QP-IFSM-23-STATISTICS-PAPER -II-051223

Time Allowed : Three Hours INDIAN FOREST SERVICE (MAINJEXAM- 2023 STATISTICS Paper - II BJKE-B-STSC 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-B-STSC Ql. (a) (b) (c) (d) (e) BJRE-B-STSC SECTION A Derive a control chart to test p = 400 given that o = 30 such that the chances of a point going out of the control limits when the process really is in control are 1 in 20 while the chances of detecting a shift of to 425 within two samples after a shift occurs are 0-99 percent. For the same chart, what are the chances of detecting a shift of u from 400 to 380 within two samples ? Define renewal process. Derive the renewal function. Represent a general linear programming problem consisting of n variables and m constraints, in its canonical and standard forms. Given the following linear programming problem : Maximize z = x; + 1-5x, subject to the constraints 2x, + 2x5 < 160 xX, + 2x. < 120 4x, + 2x. < 280 Xj, X2 20 find the solution of the problem using graphical method. Define the pay-off matrix in a two-person zero-sum rectangular game. Obtain a necessary and sufficient condition under which the game possesses a saddle point. A study about a population showed that the mobility of the population of a state to a village, town and city is in the following percentages : To From Village Town City Village 60 25 15 Town 10 70 20 City 10 30 60 What will be the proportion of population in the village, town and city after one year and two years, given that the present population has proportions of 0-50, 0-40 and 0-10 in the village, town and city respectively ? Standard Normal Probabilities PIZ < 2] 2 00 01 02 03 „04 ‘05 ‘06 07 08 09 0.0 -5000 .5040 .5080 -5120 -5160 -5199 239 -5279 -5319 -5359 0.1 .5398 .5438 .5478 5517 ಪಚ್‌ 5596 5636 .5675 714 753 0.2 793 5832. 871 5910 5948 .5987 6026 -6064 -6103 6141 0.3 -6179 6217 .6255 -6293 -6331 .6368 -6406 6443 -6480 -6517 0.4 .6554 -6591 .6628 -6664 -6700 -6736 .6772 -6808 -6844 -6879 0.5 .6915 -6950 .6985 .7019 .7054 -7088 #123 ‘7157 ‘7190 7224 0.6 ‘7257 7291 7324 ‘7357 .7389 .7422 .7454 -7486 క .7549 0.7 -7580 1 2 .7673 .7704 .7734 .7764 .7794 -7823 -7852 0.8 7881 0 .7939 .7967 .7995 .8023 .8051 -8078 .8106 .8133 0.9 -8159 .8186 .8212 .8238 .8264 -8289 -8315 -8340 .8365 -8389 1.0 8413 8438 -8461 8485 8508 8531 8554 8577 .8599 .8621 বড 8643 8665 .8686 -8708 „8729 .8749 .8770 -8790 .8810 .8830 1.2 8849 8869 :8888 8907 8925 .8944 ৪962 .8980 8997 -9015 1.3 -9032 -9049 -9066 -9082 -9099 .9115 -9131 -9147 -9162 -9177 1.4 -9192 -9207 -9222 .9236 .9251 -9265 -9279 -9292 -9306 -9319 ES -9332 .9345 9357 9370 9382 9394 .9406 9418 9429 ೬9441 1.6 9452 9463 9474 9484 9495 9505 9515 9525 -9535 -9545 1.7 -9554 ೨564 9573 -9582 -9591 9599 -9608 -9616 -9625 -9633 1.8 -9641 -9649 -9656 -9664 -9671 -9678 -9686 -9693 -9699 -9706 1.9 -9713 .9719 -9726 -9732 -9738 9744 -9750 -9756 -9761 -9767 2.0 2 .9778 .9783 -9788 -9793 -9798 -9803 -9808 -9812 -9817 2:1 -9821 -9826 -9830 .9834 .9838 -9842 -9846 -9850 -9854 -9857 2.2 -9861 -9864 -9868 9871 9875 9878 .9881 -9884 -9887 -9890 23 -9893 -9896 -9898 -9901 -9904 -9906 .9909 .9911 -9913 -9916 2.4 9918 9920 9922 9925 -9927 -9929 -9931 -9932 -9934 -9936 25 -9938 -9940 .9941 -9943 -9945 -9946 -9948 .9949 .9951 9952 2.6 9953 .9955 .9956 .9957 -9959 9960 9961 9962 -9963 -9964 2.7 -9965 -9966 -9967 -9968 -9969 .9970 .9971 -9972 -9973 -9974 2.8 -9974 9975 9976 9977 9977 -9978 9979 ೨979 -9980 .9981 2.9 -9981 -9982 -9982 .9983 -9984 ೨984 9985 -9985 ೬೨೨986 ೨೨86 3.0 ೨9987 -9987 -9987 -9988 -9988 -9989 -9989 -9989 -9990 -9990 