Statistics — UPSC Mains Optional PYQ (2023–2025)

(a) State the significance of operating characteristic (OC) curves in control chart analysis. Obtain the general expression for the OC function corresponding to the mean ( X̄ ) chart under the assumption of normal distribution for a quality characteristic. Using the expression, find the probability that a shift will be detected from μ0 to μ1 = μ0 + 2σ, when an X̄ chart is used with 3σ limits, where the subgroup size is n = 6. (Standard normal table is provided.) (b) What is meant by rectifying inspection? Explain the measures associated with rectifying inspection and derive the expressions of such measures in the case of a single sampling plan by attributes. (c) The lifetime of a semiconductor laser has a log-normal distribution with parameters μ = 10 hours and σ = 1·5 hours. (i) Find the probability that the lifetime exceeds 10000 hours. (ii) What lifetime is exceeded by 99 % of lasers? (Standard normal table is provided.) (d) A stockist has to supply 400 units of a product every Monday to his customers. He gets the product at ₹ 50 per unit from the manufacturer. The cost of ordering and transportation from the manufacturer is ₹ 75 per order. The cost of carrying inventory is 7·5 % per year of the cost of the product. Find (i) the economic lot size, (ii) the total optimal cost (including the capital cost) and (iii) the total weekly profit, if the item is sold for ₹ 55 per unit. (e) On the average, 96 patients per 24-hour day require the service of an emergency clinic. Also, on the average, a patient requires 10 minutes of active attention. Assume that the facility can handle only one emergency at a time. Suppose that it costs the clinic ₹ 1,000 per patient treated to obtain an average serving time of 10 minutes, and that each minute of decrease in this average time would cost the clinic ₹ 100 per patient treated. How much would have to be budgeted by the clinic to decrease the average size of the queue from 1⅓ patients to ⅓ patient?

(a) (i) What are control charts by variables and control charts by attributes? (ii) Derive the control limits for the construction of control charts for the mean and variability based on sample standard deviation. (b) (i) State the assumptions involved under sampling inspection plans by variables and describe the operating procedure of a single sampling plan by variables under the assumption of normal distribution for a quality characteristic. (ii) Establish the relationship between the fraction defective and the acceptance probability under a single sampling plan by variables when the quality characteristic follows a normal distribution with mean μ and variance σ², where σ² is unknown, and when an upper specification limit is specified. Using the relationship, obtain the formula for finding the parameters of the sampling plan. (c) (i) Given a system consisting of n components, define the state vector and the structure function of the system. What do they indicate? (ii) Defining (1) a series system, (2) a parallel system and (3) a k-out-of-n system, obtain the associated expressions for the structure functions and the reliability functions.

(a) A company manufactures 30 items per day. The sale of those items depends upon demand which has the following distribution: 27, 28, 29, 30, 31, 32 units with respective probabilities 0·10, 0·15, 0·20, 0·35, 0·15 and 0·05. The production cost and selling price of each unit are ₹ 400 and ₹ 500 respectively. Any unsold product is to be disposed off at a loss of ₹ 150 per unit. There is a penalty of ₹ 50 per unit if the demand is not met. Use the random numbers 23, 99, 65, 99, 95, 01, 79, 11, 16, 10 to estimate the total profit/loss for the next 10 days. If the company decides to produce 20 items per day, what is the advantage or disadvantage to the company? (b) A company has four plants P1, P2, P3 and P4 from which it supplies to three markets M1, M2 and M3. Determine the optimal transportation plan from the given data showing plant-to-market shipping costs, quantities available at each plant and quantities required at each market. (c) On January 1 (this year), brands A, B and C of a commodity had 40 %, 40 % and 20 % of the market share respectively. Market research shows that brand A retains 90 % of its customers, gaining 5 % of B’s customers and 10 % of C’s customers; brand B retains 85 % of its customers, gaining 5 % of A’s customers and 7 % of C’s customers; brand C retains 83 % of its customers and gains 5 % of A’s customers and 10 % of B’s customers. What will be each brand’s share on January 1 next year, and what will be each brand’s share in the market at equilibrium?

(a) Solve the game whose payoff matrix is [ −1 −2 8 7 5 −1 6 0 12 ]

Assume that in a population of very large number of items, proportion of defective items is 0.30. What should be the size of the sample, if a simple random sample is to be drawn from this population to estimate the percent defective within 2 percent of the true value with 95.5 percent probability? [Given P(0 ≤ Z ≤ 1.96) = 0.475; and P(0 ≤ Z ≤ 2.005) = 0.4775]

If (X, Y) follows bivariate normal BN(μ1, μ2, σ1², σ2², ρ), then obtain (A) E(e^X) (B) E(e^{X+Y}) (C) Var(e^X) and (D) Correlation between e^X and e^Y.

