QP-IFSM-25-STATISTICS-PAPER-II-241125

Indian Forest Services (Main) Examination-2025 DJSM-B-STSC STATISTICS Paper - II 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. DJSM-B-STSC I (nisM) esoivise tes107 nsibn! eSHOTRIONA Ql. (a) Define hazard function. Obtain the survival function of the following hazard function : h(t) = i t>0, O%, B, a, By > 0. 8 (og + a,0) (By +B,0)’ (b) The diameter of one end of a drive-shaft is required within 1140 + 10. The control charts for X and R-charts are initiated. After 30 sub-groups of 5 shafts each have been examined, YX = 34,290 and YR = 330 (i) Establish the mean uw’ and standard deviation o’ of the process assuming that it is in statistical control. (ii) | Determine the 3-sigma limits of X and R-charts. (iii) Determine the natural specification limits of the process. (Given that dg = 2-326, Ag = 0:58, Dg = 0 and Dy = 2-11) 3+34+2=8 (ce) Four teachers A, B, C and D are capable of teaching any of the four different courses a, b, c and d. The course preparation time (in hours) taken by four teachers are given in the table below. Each teacher is assigned only one course. How are the courses assigned to teachers so as to minimize the total course preparation hours ? 8 Courses a b c d Teachers A 7 15 14 12 B 20 9 19 13 C 18 19 21 16 D 9 20 18 14 DJSM-B-STSC 2 Q2. (d) (e) (a) (b) Two players A and B, without showing to each other, put their coin on the table. A wins = 8 when both coins show head and = 1 when both coins show tail. B wins = 3 if upper faces of the coins do not match. (i) Write pay-off matrix for both players. (ii) | Determine optimum strategies for both players and the value of the game. A woman buys three type of oils, A, B and C. She never buys the same oil on successive weeks. If she buys type A oil one week, then the next week she buys type B oil. However, if she buys either type B or type C oil, then the next week she is three times as likely to buy type A oil as the other types. Obtain transition probability matrix and determine steady-state probabilities of buying three types of oils. (i) Explain and bring out the distinction between Acceptance Quality Level (AQL) and Average Outgoing Quality Limit (AOQL). (ii) Describe the method of double sampling plan. Derive its Operating Characteristic (OC), Average Sample Number (ASN) and Average Total Inspection (ATI). Suppose that a given individual in a population has a survival time which is exponential with hazard rate 6. Each individual’s hazard rate 0 is potentially different and is sampled from a gamma distribution with density function : B oB-1 20 f() = rP OP e 8>0, T(B) A,B >0 Let X be the life length of a randomly chosen member of this population : @ Find the survival function of X. Gi) Find the hazard rate of X. What is the rough shape of the hazard function ? DISM-B-STSC 3 15 10 Q3. (c) (a) (b) (c) Explain parallel systems and their reliability function. Consider a system, consisting of n components such that the failure of the ith component occurs in accordance with a Poisson process of fixed intensity. Compute the reliability and expected life of the system under parallel system structures. Solve the following transportation problem in which cell entries represent unit costs : Wi Wo W3 W, ~~ ~Available Fy 19 30 50 10 7 Fp 70 30 40 60 9 Fs 40 8 70 20 18 Requirement 5 8 7 14 Also, find minimum transportation cost. Use simplex procedure to solve the following linear programming problem : Maximize (z) = 2x; — 3x9 + x3 subject to constraints : 3x1 + 6xq + X36 4x1 + 2xqg+xg<54 Xj —-Xg+x3S3 X1, XQ, X3 20. An item used in the manufacture of surgical equipment has an annual usage of 24,000 units, which costs = 1-25 per unit. Placing each order costs = 22:50 and the carrying cost is 5-4% per year of the average inventory. Find the economic lot size and inventory cost (including material cost). Should the company accept the offer made by the supplier of a discount of 5% on the cost price, on a single order of 24,000 units ? DJSM-B-STSC f 15 15 15 10 Q4. (a) (b) (c) (i) (ii) A random sample of size n is drawn from a Normal Population N (w’ + ko’, 0). Then show that P,= P Sample mean > pw’ +3 - Vn =1-0(3-k Vn) where ®(-) is the c.d.f. of Standard Normal Distribution. If rth sample mean is the first to exceed G + sz), then show n that E(r) = ote 15 n Determine the optimal sequence of performing 5 jobs on 4 machines. For each job machines are required in the order of A, B, C and D and the process timings are as follows : Machine Job A B Cc D 1 8 3 4 7 2 9 2 5 5 3 6 4 5 8 4 12 5 ds 9 5 7 1 2 38 Also, find minimum total elapsed time. 15 Explain censored and truncated experiment. Describe life testing censored and truncated experiments for exponential models. 10 DISM-B-STSC Q5. (a) (b) (c) (d) (e) Q6. (a) (b) (c) SECTION B What is stationary time series ? Explain Augmented Dickey-Fuller test and interpretation of the test results. Explain the utility of index number of industrial production. Describe in detail, a method of its construction. Describe the functions of Central Statistical Organisation. Describe the different methods of scaling of test scores in several tests, stating the necessary assumptions in each method. Why is population projection important for development ? Describe the mathematical method and component method for population projection. Discuss the identifiability of the following model assuming ore 0 > =| 0 S59 0 0 0 «6 Yt Bo ¥y + Yo + Yo ¥1 = Up B31 ¥y + Bgo Yo + ¥g + ¥g1 ¥y = Uy: (i) Explain Dimensional Invariance test in the case of price and quantity. (ii) | Calculate Fisher’s Ideal Index Number from the following data and verify whether it satisfies the mathematical tests for formula error. Base Year Current Year Commodity Price per Expenditure Price per Expenditure unit @ unit ® A 2 40 5 15 B 4 16 8 40 C 1 10 2 24 D 5 25 10 60 Discuss the collection of trade statistics in India with special reference to inland and internal trade of India. DJSM-B-STSC 6 15 5 10 10 Q7. (a) (b) (c) Q8. (a) (b) DISM-B-STSC What do you mean by measure of morbidity ? Explain the problems that occur while morbidity rates are constructed. Define Morbidity Incidence Rate (MIR) and Morbidity Prevalence Rate (MPR). When would MPR approximate MIR ? Find the equivalence curve of y and x if both x and y have exponential distributions with parameters 0, and 8, respectively. Further, if x and y are both normally distributed with means uw, and Uy, and variances o and oy respectively, then show that the equivalence curve method reduces to z scaling. The following table gives the mean and standard deviation of scores and the raw scores of two students in three examinations A;, Ag and Ag. Compare the overall performance of two students. Write your conclusion. Raw score of student Examination | Mean S.D. tt I Ay 150 25-4 190 160 Ag 35 10-2 30 55 Ag 95 13-5 140 110 Consider the following data on monthly expenditures for clothing and on incomes of 10 families. Assume that the variance of the disturbance term is proportional to the square of expenditure on clothing. Obtain the approximate transformation so as to make the resulting disturbance term homoscedastic. Compare the regression equation of the transformed data with that of the original data. 15 15 10 15 Expenditure on clotitinas #8): 20 25 30 40 45 50 55 60 65 Income (®) : 1500 | 2000 | 3000 | 3500 | 4500 | 4500 | 5500 | 6000 | 5500 The /, column of a certain life table for ages 0, 1, 2, 3, ..., 100 is given by the series 100, 99, 98, 97, ..., 0. In usual notations, calculate the following values : ef» T5s Lz. Aig» Pao» G25 and dzo. 7 15 (c) Define various measures of fertility. Compute GRR and NRR for the = following data : hoe Female Total no. of female | Survival rate population (100) live births (100) (per 100,000) 15-19 157680 4642 58056 20 - 24 147524 14433 55880 25 — 29 124300 14068 52971 30 — 34 ~ 105875 8429 48953 35 - 39 89365 4046 44164 40 - 44 TT797 2168 39145 45 -49 61171 698 34189 DJSM-B-STSC 10