3:1 -9990 -9991 -9991 .9991 .9992 -9992 -9992 -9992 -9993 -9993 me -9993 -9993 -9994 -9994 -9994 -9994 -9994 -9995 -9995 -9995 33 -9995 -9995 -9995 -9996 -9996 -9996 -9996 -9996 -9996 -9997 3.4 .9997 .9997 9997 9997 .9997 -9997 -9997 -9997 -9997 -9998 3.5 -9998 -9998 -9998 -9998 -9998 -9998 -9998 -9998 -9998 .9998 BJKE-B-STSC 3 Q2. (a) (b) (c) Q3. (a) BJKE-B-STSC Find a sequential plan for which, in the usual notation p, = 0-01, a = 0:05, py = 0-06 and f = 0-10. Display the plan in a table for the first 46 units. @ A system is composed of five identical independent elements in parallel. What should be the reliability of each element to achieve a system reliability of 0-96 ? (ii) Three identical units each with a reliability of 0-9 for a system are operating in parallel. What is the system reliability if at least one of the units is required for the system to be successful ? If you add another unit with identical reliability characteristics, what will be the increase in system reliability ? Consider a system where m — 1 units are standing by to support one basic unit. Obtain the system reliability assuming the units are identical. Show that for a two unit system the reliability of a standby system is better than that of an active redundant system. What is the basic philosophy behind sensitivity analysis ? What are the issues which could be answered using sensitivity analysis in linear programming problems ? For the following linear programming problem : Maximize z= 3x, + 2x» subject to the constraints X1 + 2X5 <6 2x, +X_ <8 —X,+X_ <1 X_ <2 Xy, X_g20 the optimal simplex table is obtained as 3..2°°0 0. 02:0 i | Basis | Cost | Solution By, XQ%2 By 4 8o° Bane 1] xX, 2 4/3 0 1 2% -1/3 0 O 2: 3 10/3 T 0°13" 228.020 3] 83 0 3 0 0 -1 Beier bg) 4 S4 0 2/3 0 083228 SIR Oat z 38/3 O° 20° 132 AB 0 20 where s; (i = 1, 2, 3, 4) denotes the slack variable used in the ith constraint. 15 10 (b) (c) BJKE-B-STSC (i) Find the status and worth of each resource. (ii) | Find the possible range of the cost coefficient associated with the variable x, for which the optimum solution remains unchanged. 15 Explain what is a Transportation Problem (TP). Differentiate between balanced and unbalanced transportation problems. Four petrol dealers A, B, C, D require 50,000, 40,000, 60,000 and 40,000 litres of petrol respectively. It is possible to supply these demands from three locations 1, 2 and 3 which have 80,000, 1,00,000 and 50,000 litres respectively. The costs (in rupees per thousand litres) to ship the petrol from different locations to different dealers are shown in the following table : Dealer Location Supply A B Cc D 1 70 60 60 60 8 2 50 80 60 70 10 3 80 50 80 60 5 Required 5 4 6 4 (The supply and requirement values in the table have been multiplied by 10-4 to make computations easier) Determine the amount of petrol to be shipped from each location to each dealer so that all the dealers’ requirements are satisfied and the total shipping costs are a minimum. 15 Prove that if the pair of linear programming problems Maximize Z = C'X subject to AX <b X 20 and Minimize Z* = b'Y subject to A'Y =C Y=0 have feasible vectors X and Y, then they have optimal vectors such that Maximum C’X = Minimum b’Y. 10 Q4. (a) (b) (c) BJKE-B-STSC Construct a V-mask for the following data with » = 10 and o = 1 and comment. (Given a = 0:0027, B = 0-01, 5=1) Period x. Period x. Period _ . i . i ; i i i i 1 9-45 11 9-03 21 10-90 2 7-99 12 11:47 22 9-33 3 9-29 13 10°51 23 12:29 4 11-66 14 9-40 24 11:50 5 12-16 15 10-08 25 10-60 6 10-18 16 9:37 26 11-08 7 8-04 17 