Analyse and interpret the following data concerning output of wheat per field obtained as a result of experiment conducted to test four varieties of wheat A, B, C and D under a Latin square design at 5% level of significance. [Given F(3, 6) = 4.76; F(4, 7) = 4.12] C B A D 35 33 30 30 A D C B 29 29 31 28 B A D C 29 24 27 30 D C B A 27 30 31 25

What are principal components? Show that the principal components are uncorrelated.

Consider the following five sub-questions. Answer all of them. (a) Consider a system consisting of three identical units connected in parallel. The unit reliability factor is 0·90. If the unit failures are independent of one another, and if the successful operation of the system depends on the satisfactory performance of any one unit, determine the system’s reliability. (b) Describe the procedure and some of the applications of the Cumulative Sum (CUSUM) chart for monitoring process mean. (c) Explain the following terms as used in sampling inspection plans : (i) Producer’s risk (ii) Average Outgoing Quality Limit. (d) A Linear Programming Problem (LPP) in standard form is given below : Optimise Z = CᵀX subject to AX = B, with X ≥ 0 Write down the Dual Simplex form and its iterative procedure. (e) What is Monte Carlo Simulation? State the uses and applications of Monte Carlo Simulation.

(a) Obtain the control limits for X̄-chart and R-chart and describe the significance of joint study of these charts. (b) Find the reliability and hazard functions of Weibull distribution with scale parameter θ and shape parameter β, and interpret the findings. (c) Use simplex method to solve the following Linear Programming Problem (LPP) : Maximise z = 5x₁ + 2x₂ subject to 6x₁ + x₂ ≥ 6, 4x₁ + 3x₂ ≥ 12, x₁ + 2x₂ ≥ 4, x₁ ≥ 0, x₂ ≥ 0.

(a) With respect to a given Linear Programming Problem (LPP), explain the following concepts : (i) Extreme Point Solutions (ii) Duality Theorem (iii) Complementary Slackness Principle. (b) Define a Transition Probability Matrix (TPM). When is it said to be Regular and Ergodic? Check whether the following TPM is Regular or Ergodic. Hence or otherwise obtain the limit, as n → ∞, of Pⁿ where P = [0·88 0·12; 0·15 0·85]. (c) The reliability function R(t) of a cutting assembly is given by R(t) = {(1 − t / t₀)² , 0 ≤ t ≤ t₀; 0 , t ≥ t₀} (i) Determine the failure rate. (ii) Does the failure rate increase or decrease with time? (iii) Determine the mean time to failure.

(a) A manually handled toll-booth has two tellers, who are each capable of handling an average of 60 vehicles per hour, with the actual service times exponentially distributed. Vehicles arrive at the booth according to a Poisson process, at an average rate of 100 per hour. Determine the following : (i) The probability that there are more than three vehicles in the booth at the same time. (ii) The probability that a given teller is idle. (iii) The probability that a vehicle spends more than three minutes in the booth. (b) PQR Electronics produces 300 transistors per day, which go into the inventory. It supplies 150 transistors per day to XYZ Radios. The annual demand is 37,500 units. The inventory holding cost is $0·25 per transistor per year and the set-up cost per production run is $200. Find the following : (i) Economic Order Quantity (EOQ) (ii) Production run length (iii) Number of production runs per year (iv) Maximum Inventory Level (c) (i) Explain the terms ‘chance causes’ and ‘assignable causes’ of variation in quality control. Also provide some principal advantages of statistical quality control. (ii) Describe the procedure of obtaining OC curve for a single sampling plan.

Suppose that the life in hours of an electric gadget manufactured by a certain process is normally distributed with parameters μ = 160 hours and some σ. What would be the maximum allowable value of σ if the life X of the gadget is to have a probability 0.80 of being between 120 hours and 200 hours? (Normal distribution Table is given at the end.)

State and discuss the important points which have to be considered in the construction of index numbers.

Explain the principle of least squares. How is it used in fitting trend in time-series analysis? Explain the fitting of trend for the curve y = a bᵗ c^{t²}.

Derive, by starting from a suitable functional form for lx, the formula (i) Lx = (lx + lx+1)/2 and (ii) Lx = (lx − lx+1)/(log lx − log lx+1) = − dx / log px (iii) e0x = 1/2 + Σ_{i=1}^{∞} i dx+i / lx where lx = members of the cohort alive at age x Lx = number of years lived, in the aggregate, by the cohort of l0 persons between age x and (x + 1) dx = number of persons dying between age x and (x + 1) = lx − lx+1 px = probability that a person of age x will survive till age (x + 1) e0x = expectation of life at age x

Explain the method of collection of agriculture data. Describe the (i) official publications for data collection, and (ii) statistics collected by the various official agencies pertaining to agriculture production.