UPSC IFS 2018 Question Paper for Statistics Paper - I

EXAM-(M) 2018
FSIT-STSC

STATISTICS
Paper - I

Time Allowed: Three Hours Maximum Marks : 200

Question Paper Specific Instructions

Please read each of the following instructions carefully before attempting
questions:

There are EIGHT questions in all, out of which FIVE are to be attempted.

Questions no. 1 and 5 are compulsory. Out of the remaining SIX questions, THREE are
to be attempted selecting at least ONE question from each of the two Sections A and B.

Attempts of questions shall be counted in sequential order. Unless struck off attempt of a
question shall be counted even if attempted partly. Any page or portion of the page left
blank in the Question-cum-Answer Booklet must be clearly struck off.

All questions carry equal marks. The number of marks carried by a question/part is
indicated against it.

Answers must be written in ENGLISH only.

Unless otherwise mentioned, symbols and notations have their usual standard meanings.

Assume suitable data, if necessary and indicate the same clearly.

FSI P STSC 1

SECTION A

Q1. (i) For n events A1, A2, ..., An, show that
n

n Ai
i=1
- (n - 1).

(ii) Let {An} be an increasing sequence of sets (events), then show
that
00
(
lim P(A ) = P lim A =P U An 3+5
n->00 -) n
1 /4‘n =1

Let K2 = {1, 2, 3, 4} and pi = PM, i = 1, 2, 3, 4.

Assume that

3
and
P3= 4 - 2
1
P4 = •

Define the events
E1 = (1, 3), E2 = (2, 3) and E3 = (3, 4).
Check whether E1, E2 and E3 are mutually independent. 8

Suppose that X1, ..., Xr, form a random sample from a Uniform
distribution on the interval [01, 021 where both 0/ and 02 are unknown
(-00 < 01 < 02 < co). Find the maximum likelihood estimators of
01 and 02. 8

Suppose that X1, ..., X. form a random sample from a Gamma
distribution for which the value of parameter a is unknown (a> 0) and
value of parameter f3 is known. Show that the joint probability density
function of X1, ..., X. has a Monotone Likelihood Ratio (MLR). 8
FSI-P-STSC 2

(e) (i) Le (;) be a sequence of events such that

p[xn = n2 _ = ,
nz

P = = 1 -- .
n`
Check whether Strong Law of Large Numbers holds. Also
comment about Weak Law of Large Numbers.
(ii) Let
xn prob.
, X and ; prob.
> Y, then show that

XI in probability. 8

Q2. (a) Suppose that a diagnostic test for HIV(+) status has both sensitivity
(P(Test positive I Disease)) and specificity (P(Test negative I No disease))
equal to 0.95 and the real possibility (P(Disease)) is 0.005. Find the
probability that a subject is truely HIV(+) given that the diagnostic test
is positive. 8
(b) Let a continuous random variable X have probability density function
(pdf) given by
2x
f(x) 712 '
0, otherwise

Find the probability density function of Y = sin X. 10

(c) Let (X, Y) be uniformly distributed over
R y) I x2 + y2 y

Find
the distributions of (XI Y = y) and (Y IX = x),
E(Y I X = x) and E(X I Y = y), and
Correlation coefficient pxy. 12

FSI-P-STSC 3

(d) Let X follow Pareto distribution with parameters a and 0 with
probability density function (pdf)

a Oa
f(x) = , x > 0,
(x + +

then show that

Y = ln(--
X-)
0

follows Logistic distribution. /0

Q3. (a) Suppose that x l, xn. form a random sample from a Beta distribution
with parameters a and 13 where the value of a is known and the value of
p is unknown (p > 0). Obtain a sufficient statistic for f3. 8

Let xl, xn. be a random sample from P(0). Find Uniformly Minimum
Variance Unbiased Estimator (UMVUE) of e-9. Compute the estimator
based on the following sample observations : 12
1, 3, 0, 8, 5,6, 9,2, 7,5

The lifetimes of fluorescent lamps are independent exponential random
variables with parameter p. Suppose that 13 has a prior distribution
Gamma with parameters 4 and 20,000. After we observe 5 lamps with
lifetimes 2911, 3403, 3237, 3509 and 3118 (in hours), we want to predict
the lifetime X6 of the next lamp. Obtain the predictive distribution of Xo. 10

Suppose X1, X2, ..., Xn N(a, a2) where a2 is known.

