QP-IFSM-24-STATISTICS-PAPER-I-021224

Inaian Forest Service (Main) Examination, 2024 JBNV-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. Question Nos. 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. JBNV-U-STSC/50 1 [ P.T.O. SECTION—A 1. (a) The lifetime of a mobile charger (in hours) has the normal distribution with mean (u) =100 and variance (67) = 400. () What is the probability that the mobile charger lasts at least 125 hours? (ii) If the mobile charger has already lasted for 105 hours, what is the conditional probability that it will last another 20 hours? (Normal Distribution Table is given in Page Nos. 10 and 11) 4+4=8 (b) (i) State Lindeberg condition for non-identically distributed independent variables to hold central limit theorem (CLT). 4 (ii) Examine whether CLT holds for the sequence {X,,}, where ax, =2()}=3 2” 2 4 (c) Suppose X,, Xj,---, X, are independently identically distributed (iid) observations from a location parameter family with cumulative distribution function F(x-®),-%<®<, Show that R=X;,)—X,q) is ancillary statistic, where X(,) =max{X;} and Xj) = min{X;}. 8 wt u (d) Consider the problem of testing Hy : 8 =1 versus H, : 8 = > where 0 is the mean of a Poisson random variable. Let X and Y be a random sample from Poisson (6) distribution. Consider the following test procedure : Reject Hg if X =1 or (Y=1 and X+Y <2), otherwise accept Hp Determine the probability of type I and type II errors. 8 (e) In an ecological study of the feeding behaviour of birds, the number of hops between flights is counted for several birds : No. of hops Observed frequency 48 31 20 a BOEePAMINaANAWNHH PNOrPrPNAUA OO 12 Total 130 Assuming that the data are generated by a geometric (p) model and take a uniform prior for p, what is the posterior distribution of parameter p? What are the mean and the standard deviation of the posterior distribution? 4+2+2=8 JBNV-U-STSC/50 2 2. (a) (b) ( 3. () (9 Suppose that the transition probability matrix of a Markov chain model is given by 0-7 O- P= 3 0.4 0.6 Compute n-step transition probability matrix (07), 10 (i) A random variable X has mean () = 40 and standard deviation (6) = 2. Apply Chebyshev’s inequality to estimate P(25 < X < 55). 5 Let 2], 202, 203, --- be a sequence of independent random variables, where the probability mass function (PMF) of the random variables 200, 71 = 1, 2, 3, --- is given by p(x, = + | ~ 1 2 2 Does the law of large numbers hold for this sequence of random variables? 10 (9 Let X follow binomial 96 9) distribution, 0 < 6 < 1. To test the hypothesis Ho: ‡ 56 <2 versus 77 : 6 < or 6 > 2 with size 0-3, the test function is based on the following procedure : Reject Hg with probability y,, if X =1 Reject Hg with probability (2) if X = 3 Determine the constants 1] and 1/2. Compute power of the test when 6 = 0-2. 10 ( Let 2], 22, -.., X, be independently identically distributed (iid) random variables from exponential distribution with mean 9 > 0. Define Xn = 42 + + Xp Find ‎ [900]. 5 Two random variables X and Y have the following joint probability density function : k(4-x-y); 0૬2૨2, 0૨૪૨૬2 { { = | . 0 ; otherwise Find the following : (0 Constant k (1) Marginal density function of and Y (1) > (> |¥ = 1) (iv) V(X|Y = 1) () cov(X, Y) 20 JBNV-U-STSC/50 3 | P30. (b) A study is run to evaluate the effectiveness of a physical exercise in reducing systolic blood pressure (SBP) in 15 patients. SBPs are measured before joining the physical exercise and after six weeks of joining the physical exercise. The data are shown below : Patient SBP before exercise SBP after exercise il 127 120 134 136 3 140 132 4 132 134 5 129 128 6 130 137 7 122 126 8 127 107 9 129 132 10 138 142 11 137 128 12 134 130 13 141 134 14 138 132 1S 133 125 Use Wilcoxon signed-rank test to test the difference in SBP after participating in physical exercise as compared to before at 5% level of significance. [ Given that Wis, 0.05) =25, Waa, 0-05) =21 for two-tailed test ] 10 (c) (j A 24-hour advance prediction of a day’s high temperature is ‘unbiased’ if the long-term average of the error in prediction (true high temperature minus predicted high temperature) is zero. The errors in predictions (x) made by one meteorological station for 20 randomly selected days were recorded. The results were 20 dx; =-15, dx? =35 i=l i Assume normal distribution of errors and test the null hypothesis that the predictions are unbiased versus the alternative that they are biased at 1% level of significance. Verify whether the decision would be the same at 5% and 10% level of significance. [Given that tho, 0.005) = 2-861, tao, 0-025) = 2093, tao, 0.05) = 1-729, where P(t, > tn, g)) =o] 3+1+1=5 (i) Distinguish between Wald’s SPRT and the test based on Neyman-Pearson theory to test simple null versus simple alternative hypotheses. 