
and: E(T2) = E1 X4 = 3 = :
Since =3 < 3 we prefer T2 because the variance (and hence mean squared error) is smaller.
(c) The random variable X is normally distributed with a mean of 1 and a variance
of 4. Calculate:
i. P X < 2X < 4 and ii. P X < 2jXj < 4 :
(5 marks)
Reading for this question
Section 2.7.3 of the subject guide denes conditional probability, and Section 4.5.3 introduces the normal distribution, including the standardisation transformation.
Approaching the question
i. We standardise and nd:
P(X < 2) P(Z < 0:5) 0:6915 P(X < 4) P(Z < 1:5) 0:9332
ii. Similarly, we nd:
P X < 2jXj < 4 = P( 4 < X < 2)
P(Z < 0:5) P(Z < 5=2) P(Z < 1:5) P(Z < 5=2)
0:6915 0:0062 0:9332 0:0062
= 0:7392:
(d) Two fair six-sided dice are thrown. If their total score is strictly less than 4, they are thrown once more. What is the probability the total score is odd?
(5 marks)
Reading for this question
This probability problem makes use of classical probability, which can be found in Section 2.6 of the subject guide.
Approaching the question
The total score is odd if the total score of the rst throw is 5, 7, 9 or 11, or the total score of the rst throw is 2 or 3 and the total score of the second throw is odd. We hence nd (for example, by a quick table) that the required probability is given by:
36 + 36 + 36 + 36 + 1362 2 = 72 = 0:4861
since the probability that the sum of the outcomes of two dice being odd is equal to 1/2 as this corresponds to one of the dice being even and the other odd.
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ST104b Statistics 2
(e) The random variable X has the probability density function given by:
( f(x) = kx 4
for x 1
otherwise:
i. Determine the value of k.
(3 marks)
ii. Compute E(X) and Var(X).
(5 marks)
iii. Derive the cumulative distribution function of X.
(3 marks)
Reading for this question
Section 3.5 of the subject guide covers all required aspects in this question related to continuous random variables.
Approaching the question
i. The probability density function should integrate to 1 and hence:
so k = 3.
ii. We compute:
and:
hence:
Z 1 1
1 = k 1 x 4 dx = k 3x3 1 = 3
E(X) = 3Z 1 x 3 dx = 3 1
Z 1
E(X2) = 3 x 2 dx = 3 1
2 Var(X) = E(X2) (E(X))2 = 3 2 = 4:
iii. For the cumulative distribution function, note that for x < 1 we have P(X x) = 0. For x 1 we get: Z x
P(X x) = f(t)dt = 1 x 3: 1
Hence in full: (
F(x) = 1 x 3
for x < 1
for x 1:
Section B
Answer all three questions from this section.
Question 2
(a) A chess player who plays with white pieces will win any particular game with probability 0.4, draw with probability 0.4, and lose with probability 0.2. The corresponding probabilities when she plays with black pieces are 0.2, 0.5 and 0.3, respectively. Before each game a fair coin is tossed to decide whether she will play with the black pieces or the white pieces. Hence the probability she plays with the white pieces is 0.5. Assume that the results of dierent games are independent.
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Examiners’ commentaries 2021
i. Given that she won her last game, what is the probability she was playing with the white pieces?
(2 marks)
ii. Given that she did not lose either of her last two games, what is the probability she was playing with the same colour of pieces in both games?
(3 marks)
iii. Considering the next 10 games, what could you say about the distribution of the number of these games that she wins? Your answer should include a sketch plot of the probability distribution.
(5 marks)
Reading for this question
This question makes use of the total probability formula, covered in Section 2.7.5 of the subject guide, and recognition of the applicability of the binomial distribution, covered in Section 4.4.3.
Approaching the question
i. The probability she wins any given game is equal to 0:50:4+0:50:2 = 0:3. Hence the required probability is:
0:50:4 2
0:3 3
ii. The probability she loses any given game is 0:50:2+0:50:3 = 0:25. Hence the probability of her being undefeated in two consecutive games is (0:75)2 = 9=16. The probability she was undefeated in two games while playing white is (0:50:8)2 = 0:16 whereas this probability while playing black is (0:50:7)2 = 0:1225. Hence the required probability is:
0:1225 = 0:2178:
iii. If she plays 10 games, due to the independence assumption the number of games she will win follows a binomial distribution with n = 10 and = 0:3. Possible outcomes are f0;1;2;:::;10g and the respective probabilities are given by n x(1 )n x. The sketch plot should indicate a clear asymmetry.
(b) Let fX1;X2;:::;Xng be a random sample from the continuous uniform distribution dened over the interval [0;5].
i. Derive the maximum likelihood estimator of .
(4 marks)
ii. Given that the method of moments estimator of is 2X=5, check whether the method of moments estimator is consistent. Justify your answer.
(6 marks)
Hint: You may use any results on the formula sheet at the end of the question paper and you may state expressions for E(X) and Var(X) without proof.
Reading for this question
Method of moments and maximum likelihood estimation are covered in Sections 7.5 and 7.7 of the subject guide, respectively. Details of the continuous uniform distribution can be found in Section 4.5.1.
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ST104b Statistics 2
Approaching the question
i. Due to independence, the likelihood function is:
n
L() = (5) 1 = 5 n n
i=1
which is decreasing in , for X(n), and L() = 0 for < X(n), where X(n) is the maximum value in the random sample. Hence the maximum likelihood estimator of 5 is X(n), so by the invariance principle the maximum likelihood estimator of is:
= X(n) :
ii. Using the formula sheet (with a = 0 and b = 5), we have:
E(X) = 0+5 = 2 :
Since: E() = E5X = 2 E(X) = 2 E(X) = 2 5 =
then is an unbiased estimator of . We have, from the formula sheet (with a = 0 and b = 5):
Var(X) = (5 0)2 = 2122 :
Therefore:
b 2 4 4 Var(X) 4 252 2 5 25 25 n 25 12n 3n
Since is an unbiased estimator:
2 MSE() = Var() = 3n ! 0
as n ! 1, hence is a consistent estimator of .