Find the Likelihood Ratio Test (LRT) for H0 : µ0 vs Hi : i>

Show that the test in (i) is a UMP test. 10

FSI-P-STSC 4

Q4. (a) For any sequence {A., n 1} of events with Ep(Ak ) <cc , comment on
k =1
the following with justification: 4+6

P UAk I p(Ak)
=n k=n

P( lim sup An) = 0

Suppose that an examination contains 99 questions arranged in a
sequence from the easiest to the most difficult. Suppose that the
probability that a particular student will answer the first question
correctly is 0.99, the probability that he will answer the second question
correctly is 0.98 and in general, the probability that he will answer the
ith question correctly is 1 —, for i = 1, 2, ..., 99. It is assumed that
100
all questions will be answered independently and that the student must
answer at least 60 questions correctly to pass the examination What is
the probability that the student will pass? 10

(Tables 1(a) and 1(b) are provided at the end.)

Let {X0, X1, ...} be a Markov Chain (MC) with transition probability
matrix
1 2 3
1 -0 1 0-

P=2 0 0 1 , 0 <p< 1

3 p 1—p 0_

0, x=1
Let g(x) =
1, x = 2, 3.

If Yn. = n 0, show that {Yo, Y1, ...} is not a Markov Chain (MC). 10

Derive Kolmogorov — Smirnov test for two samples and illustrate. 10
FSI-P-STSC 5

SECTION B

Q5. (a) Let N be the incidence matrix of a Balanced Incomplete Block Design
(BIBD) of order b x v. Show that b v. 8

(i) A plane is fitted to n = 33 observations on (X1, X2, Y) and it is

found that overall regression is just significant at a = 0.05 level.
Find out R2 based on the available information.
(Tables 2(a) and 2(b) are provided at the end.)
(ii) In a one-way layout, show that for all values of i, and j,
j= 1, 2, ..., n,
= 1, 2, ..., p,

wi =

=

w3 = yi..

are uncorrelated with each other (under usual assumptions). 3+5

Let Y follow N2 (lit, 1.t2, cs, 4 , p). Find distribution of Z = CY where

C is a 2 x 2 non-singular matrix. Also give an explicit form of matrix C
such that CEC' = I, where E is the dispersion matrix of Y. 8

Construct a 23 design in two blocks where ABC is confounded. 8

Find the condition under which systematic sample mean is more
efficient than a simple random sample mean. 8

Q6. (a) Let X1, X2, ..., X. be a random sample from Np(u, D. Consider the
hypothesis Ho : = kuo, where E and uo are known. Derive the

MLE for k. Show that - 2 log likelihood ratio is

nYC - (Filo µo)-1 tilo )E-1R.

Deduce the distribution of the statistic. i 10

FSI-P-STSC 6

(b) Suppose that a chemical engineer considers the time of reaction for a
chemical process as a function of the type of catalyst used. Four
catalysts are being investigated and the procedure consists of selecting a
batch of raw materials The observations recorded are as shown below :
Batch of Raw Materials
Catalyst
1 2 3 4
1 73 75 68 —
2 75 — 72 75
3 73 74 71
4 — 75 67 72
Identify the design and analyse it. 10
(Tables 2(a) and 2(b) are provided at the end.)
Let X1 and X2 be independent random vectors of order (n1 x p) and
(n2 x p) respectively and let ni rows of Xi (i = 1, 2) be independently and
identically distributed as Np(pi, Ei).

Show that for ui = 122 and E l = E2,
n1n 2 D2 T
2 (p, n — 2),

where n = n1 + n2, D2 denotes sample Mahalanobis distance statistic
and T2 denotes the Hotelling's T2. 10

Find an unbiased estimator of the population mean under probability
proportional to size (PPS) sampling with replacement. Find the variance
of this estimator and also give an estimator of this variance. 10

Q7. (a) In a simple linear regression problem Y = po + ppc. + E, 6 - NW, (32), in
which a patient's response Y to a new drug B is to be related to his
response X to a standard drug A. Suppose 10 pairs of observations
xi), i = 1... 10 are obtained.
Determine the MLEs of fA30, 131 and a
A 2 of their corresponding
parameters,
A A
Obtain variance (so), variance C131) and corr ( Po, Pi). 10
FSI-P-STSC 7