5 JBNV-U-STSC/50 4 4. (a) Let and Y be independent Poisson variables with V(X +Y)=9 and 5 [# = 3|27 +7 = 6} = Pl | == Obtain E(Y). 10 (b) Suppose X and Y are independent identically distributed as exponential variates with mean = 1. Obtain the characteristic function of X and X-Y. Hence, deduce the distribution of Z = X - . 10 (c) Suppose that X is a discrete random variable with = )2-3 and‏ _o 26 19 2-21-9 9 ‎ 1( ’ 22 2) 3 3 where 9 € [0, 1] is the parameter. The following 10 independent observations were taken from such distribution : 8, 0, 2, 1, 3, 2, 1, 0, 2, 1} (0 Find the moment estimate of 6. (i) Find the approximate standard error for your estimate. (iii) What is the maximum likelihood estimate (MLE) of 9? (iv) What is the approximate standard error of the MLE of 5+5+5+5=20 SECTION—B 5. (a) For the model =00 8,255, = N(0, 2), a regression line is fitted on the basis of n paired observations ((x;, {£}, i=1, ---, n). The fitted line is y = ‎ + b,x, where 2 var (bo) , ) = and _ (, ) =-——" ~ : ( - x)? > -) ( - x)? i=l] i=] i=1 For a given %= 20, one can predict the value of a new observation 70 as Yo = 50 + b,x. Find an expression for var(Y) ¬ ) and compare it with var(Y9). - 0 Find , the standard deviation of - 8 JBNV-U-STSC/50 5 (0. () () (a) Consider a pdimensional random vector X with mean u and variance-covariance matrix ». Let (uy, oy Up) be an orthonormal system of eigenvectors of { with corresponding eigenvalues A, >Ag >--- 2A p: Denote U=(uj, --, Up). Let Y =(¥;, ---, Y,)’ be the vector of principal components of X. Then for any pair (i j)e {1, ---, p}, show that cov(X;, Yj) = ug; win. (Xj, Y;) =U Ness {var (X;) where uj =(u;); =( jth element of U. From the following population based on SRSWOR scheme with n = 3, verify that sample mean y and sample variance s? are unbiased for population mean Y and variance S?. Also show that the sampling variance of y agrees with the expression for V(Y) = sh. = : i ¥; 1 5 2 8 3 3 4 11 5 9 Define main effects and interaction effects in a 2? factorial experiment. Consider an experiment with two factors, reactant concentration (A) and catalyst (B). Let the two levels of factor A be 15 percent (ag) and 25 percent (a;) concentration. The two levels of factor B are 2 pounds (b,) and 1 pound (bo) of the catalyst. The order in which the runs are made is random. The data obtained are as follows : Treatment Replication <n es Total combination Fi IT vii Agbo 28 25 27 80 abo 36 32 32 100 aAgh 18 19 23 60 a,b, 31 30 29 90 Obtain the best estimates of main effects and interaction effects. Test for the significance of main effects and interaction effects. Set up the ANOVA table. JBNV-U-STSC/50 6 8 (0) Observations 7], --., described by the model Y; = 8202 4-5, where ‎ " X, are fixed constants and ], ---, are iid Normal (0, 2). Find the following : ( Least squares estimate of |} ‎ Maximum likelihood estimate of |} (iii) Best unbiased estimate of |} 8 6. (a) The sample mean vector and covariance matrix, computed on the basis of a random sample of size 20 from a bivariate normal population ‎ (11, 2), are given below : _ (10 40 -50 x= $ S= 20 -50 100 i) 7 8 Evaluate the test statistic for testing Hp : = (11) -(, i} 2 (1) Test 770 at 5% level of significance. What conclusion can be reached? (iii) Find the simultaneous confidence interval for ||]. (Fy, 19, 0.05 = 4-38 ‌ 19, 0.01 = 8-18 ‎ 18, 0.05 > 355 22, 18, 0.01 = 6:01) 15 (b) What is orthogonal factor model in factor analysis? Write down an orthogonal factor model with m common factors together with the covariance structure. Consider three standardized random variables 21, 22 and Z3 with a single factor (71 = 1) : 21 0-9 +£] 22 =0-7F, 2 23 =0-SF, +£ 3 where var(F,) = 1, 00 (, ) =0 and 0.19 0 0 = () = | 0 051 0 0 0 0-75 Obtain the covariance matrix. 10 ( Suppose an analyst uses the model y; = 80 + 01 x; +€; instead of the true model =Bo +B x; 40327 +e; with cov(y) = 1], ‎ = (1, . 7)". (0 Obtain (80) and (87) (in terms of true model parameters), if the observations are taken at x = -3, - 2, =], 0, 1, 2, 3. Note here that Bo and By are the least square estimates of Bg and 01. (ii) Find E(s?) for the same set of values of x, where 5 is the sample variance for the true model. 