QP-IFSM-23-STATISTICS-PAPER -I-051223

INDIAN FOREST SERVICE (WAINJEXAM. 2073 BJKE-U-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. Unless otherwise mentioned, symbols and notations have their usual standard meanings. Assume suitable data, if necessary and indicate the same clearly. Answers must be written in ENGLISH only. BJKE-U-STSC 1 Ql. (a) (b) (c) (d) (e) BJKE-U-STSC SECTION A An urn U contains 3 white and 4 black balls and another urn V contains 5 white and 3 black balls. One ball is chosen at random from urn U and transferred to urn V. If two balls are chosen without replacement from V, what is the probability that they are of the same colour ? Let X be a random variable with the following probability density function : aa O<x<l 2 ae Lx < 2 flx)={ 2 a 25 S<3 2 0, elsewhere Determine the cumulative distribution function of X and hence obtain the value of P(X > >). (i) Define an unbiased estimator of a population parameter 6. Give an example to show that unbiased estimator is not unique. Let Xj, ..., X, be a random sample from N(0, 62). Check whether X is an ancillary statistic with respect to parameter o2. Gi) (i) The incidence of occupational disease in an industry is such that the workers have a 20% chance of suffering from it. What is the probability that out of six workers, five or six will catch the disease ? Gi) Suppose X,, Xo, ... are independent and identically distributed random variables from B(1, p). Find a consistent estimator of p. Let Xj, ..., X, be independent N(u, o2) random variables with known o2. Using Monotone Likelihood Ratio Approach, find a Uniformly Most Powerful (UMP) test for testing Ho:=po versus Hy: p> pp. Write the UMP test for n = 5, o2 = 9, a = -01 and pig = 5. (Tables are provided.) Normal Distribution Table 1.96. z z 0.09 -0.08 -0.07 -0.06 -0.05 --0.04 - 0.03 -0.08 -0.01 0.00 2 -3.80 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 .0001 --3.80 8.70 .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 -3.10 .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 -2.70 260 .0036 .0037 .0038 .0039 .0040 .0041 .0043 .0044 .0045 .0047 —2.60 -2.50 .0048 .0049 .0051 .0052 .0054 .0055 .0057 .0059 .0060 .0062 --2.50 —240 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 20 .0110 .0113 .0116 0119 .0122 .0125 .0129 .0132 .0136 .0139 --2.20 -2.10 .0143 40146 .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 .0289 .0244 .0250 .0256 .0262 0268 .0274 .0281 .0287 —1.90 - 180 .0294 ,0801 .0307 .0314 .0322 .0329 .0336 .0344 ,0851 .0359 —1.80 -1.70 .0367 .0375 .0384 .0392 .0401 .0409 .0418 .0427 04396 .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 10398 1056 .1075 .1093 .1112 .1131 .1151 —1.20 - 110 .1170 .1190 .1210 .1230 .1251 .1271 1292 .1314 .1335 .1857 —1.10 -1.00 .1379 1401 1423 146 1469 1492 1515 1599 1562 .1587 —1.00 -0.90 .1611 1695 1685 1711 .1796 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 - 9.70 0.60 2451 2489 2514 .2546 .2578 .2611 .2643 266 .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 BJKE-U-STSC 3 Normal Distribution Table 2 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 2 0.00 5000 5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319 .5359 0.00 0.10 5398 5438 .5478 .5517 5557 .5596 5696 5675 .5714 .5753 0.10 0.20 5799 .5832 .5871 .5910 5948 5987 .6026 .6064 6103 .6141 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 .6950 .6985 7019 .7054 .7088 .7123 7157 .7190 .7224 0.50 0.60 .7257 .7291 .7324 .7357 .7389 7422 .7454 7486 .7517 .7549 0.60 0.70 .7580 .7611 .7642 .7673 .7704 .7784 .7764 .7794 .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 .8830 1.10 1.20 .8849 8869 .8888 .8907 .8925 .8944 962 .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 1.40 150 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441 1.50 1.60 .9452 .9463 .9474 .9484 .9495 .9505 .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 .9838 9842 .9846 .9850 .9854 .9857 2.10 2.20 .9661 .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 .9934 .