In a two-way layout with k observations in each cell (k 2), construct a
test of the null hypothesis that all the interactions are zero. (Model and
all assumptions are required to be specified in detail). 8

For the covariance matrix given by
1 4
4 100,

obtain the proportion of the total population variance explained by the
first principal component. 10

In simple random sampling where n paired observations
(yi, i = 1 n are drawn, obtain the regression estimators and derive
its large sample variance. 12

Q8. (a) Use the method of Lagrange's multipliers to show that for a least
squares problem,
T = (Y — Xf3)/ (Y— X8) + X'(d — Cf3)
is minimized with respect to 13 and A, where
= b + oc' xyl c' [coc'x)-10-1- (d — Cb)
where b is unrestricted least squares estimator of J3. 10

(b) Let there be two populations II1 and 112. It is known that about 30% of
all objects belong to II2 and
0(211) : cost incurred when a 11 observation is incorrectly classified as
H2 observation = 15;
0(112) : cost incurred when a 112 observation is incorrectly classified as
n1 observation = 10.
Suppose the two density functions f1(x) and f2(x) (corresponding to
1-11 and 112) are evaluated at a new observation xo and f(x0) = 0.32,
f2(x0) = 0.56.
Can the new observation be classified from 111 or H2 ? 10

FSI-P-STSC 8

(c) A chemical experiment was performed to investigate the effect of
extrusion temperature X1 and cooling temperature X2 on the
compressibility of a finished product. Knowledge of the process
suggested that a model of the form Y = pp + f31; + f32X2 + B12 X1X2 +
would satisfactorily explain the variation observed. Two levels of
extrusion temperature and two levels of cooling temperature were
chosen and all four of the combinations were performed. Each of the four
experiments was carried out four times and the data yielded the
following information :

ANOVA

S.V d.f S.S MSS
Due to reg. — 881.2500
Po 1 798.0625
Pi 1 18.0625
P2 - -

P12 5-0625
Residual — —
Total 16 921.000

Using a = 0.05, examine the following questions: 10
Is the overall regression equation statistically significant?
Are all P significant?
(Tables 2(a) and 2(b) are provided at the end.)

(d) For apxp Latin square with rows (a1), columns (Pk) and treatments (Tj)
fixed, obtain least squares estimators of a1, pk and j, k = 1, p.
Derive the missing value formula (when just one observation is missing)
for the Latin square design. 10

FSI-P-STSC 9

TABLE 1(a)
A-38 APPENDIX STATISTICAL TABLES
TABLE D Normal Curve Areas P(z Sz). Entries in the Body of the Table are
Areas Between-cc and z