10-62 27 10:38 8 11-46 18 10°31 28 11-62 9 9-20 19 8-52 29 11:31 10 10-34 20 10-84 30 10°52 Distinguish between deterministic and probabilistic inventory problems. Discuss the continuous case of a probabilistic inventory model with instantaneous demand and no set-up cost. A sweet company sells chocolate-based sweets by the kilogram (kg). On every kg of these sweets sold on the day of its preparation, the company makes a profit of = 0-55 per kg. The company disposes of all the chocolate-based sweets not sold on the date of its preparation, at a loss of € 0-22 per kg. If the demand is known to be rectangular between 2000 and 4000 kg, determine the optimum daily amount of sweets prepared. Obtain the steady-state solution for P,, the probability of n customers at any arbitrary point of time in the (M/M/C) : (oo/FIFO) queue system. Hence, find the formulae for (i) Traffic intensity, (ii) The probability that an arriving customer has to wait, and (iii) The average number of customers in the queue. 15 10 15 SECTION B Q5. (a) What is an index number ? Describe the general lines on which you proceed to construct a cost of living index for factory workers in an industrial area. 8 (b) Explain seemingly unrelated regression model. Obtain the generalized least squares estimator of the parameter vector when the covariance matrix is diagonal. 8 (c) Explain the two series of acreage statistics available in India. 8 (d) What are the defects in comparing mortality situations between two regions by means of their crude death rates ? Explain how these defects can be eliminated. 8 (e) Describe different methods of converting raw scores into some equivalent comparable scores in Psychometry. 8 Q6. (a) (i) Obtain the autocorrelation function of AR(2) process and show that the process is covariance stationary. 8 Gi) Obtain the best forecast in the mean square sense for ARIMA (p, d, q) process. 7 (b) For a model : Yyt = Byo Yo + M11 Xe + Yat Yor = Boi Vit + Yoo Xot + Yo3 Xat + Use 1 O 0 5 610 XX=|0 20 0] and XY=|40 20 0 oO 10 20 30 Determine the identifiability of each equation and estimate the parameters by using an appropriate method. 15 (c) | Explain the existing machinery for collection of births and deaths in India and its shortcomings. 10 BJKE-B-STSC 7 Q7. (a) (b) (c) Q8. (a) (b) BJKE-B-STSC Describe the major sources of demographic data in India. 15 Calculate (i) General Fertility Rate (ii) Total Fertility Rate, and (iii) Gross Reproduction Rate from the following data. Age group of females | No. of females (000) | Total Births 15-19 16-0 260 20 — 24 16-4 2244 25 — 29 15:8 1894 30 — 34 15-2 1320 35 — 39 14:8 916 40 — 44 15-0 280 45 — 49 14-5 145 Assume that the proportion of female births is 46-2 percent. 15 Explain the concept of validity of a test. Discuss the different types of validity. 10 Compute Laspeyres’ quantity and price index numbers for 2005. : 1995 2005 Commodity : . Quantity Value Quantity Value A 50 350 60 420 B 120 600 140 700 Cc 30 330 20 200 D 20 360 15 300 E 5 40 5 50 Also show that Fisher’s ideal index number satisfies factor reversal test. 15 The model appropriate for the following data is given by Y; =at BX, + u; where v(u;) = x, pee Donne m. Obtain the generalized least square estimates and their variances. 15 marcas, 6. le 1011 X,: 23 4 5 6 Ae ee Gee a a (c) BJKE-B-STSC A part of a life table is given with most of the entries missing. On the basis of the available figures, complete the missing ones. Age x l, dy dx L, Ty 87 34 — | 0:2353 - - 88 - — | 0-2308 ~ - 89 - — | 0-2500 ~ - 90 - — | 0:2667 — | 35-5 9 10