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 .9964 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 .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 3.50 .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 3.80 9999 .9999 .9999 .9999 .9999 .9999 9999 .9999 .9999 .9999 3.80 BJKE-U-STSC 4 Q2. (a) (b) (c) Q3. (a) (b) BJKE-U-STSC Let {A,, n =1} be a sequence of events on a probability space (0, #,P)andletF,= U Aj forn > 1. Then show that k=n Pim infA,) < lim P(F,). n> Let {A,, n= 1} be a sequence of events on a probability space (Q, FF, P). Prove or disprove : Pim sup A,)< lim > P(A,) n>? k=n (i) Let X, and X, be independent and identically distributed as Poisson with parameter 4. For a > 1 (an integer), check for sufficiency of the estimator T=X,+oX, for parameter A. (ii) Let X,, Xo, .... X, be a random sample from U(0, 6). Show that T = max (Xj, ..., X,) is complete for 0. (iii) Let Xj,...,X, be independent and identically distributed (iid) N(u, 6), with both p and o2 unknown. Find a minimal sufficient statistic for (1, 6”). Suppose that X,, Xo, ..., Xg are observed as 1-48, 0-25, 1-99, 0-40, 0-58, 0-78, 1-5, 1-2. Using Kolmogorov — Smirnov’s goodness-of-fit test, check whether the data were randomly drawn from a U(0, 2) distribution. (Critical value is 0-46 for a = 0-05) (i) | Describe Wald’s Sequential Probability Ratio Test (SPRT). (ii) Let X,...,X, be independent and identically distributed as N(u, 02), where o? is known. Write the 100(1 — «)% UMA lower confidence bound for testing Hp: =o versus Hy: p> ho. 5 10 10 10 (c) Q4. (a) (b) BJKE-U-STSC (i) (ii) (i) (ii) (i) Consider a sequence {X,, k > 1} of independent and identically distributed random variables, each with finite mean p. For k = 1, let Ty yt, wt KK and Ye oe Show that Y, converges in distribution to a random variable Y with P(Y = 0) = 1 as k tends to infinity. Three coins whose faces are marked 1 and 2, are tossed. Find the mean and variance of total value of numbers on their faces. Let Xj, ..., be a random sample from U(6,, 65). Find the 1 1 2: maximum likelihood estimators of 8, and Oo. Suppose 47, 40, 50, 14, 30, 25, 37, 60 is a random sample from discrete uniform distribution with probability mass function as PK=*)= + ,x=1,2,..,N. Find the value of estimate of N using the method of moments. Let {X,,n = 0} be a Markov chain with state space {1, 2, 3, 4} and the following transition probability matrix : sae | oOo NIE FF wile O28 So wily o NIP wl oO ee) (1) Which states of this Markov chain are recurrent and which states are transient ? (2) Obtain the mean recurrence time in case of these identified recurrent states. (3) | Examine whether these identified recurrent states are 12 ergodic. Justify. 5+5+5=15 (ii) | Consider a Markov chain with state space {1, 2, 3} and transition probability matrix P = ((pj)) such that the probabilities corresponding to cells (1, 2) and (1, 3) are equal; (2, 1) and (3, 1) are equal and all other cells have zero probability. If p\™ denotes the n-step transition probability of state j, compute py and ey 5 (c) (i) Let X, ..., X, follow Poisson distribution with parameter 6 and let g(0) =e 9 (1+). Find the Cramer — Rao (CR) lower bound for estimator of g(0). 6 (ii) Let X follow B(, p) and let the loss function be L(p, 5(x) = [p - &(x)1?. Ifn(p)=1, 0<p<1 is the prior pdf of p, then find Bayes’ estimator of 5(X). 4 BJKE-U-STSC 7 Q5. (a) (b) (c) (d) BJKE-U-STSC SECTION B What is curvilinear regression and when is it used ? Give some examples of curvilinear regression. How will you use scatter plot and adjusted R-squared value to identify curvilinear regression ? If X,, X_ and Xz are uncorrelated variables each having zero mean and the same standard deviation, then show that the correlation coefficient between (X, + Xp») and (KX, + Xg) is equal to (1/2). Why is the correlation coefficient not zero ? Further if we let Y, = X, + Xo, Yo = Xq + Xs and Y3 = Xg + Xj, then obtain : (i) Partial correlation coefficient Ty, ¥o+¥g. (ii) Multiple correlation coefficient R2 5 ¥1-Yo¥g Is there any relation between partial