.9750

0 1.96
z -0.09 -0.07 -0M6 -0M5 -0M4 -0M3 -0M2 -0.01 OMO z
-3.80 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 -3.80
-3.70 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 -3.70
-3.60 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0002 .0002 -3.60
-3.50 .0002 .0002 .0002 .0002 .0002 .0002 .0002 .0002 .0002 .0002 -3.50
-3.40 .0002 .0003 .0003 .0003 .0003 .0003 .0003 ,0003 .0003 .0003 -3.40
-3.30 .0003 .0004 .0004 .0004 .0004 .0004 .0004 .0005 .0005 .0005 -3.30
-3.20 .0005 .0005 .0005 .0006 .0006 .0006 .0006 .0006 .0007 .0007 -3.20
-3.10 .0007 .0007 .0008 .0008 .0008 .0008 .0009 .0009 .0009 .0010 -3.10
-3.00 .0010 .0010 .0011 .0011 .0011 .0012 .0012 .0013 .0013 .0013 -3.00
-2.90 .0014 .0014 .0015 .0015 .0016 .0016 .0017 .0018 .0018 .0019 -2.90
-2.80 .0019 .0020 .0021 .0021 .0022 .0023 .0023 .0024 .0025 .0026 -2.80
-2.70 .0026 .0027 .0028 .0029 .0030 .0031 .0032 .0033 .0034 .0035 -2.70
-2.60 .0036 .0037 .0038 .0039 .0010 .0041 .0043 .0044 .0045 .0047 -2.60
-2.50 .0048 .0049 .0051 .0052 .0054 .0055 .0057 .0059 .0060 .0062 -2.50
-2.40 .0064 .0066 .0068 .0069 .0071 .0073 .0075 .0078 .0080 .0082 -2.40
-2.30 .0084 .0087 .0089 .0091 .0094 .0096 .0099 .0102 .0104 .0107 -2.30
-2.20 .0110 .0113 .0116 .0119 .0122 .0125 .0129 .0132 .0136 .0139 -2.20
-2.10 .0143 .0146 .0150 .0154 .0158 .0162 .0166 .0170 .0174 .0179 -2.10
-2.00 .0183 .0188 .0192 .0197 .0202 .0207 .0212 .0217 .0222 .0228 -2.00
-1.90 .0233 .0239 .0244 .0250 .0256 .0262 .0268 .0274 .0281 .0287 -1.90
-1.80 .0294 .0301 .0307 .0314 .0322 .0329 .0336 .0344 .0351 .0359 -1.80
-1.70 .0367 .0375 .0384 .0392 .0401 .0409 .0418 .0427 .0436 .0446 -1.70
-1.60 .0455 .0465 .0475 .0485 .0495 .0505 .0516 .0526 .0537 .0548 -1.60
-1.50 .0559 .0571 .0582 .0594 .0606 .0618 .0630 .0643 .0655 .0668 -1.50
-1.40 .0681 .0694 .0708 .0721 .0735 .0749 .0764 .0778 .0793 .0808 -1.40
-1.30 .0823 .0838 .0853 .0869 .0885 .0901 .0918 .0934 .0951 .0968 -1.30
-1.20 .0985 .1003 .1020 .1038 .1056 .1075 .1093 .1112 .1131 .1151 -1.20
-1.10 .1170 .1190 .1210 .1230 .1251 .1271 .1292 .1314 .1335 .1357 -1.10
-1.00 .1379 .1401 .1423 .1446 .1469 .1492 .1515 .1539 .1562 .1587 -1.00
-0.90 .1611 .1635 .1660 .1685 .1711 .1736 .1762 .1788 .1814 .1841 -0.90
-0.80 .1867 .1894 .1922 .1949 .1977 .2005 .2033 .2061 .2090 .2119 -0.80
-0.70 .2148 .2177 .2206 .2236 .2266 .2296 .2327 .2358 .2389 .2420 -0.70
-0.60 .2451 .2483 .2514 .2546 .2578 .2611 .2643 .2676 .2709 .2743 -0.60
-0.50 .2776 .2810 .2843 .2877 .2912 .2946 .2981 .3015 .3050 .3085 -0.50
-0.40 .3121 .3156 .3192 .3228 .3264 .3300 .3336 .3372 .3409 .3446 -0.40
-0.30 .3483 .3520 .3557 .3594 .3632 .3669 .3707 .3745 .3783 .3821 -0.30
-0.20 .3859 .3897 .3936 .3974 .4013 .4052 .4090 .4129 .4168 .4207 -0.20
-0.10 .4247 .4286 .4325 .4364 .4404 .4443 .4483 .4522 .4562 .4602 -0.10
0.00 .4641 .4681 .4721 .4761 .4801 .4840 .4880 .4920 .4960 .5000 0.00

FSI-P-STSC 10

TABLE 1(b)

APPENDIX STATISTICAL TABLES

TABLE D (continued)

