Statistics UPSC PYQ 2024

(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. <!--qid:MAINS_2024_Statistics-II_Q2-->

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. <!--qid:MAINS_2024_Statistics-II_Q1-->

(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 $200. Find the following : (i) Economic Order Quantity (EOQ) (ii) Production run length (iii) Number of production runs per year (iv) Maximum Inventory Level (c) (i) Explain the terms ‘chance causes’ and ‘assignable causes’ of variation in quality control. Also provide some principal advantages of statistical quality control. (ii) Describe the procedure of obtaining OC curve for a single sampling plan.3a:["$"," <!--qid:MAINS_2024_Statistics-II_Q4-->

(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. <!--qid:MAINS_2024_Statistics-II_Q3-->