and multiple correlation coefficients you have obtained ? Let x ~ Ng (p, =) where p’ = (1, — 1, 2) and 4 0 -1 X=/0 5 0 -1 0 2 Obtain the conditional distributions of : Gi) Xj, given Xg = xg, and (ii) —-X4, Xo, given Xg = xg. Derive and compare the variances of the ratio and regression estimators of the population mean and state when the ratio estimator is as good as the regression estimator. (6) |Arandomized block design gave the following results : Degrees of Freedom Sum of Squares Blocks 5 620-4 Treatments 3 415-03 Error 11 121-50 How many replications per treatment are required for a completely randomized design to be as effective as the randomized block design using the same experimental material ? 8 66. (a) Explain the concept of interval estimation for the parameters of linear regression model $; = 01+ 022 For the weekly consumption expenditure and weekly income data, the following results were obtained : , = 24-4545, Var () = 41-1370 and 01) = 6-4138 A A A 02 = 0-5091, Var (92) = 0:0018 and ) = 7‎ A A x Cov (, 02) = ¬ 0-2172, 2 = 42-1591 12 = 0-9621, r = 0-9809, df = 8 Write estimated linear regression model. Obtain 95 percent confidence A A intervals for 3] and 3) . Interpret the results. (/2 = 0.025 = 2°306 for 8 df) i BJKE-U-STSC 9 (b) (c) Q7. (a) BJKE-U-STSC Find the principal components and the proportion of the total variance explained by each principal component of the standardized covariance matrix, when the covariance matrix 2 = (04); oi = o” for i= 1, 2,3,4 and Oij = po? for i j;i,j =1, 2, 3, 4. 15 Consider the data matrix : -1 3 -2 (i) Calculate the matrix of deviations (residuals), (X — 1X’). Is this a non-singular matrix ? 1 1% — 1x’) (& —- 1x’) and calculate the PAE (ii) Determine S = generalised sample variance |S]. (iii) | Calculate the total sample variance. 10 (i) Define the Horvitz Thompson (HT) estimator. In a PPSWOR sampling, show that the HT estimator is unbiased for the population total. Gi) Calculate the inclusion probability for the following population where x is the size variable when two units are selected without replacement with PPS at each step. 5+10=15 S.No. yj x; 1 2 a 2 6 2 3 8 it 4 24 10 10 (b) BJKE-U-STSC (i) | How are the three principles of experimentation implemented in a Latin square ? (ii) A scientist needed to test the strength of four different types of polymers A, B, C, D and a control E. He made five samples of each polymer. He suspected there was a temperature gradient from front to back and top to bottom in the thermal furnace. He conducted the experiment using shelves as rows, and position in shelves as columns to obtain the following data. Analyze the data and give your conclusion on his belief at both 5% and 1% significance level. 5+10=15 Front > Back Top 33-8(A) 33-7(B) 30-4(D) 32-7(C) 24-4(E) L 37:0(D) 28-8(E) 33:5(B) 346(A) 33-4 (C) 35°8(C) 35:6(D) 36:9(A) 26-7(E) 35-1(B) 33-2(E) 37:1(A) 37-4(C) 381(B) 341(D) Bottom 34:8(B) 39:1(C) 32:7(E) 37:4(D) 36-4(A) [Tables are provided] 11 F-distribution Table +0.95 Degrees of Numerator Degrees of Freedom Freedom 1 2 3 4 5 6 7 1 ଵା லும்‌ દત ତପ তা 161.4 199.5 215.7 224.6 230.2 234.0 2368 238.9 240.5 18.51 19.00 19.16 19.25 19.30 19.33 19,35 1937 19.38 1013 955 928 912 901 8594 889 685 81 7.71 6.94 659 639 626 616 609 604 6.00 1 2 3 4 5 661 579 S541 8ا5‎ 505 495 488 482 477 6 7 8 9 599 3.14 476 453 439 428 421 415 4.10 ૩59 4.74 435. 412 397 387 3769 37 3.68 532 446 407 384 369 358 350 3.44 3.39 ૩12 20 1080 303 ૩8 ૩5/ ୨୭୭ 52 3.0 10 496 410 3.71 348 333 3.22 3.14 3.07 3:02 11 ୭ ୫୬୩୭ ஆலு. 3536 ୨୫୩ কৰো ଓ ୩୦୭ হা 12 475 389 349 326 341 300 291 285 2:86 13 467 381 341 31 300 292 28) 277 2) 14 கல 3.744. 334 3:11 296 28 276 29 2:65 15 क 368 38 306 290 279 22; 38 239 16 449 363 324 301 285 274 266 259 2.34 17 శక్‌ 358 320 296 28) 22 ୪୭୮୮ 255 249 18 441 $55 816 298 277 266 256 251 6 19 4858 ॐ 313 296 274 20 ડઝ 2 242 20 4.33: : 360 310 4897 270: - 200 22 245 “238 21 432 SAT হয়া 29 268 257 240 242 ಯ 22 430 3.44 305 282 266 255 246 240 2.34 23 4.28. 