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 z
0.00 .5000 3040 .5080 .5120 .5160 .5199 ,5239 .5279 .5319 .5359 0.00
0.10 .5398 .5138 3478 .5517 3557 ,5596 3636 .567i .5714 .5753 0.10
0.20 .5793 3832 .5871 .5910 .5948 ,59117 6026 .6064 .6103 .6 I 41 0.20
0.30 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480 .6517 0.30
0.40 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844 .6879 0.40
0.50 .6915 .6930 .6985 .7019 .7054 .7088 .7123 .71,57 .7190 .7224 0.50
0.60 .7257 .7291 .7324 .7357 .73149 .7422 .7454 .7486 .7517 .7549 0.60
0.70 .7580 .7611 .7612 .7673 .7704 .7734 .7764 .7791 .7823 .7852 0.70
0.80 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 8133 0.80
0.90 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 0.90
1.00 .8413 .8438 .8461 .8485 .8508 ,8531 .8554 .8577 .8599 .8621 1.00
1,10 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810 .11830 LID
1.20 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997 .9015 1.20
1.30 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162 .9177 1.30
1.40 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306 .9319 1.40
1.50 .9332 9345 .9357 .9370 .9382 .9394 .9406 .94 t El .9429 .9441 130
1.60 .9452 .9463 .9474 .9484 .9495 .9305 .9515 .9525 .9535 .9545 1.60
1.70 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633 1.70
1.80 .9641 ,9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699 .9706 1.80
1,90 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767 1,90
2.00 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812 .9817 2.00
2.10 .9821 .9826 .9830 9834 .9858 .9842 9846 .9850 9854 .9857 2.10
2.20 .986 I .9864 ,9868 .9871 .9875 .9878 .9881 .9884 .9887 .9890 2.20
2.30 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913 .9916 2.30
2.40 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9931 .9936 2.40
2.50 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951 .9952 2.50
2.60 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963 $964 2.60
2.70 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973 .9974 2.70
2.80 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980 .9981 2.80
2.90 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986 .9986 2.90
3.00 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990 .9990 3,00
3.10 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993 .9993 3.10
3.20 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995 .9995 3,20
3.30 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996 .9997 3.30
3.40 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9998 3.40
330 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998 .9998 3.50
3.60 .9998 .9998 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 3.60
3.70 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 .9999 3.70
5.80 .9999 .9999 .9999 49999 .9999 .9999 .9999 .9999 .9999 .9999 3.80

FSI-P-STSC 11

TABLE 2(a)
A-48 APPENDIX STATISTICAL TABLES

TABLE G (continued)

Fs3

Denominator
Degrees of Numerator Degrees of Freedom
Freedom 1 2 3 4 5 6 7 8 9
1 161.4 199.5 215.7 224.6 230.2 234.0 236.8 238.9 240.5
2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38
3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81
4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.09 6.00
5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77
6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10
7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68
8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39
9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18
10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02
11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90
12 4.75 3,89 3.49 3.26 3.1 1 3.00 2.91 2.85 2.80
13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71
14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65
15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59
16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54
17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49
18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46
19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42
20 4.35 149 3.10 2.87 2.71 2.60 2.51 2.45 2.39
21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37
22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34
23 4.28 3.42 3.03 2.89 2.64 2.53 2.44 2.37 2.32
24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30
25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28
26 4.23 3.37 2.98 2.74 2.59 247 2.39 2.32 2.27
27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25
28 4,20 3.34 2.95 2.71 2,56 2.45 2.36 2,29 2.24
29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22
30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21
40 4,08 3.23 2.84 2,61 2.45 2.34 2.25 2.18 2.12
60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04
129 3.92 3.07 2.68 2.45 2.29 2.17 2.09 2.02 1.96
co 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88

FSI-P-STSC 12

TABLE 2(b)
APPENDIX STATISTICAL TABLES
TABLE G (continued)

Denominator
Degrees of Numerator Degrees of Freedom
Freedom 10 12 15 20 24 30 40 60 120 cc
1 241.9 243.9 245.9 248.0 249,1 250.1 251.1 252.2 253.3 254.3
2 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.50
3 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.53
4 5.96 5.91 5.86 5,80 5.77 5.75 5.72 5.69 5.66 5.63
5 4.74 4.68 4.62 4.56 4.53 4.50 4,46 4.43 4.40 4.36
6 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67
7 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23
8 3.35 3.28 3.22 3.15 3,12 3.08 3.04 3.01 2.97 2.93
9 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71
30 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54
11 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.40
12 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30
13 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21
14 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13
15 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.1I 2.07
16 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01
17 2,45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.96
18 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1,97 1.92
19 2.38 2.31 2,23 2.16 2.11 2.07 2.03 1.98 1.93 1.88
20 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84
21 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81
22 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78
23 2.27 2.20 2.13 2,05 2.01 1.96 1.91 1.86 1.81 1.76
24 2.23 2.18 2.11 2.03 1.98 1.94 1.89 1.84 139 1.73
25 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71
26 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.69
27 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.67
28 2,19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.65
29 2.18 2.10 2.03 1.94 1.90 1.85 1,81 1.75 1.70 1.64
30 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62
40 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 3.58 1.51
60 1.99 1.92 1.84 1.73 1.70 1.65 1.59 1.53 1.47 1.39
120 1.93 1.83 135 1.66 1.61 1.55 1.50 1.43 1.35 1.25
CO 1,83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00

FSI-P-STSC 13

I

1