Question 3
(a) Let fX1;X2;:::;X25g be a random sample of size n = 25 from N(;2), where 2 = 3:24 is known. A researcher decides to test:
H0 : = 15 vs. H1 : < 15
using a 1% signicance level.
i. Calculate the power of the test when = 14:7.
(6 marks)
ii. Briey explain in two dierent ways how you could increase the power of this test.
(4 marks)
Reading for this question
Section 9.8 of the subject guide denes a power function, from which the power of a test can be calculated. Section 9.5.4 explains the hypothesis testing procedure for one-sided tests for normal means.
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Examiners’ commentaries 2021
Approaching the question
i. The test statistic under H0 is:
3:24=25 N(0;1)
and the critical value at a 1% signicance level is z0:01 = 2:326. Hence we reject H0 if:
p3:24=25 < 2:326 equivalently x < 15 2:3260:36 = 14:163:
When = 14:7, the power of the test is:
P(X < 14:163j = 14:7) = P Z < 14:163 14:7 = P(Z < 1:49) 0:0681:
ii. The power of the test could be increased either by increasing the sample size or by using a larger signicance level, such as 5%. Both ways would increase the probability of rejecting the null hypothesis. (Note although 2 aects the power, this is not under the control of the researcher.)
(b) A company produces copper wire to a particular specication of breaking strength using four dierent types of machines (A, B, C and D). One machine of each type is selected at random and ve copper wire samples are measured from each machine. The measurements were:
Machine type
A B C D
Sample wire 1 Sample wire 2 Sample wire 3 Sample wire 4 Sample wire 5 15.2 16.4 15.5 15.9 16.1 13.7 13.9 14.0 12.9 13.0 15.0 14.9 14.7 15.2 14.5 14.8 13.8 12.9 13.5 14.0
Sample mean 15.82 13.50 14.86 13.80
You are given that:
X X
xij nx = 21:0095
j=1 i=1
where the overall sample mean is x = 14:495.
i. Test the null hypothesis that the mean breaking strengths of the four types of machines are the same. Use a 1% signicance level.
(6 marks)
ii. Compute a 95% condence interval for the dierence of the mean breaking strengths between machine types A and B.
(4 marks)
Reading for this question
Section 10.5 of the subject guide covers all aspects concerning one-way analysis of variance necessary to perform the required F test and to construct a condence interval for the dierence of means.
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ST104b Statistics 2
Approaching the question
i. We test:
vs.
H0 : There is no dierence in mean breaking strengths
H1 : There is a dierence.
The sum of squares between machines is:
5((15:82 14:495)2 +(13:50 14:495)2 +(14:86 14:495)2
+(13:80 14:495)2)
= 16:8095
and so the error sum of squares is:
21:0095 16:8095 = 4:20:
The test statistic value is: 16:8095=(4 1) 4:20=(20 4)
The 1% critical value is F0:01;3;16 = 5:29. Since 5:29 < 21:35 we reject H0. There is (strong) evidence of a dierence in means.
ii. A 95% condence interval for A B is computed using:
s
xA xB t0:025;16 b2 + : A B
Therefore, we have:
s
15:82 13:502:120 20: 4 5 + 5
which gives (1:633;3:007).
Question 4
Suppose X and Y are two independent random variables with the following probability distributions:
X = x 1 0 1
P(X = x) 0.20 0.60 0.20
and
Y = y 1 0 1
P(Y = y) 0.30 0.40 0.30
The random variables V and W are dened as:
V = X2 + Y 2 and W = X + Y:
(a) Construct the table of the joint probability distribution of V and W.
(8 marks)
(b) Calculate the following quantities:
i. Var(W), given that E(W) = 0.
(2 marks)
ii. Cov(V;W).
(3 marks)
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Examiners’ commentaries 2021
iii. E(V jW = 0).
(4 marks)
(c) Are V and W uncorrelated? Are V and W independent? Justify your answers. (3 marks)
Reading for this question
Sections 5.4 to 5.7 of the subject guide are relevant for this question, which span joint, marginal and conditional distributions, as well as covariance and correlation.
Approaching the question
(a) The joint probability distribution of V and W is:
V
0 1 2
2
1 W 0
1 2 0 0
0.24 0 0 0 0.26 0 0.26 0 0.06 0 0.12 0 0.06
(b) i. Since E(W) = 0, we have:
Var(W) = E(W2)
2
= w2p(w)
w= 2
= ( 2)2 0:06+( 1)2 0:26+02 0:36+12 0:26+22 0:06
= 1:
ii. We have that:
2 2
E(VW) = vwp(v;w) = ( 40:06)+( 10:26)+1 0:26+4 0:06 = 0: v=0 w= 2
Since E(W) = 0, then:
Cov(V;W) = E(VW) E(V)E(W) = E(VW) = 0:
iii. We have:
2
E(V jW = 0) = v=0 vpV jW=0(vjw = 0) = 0 0:36 +2 0:36 = 3:
(c) The random variables V and W are uncorrelated, since Cov(V;W) = 0. However:
P(W = 2) = 0:06 and P(V = 0) = 0:52 ) P(W = 2)P(V = 0) = 0:0312
but:
P(fW = 2g\fV = 0g) = 0 = P(W = 2)P(V = 0)
which is sucient to show that V and W are not independent.
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