342 30 260 2 255 24 237 232 24 536 1940 ਹੋਰ 238 20: 3231 ୪୯୩ ૩38 ୧୨୫ 25 4.24: 339 299 276 260 249 240 234 22 26 423. 857 298 274 259 ಯ್ಯ 239 232 227 27 491 SS) 296 S373 257 246 ୦୬ 2.25 28. 420 $34 285 271 2546 245 296 229 224 29 ୫୫୫ 335 208 220 255 2465 255 228 222 30 417 332 292 269 253 242 238 227 221 40 ୫୩୫୭ ସସ 284 ୩୫ 245 23% ইল 2 232 60 400 ૩315 228 258 22 225 217 ௨19 20: 120 ୬ 300 ડેલી ^ રેટ? થી? ಲಯ ಯ 2020 100 ০ 9 500 پوت‎ 237 এওঁ 210 ~ 14 109 BJKE-U-STSC 12 F-distribution Table Fo.99 D a Numerator Degrees of Freedom Freedom 10 12 15 20 24 30 640 60 120 o@ 241.9 243.9 245.9 248.0 249.1 250.1 251.1 252.2 253.3 254.3 19.40 1941 19:43 1945 19.45 19.46 19.47 19.48 19.49 19.50 8.79 8.74 870 866 864 862 859 857 8.55 8.53 5.96 591 5.86 5.80 577 5.75 5.72 569 566 5.63 1 2 Ss 4 5 4.74 468 462 456 453 450 446 443 440 436 6 7 8 9 4.06 4.00 3.94 3.87 384 381 3.77 3.74 3.70 3.67 3.64 357 3.51 344 341 338 334 330 3.27 3.23 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 297 2.93 3.14 $.07 3.01 294 290 286 283 279 275 271 10 298 291 285 2.77 2.74 2.70 266 262 258 254 1 285 279 2.72 265 261 257 253 249 245 240 12 2.75 269 262 254 251 247 243 238 234 230 13 2.67 260 253 246 242 238 234 230 225 221 14 2.60 253 246 239 235 231 227 222 218 213 15 254 248 240 233 229 225 220 216 211 2.07 16 249 242 235 298 294 219 215 211 206 201 17 245 238 231 223 219 215 210 206 201 1.96 18 241 234 297 219 215 211 206 2.02 197 1.92 19 238 231 223 216 211 207 203 198 1.93 1.88 20 235 228 220 2.12 208 204 199 195 190 184 21 232 225 218 210 205 201 1.96 192 187 L8l 22 230 223 215 2.07 203 198 1.94 189 1.84 1.78 23 227 220 213 205 201 196 191 186 181 1.76 4 225 218 211 203 1.98 194 189 1.84 1,79 1.73 25 22 216 209 20f 196 192 187 182 1.77 171 26 222 215 207 199 195 190 185 180 1.75 1.69 27 220 213 2.06 1.97 193 188 184 179 4173 1.67 28. 219 212 204 196 191 187 182 1.77 171 165 29 2.18 2.10 203 194 190 185 181 175 1.70 1.64 30 2.16 209 2.01 193 189 184 179 174 168 1.62 40 208 2.00 1.92 184 179 1.74 169 1.64 158 1.51 60 199 192 1.84 175 170 165 159 153 147 139 120 191 183 1,75 166 161 155 150 143 135 1,25 ps 183 1.75 167 157 152 146 189 132 1.22 1,00 BJKE-U-STSC 13 (c) Q8. (a) (b) (c) BJKE-U-STSC A farmer packing pineapples used 600 baskets to collect the fruit from all his farms. Each basket could hold 40 fruits. But some baskets held lesser number of the fruits. A simple random sample of 50 baskets was examined and the data are given below : No. of fruits 40 89 86 32 29 27 23 19 No.of baskets-"21 37.4 27 2" 42294 No. of fruits 1G's45: 14, Ad--1A0s9 fer 6b 42-8 No, of baskets; 1°. 2:22, 1°: eel flee 3o Oe, Give the 80% confidence interval for the total number of pineapples he has harvested from his farms. (Use table value t = 1-28, if required) 10 Discuss all the assumptions and their relative importance for Classical Linear Regression Model (CLRM) in the context of the two-variable linear regression model. 15 (i) (ii) Consider two factors A and B at three levels. Under a 32 factorial experiment, give the contrast for estimating the linear and quadratic effects of A and B and the interaction effects. Give the assignment of treatments, along with breakdown of the total degrees of freedom, in a 3° factorial experiment, into blocks by confounding the interaction effect AB. 5+10=15 A rural block in a district is divided into three strata comprising 10, 8 and 25 villages, respectively. The following table gives the number of villages, area of cultivation of pepper and standard deviation of area under cultivation of pepper for different strata. Stratum | Number of Area Under Standard No. Villages Cultivation (hectares) Deviation £ 10 425 90-5 2 8 470 40:7 3 25 855 100-8 14 @) (ii) BJKE-U-STSC A stratified sample of 10 villages is selected and yields are given below : oo Yield in quintals for the selected villages 1 8-5 10-2 2 76 9-3 8-5 3 74 10-1 8:3 8-5 8-9 Give an unbiased estimate of average yield based on the above stratified design. Do you think that the stratification is more effective than a simple random sampling in this case ? 15 10 કા