15 JBNV-U-STSC/50 7 [ P.T.O. 7. (a) (i For the following design, write the C-matrix (information matrix of the design) and obtain independent estimable treatment contrasts : aN WNNWHN Prep LH The number denotes the treatments. Also examine whether it is a connected design. 10 (ii) Define confounding. Suppose that 2* =16 treatments cannot be run using one batch of raw materials. The experimentor can run eight treatment combinations from a single batch of materials. A 2* confounded in two blocks is run. Construct a design with two blocks of eight observations each with ABCD confounded. 5 (b) (i Describe the probability proportional to size sampling with replacement scheme. Obtain an unbiased estimator for the population total along with its sampling variance under this scheme. 10 (ii) Estimate the gain in efficiency due to PPS sampling compared to SRS sampling based on SRS sample. 5 (c) In sampling with unequal probabilities, without replacement a sample of size 2 is drawn. The first unit is drawn with PPS and the second unit with PPS of remaining units. Show that Yates, Grundy and Sen’s variance estimator is always positive for this sampling system. 10 8. (a) (i) Let a’ais a vector of (v -t) independent estimable treatment contrasts from an incomplete block design. Obtain the test statistic for testing the hypothesis Ho : a’a =0. 7 (ii) Obtain the best estimate of treatment effect and derive the test statistic for testing the linear hypothesis of equality of treatment effects given yield from a balanced incomplete block design. 8 (b) Describe two-stage sampling. Obtain the estimator of population total when SRSWOR is used at both the stages. 10 JBNV-U-STSC/50 8 (c) Show that under the usual assumptions for one-way ANOVA model 967 ರಟ ಫರ್ಮು 8-0, var(e;;)=07 ೦೦೪೭), 811) = 0, £ × ৮, ] * J, 8015 are iid NO, ୦2) Vi, j abet k নি عو‎ evi 40/57 ക. 93 छ टर्छ, + Ao sys = 1-0 ੧੨194 ୮ =1 छि) simultaneously for all ৫ = (लु, ` ` `, cx), where 2 1 2 ଶ୍ର 7 Ay = கவட নি › ॐ ರಾಚ ல £, ऊ (౧-1), ८ 0 -1) हन] 7=1 i=1 টু JBNV-U-STSC/50 9 15 [ P.T.O. Normal Distribution Table 2 2 20:09 20.08 20೦.೦7 20.06 20೦.೦5 20.೦4 20.03 20.02 -0-01 0-00 2 -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 ০0002 ০0002 - 3.50 - 3:40 "0002 “0003 “0003 0003 0003 -0003 0003 ০0003 ০0003 ০0003 - 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 ০0007 - 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 ০013 -3-00 ൦ 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 “୦୦36 “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 - 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 ০0116 -0119 «0122 *9125 "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 ০0418 “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 81 - 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 1ء 1ء 1230- 41210 1190" 0ء‎ +1292 1314 +1335 +1357 -1:10 -1:00 1379 -1401 -1423 -1446 -1469 "1492 "1515 +1539 "1562 "1587 -1-00 -0.90 1 +1635 -1660 -1685 -1711 -1736 -1762 -1788 -1814 +1841 -0-90 - 9-80 -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 - 0:66 1 2483 -2514 2546 - 2578 2611 -2643 -2676 -2709 ‘2743 -0-60 ൦0-50 :2776 “2810 2843 2877 -2912 ` 2946 -2981 -3015 -3050 -3085 -0-50 - 0:40 1 "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 ೭4723 -4761 - 4801 পা 4081 3721 476] :4801 4840 -4880 "4920 -4960 :5000 0.00 _ JBNV-U-STSC/50 10 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 - 5636 -5675 - 5714 -5753 0-10 0-20 - 5793 - 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 ˆ 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 - 7734 ` 7764 - 7794 7823 - 7852 9-70 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 0 - 8830 1-10 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 “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 - 9861 -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 “922 - 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 94974 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 -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 × # JBNV-U-STSC/50 11 25SS—96‏ ہے جج 111 ೩೨41114‏ & ' le ಎಂಟ್‌ | ४ | ॥- ନ୍‌ = ef ந + ५ = # ~ = ¦ 1४ नि 1 त wt এ! ےر‎ ಕ جا‎ ॥ ^ + ର _ १, ഇ ہجام‎ कि ட पतभ চকা a + ˆ 7 வன்‌ ٤ ঢু 2 ۱ ୮ लि [हि ள்‌ ¬ ఆ ۰ ತಂ + ॥ ঢু ॥ 1 ५.५ 5 ச, + ۲ ۲ মৰ پ ہس‎ nf ۳ 3 ای‎ | + ঢু ॥ ୮ °: 76 உ حفٹت‎ = তল ne “= ۰ ட ०६ = ¢ ક ପକ ఆ oa তু య్య છ ঢ় 7ب‎ | ੧ கக = ह 5 রয়ে પા | है लि ٠-٤ হু लि ர પ യം ~ କ ‘ ॥ ఆ : ५ ४ ५५ a | ঢ় ਨ ୬୨୧ ٭‎ छ, 1 உண்‌ 55 ¦ „¥ ¦ ৷" ¬+ ঢ় न 15 : ۲ bet ತ ۲ +. rig? ୧ + ¢