UPSC Indian Forest Service (IFoS) Main Previous Year Question Paper 2024 Statistics-II-Free PDF Download

UPSC Indian Forest Service (IFoS) Main Previous Year Question Paper 2024 Statistics-II

1. (a) A subgroup of 5 items each are taken from a manufacturing process at a regular interval. A certain quality characteristic is measured and X̄ and R values are computed. After 25 subgroups, it is found that Σ X̄ = 357.50 and Σ R = 8.80. If the specification limits are 14.40 ± 0.40 and if the process is in statistical control, what conclusion can you draw about the ability of the process to produce items within specifications. (Given that, for subgroup of 5 items, d₂ = 2.326, c₂ = 0.8407, d₃ = 0 and d₄ = 2.11) 8

(b) Give the relationship between survival function S(t), probability density function f(t), and hazard function h(t).

Also find : (i) survival function and hazard function for a density function f(t) = e⁻ᵗ, t ≥ 0 and (ii) probability density function and hazard function for the survival function S(t) = exp(−tʳ). 8

UPSC Indian Forest Service (IFoS) MAIN Previous Year Question Paper 2024 – statistics-II

(c) For a homogeneous Markov chain, define the probability of the first return to state i in n steps. Compute the same for a Markov chain with state space {1, 2, 3} and the following transition probability matrix, when i = 2 and n = 3.

[IMAGE: transition probability matrix missing/not visible]

In addition, if the initial probability P(X₀ = 1) = 1/2, P(X₀ = 2) = 1/4 and P(X₀ = 3) = 1/4, compute the probability P(X₁ = 1 | X₀ ≠ 2). 8

(d) Get an estimate of the integral ∫₀⁴ e⁻ˣ / (x² + e⁻⁴ˣ) dx through Monte Carlo simulation, using the following random numbers : 8
0.463, 0.802, 0.607, 0.455, 0.37
0.839, 0.401, 0.029, 0.843, 0.811

(e) Solve the following linear programming problem graphically.

Maximize z = 3x₁ + 2x₂ + 2x₃
subject to 2x₁ − 3x₂ ≤ 6
10x₁ + 2x₂ + 3x₃ ≤ 40
3x₁ + x₃ = 10
x₁ ≥ 0, x₂ ≥ 0 and x₃ is unrestricted.

Obtain the optimal values of x₁, x₂ and x₃. What is the optimal z?

Q2. (a) Suppose that 50 units are placed on life test (without replacement) and the test is to be truncated after 10 of them have failed. It was further supposed that the first 10 failure times are 65, 110, 380, 420, 505, 580, 650, 840, 910 and 950 hours. Estimate the mean life of the component and its failure rate. Also calculate a 90% confidence interval for mean time ‘μ’. (Assume that the exponential model holds) 15

Given that :

v | χ²₀.₀₅ | χ²₀.₉₅
20 | 31.41 | 10.85
21 | 32.67 | 11.59
31 | ? | ?

(b) The following table gives the number of missing rivets at aircraft final inspection :

Aircraft Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
Number of missing rivets | 8 | 16 | 14 | 19 | 11 | 15 | 8 | 11 | 21 | 12 | 23 | 16 | 9

Aircraft Number | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25
Number of missing rivets | 2 | 5 | 15 | 9 | 9 | 14 | 11 | 9 | 10 | 22 | 7 | 28 | 9

Find c̄, compute trial control charts and plot control chart for c. What value of c′ would you suggest for the subsequent period? 15

Q3. (a) Solve the following linear programming problem using two-phase method : 15

Maximize z = 2x₁ + x₂
subject to x₁ + 2x₂ ≥ 2
x₁ + x₂ ≤ 4
x₁, x₂ ≥ 0

(b) The cost of a new machine is ₹ 5,000. The running cost of nth year is given by Rₙ = 500 (n − 1), n = 1, 2 ... Assuming an interest rate of 5% per year, after how many years will it be economical to replace the machinery with a new one? 15

(c) The probability distribution of monthly sales of a certain item is as follows :

Monthly sales : 0 1 2 3 4 5 6
Probability : 0.04 0.06 0.25 0.35 0.15 0.09 0.06

The cost of carrying inventory is ₹ 20 per unit per month. The current policy is to maintain a stock of 4 units at the beginning of each month. Assuming that the cost of shortage is proportional to both time and quantity short, obtain the attributed range of cost of a shortage of one item for one unit of time. 10

Q4. (a) The following data is on defective electronic machine parts from a standard brand of manufacturer. Sample sizes and number of defectives in each sample are given below. Using suitable control limits, test whether the process is statistically controlled or not. 15

Sample Size | 115 | 220 | 210 | 220 | 220 | 255 | 440
Number of defectives | 15 | 18 | 23 | 22 | 18 | 15 | 44

Sample Size | 365 | 255 | 300 | 280 | 330
Number of defectives | 47 | 13 | 33 | 42 | 46

(b) Two companies A and B manufacture similar products and they are competing for an increased market share. The payoff matrix shown in the following table, describes the increase in market share of company A and decrease in market share of company B.

Company B →
Company A ↓ | Give Coupons | Decrease Price | Maintain Present Strategy | Increase Advertising
Give Coupons | 2 | -2 | 4 | 1
Decrease Price | 6 | 1 | 1 | 2
Maintain Present Strategy | -3 | 2 | 0 | 6
Increase Advertising | 2 | -3 | 7 | 1

Determine the optimal strategies for both the manufacturers and the value of the game. 15

(c) Mention the names of different sets of tables under Dodge-Romig system. Explain the CSP-1 as developed by Dodge. 10

5. (a) Define Crude Death Rate. Write merits and demerits of Crude Death Rate. Is Crude Death Rate an accurate measure of the mortality of population of a country? If yes, explain why. If not, how will you modify it to give reliable results? 8

(b) What do you mean by stable and stationary population? Derive the age distribution of a life table for stationary population. 8

(c) Explain the system of collection and publication of Industrial Statistics in India. 8

(d) Explain ratio to trend method for measurement of seasonal variation in a time series data. Discuss its merits and demerits. 8

(e) If L(p) and P(q) represent Laspeyres’ Index Number for prices and Paasche’s Index Number for quantities respectively, show that

L(p) × P(q) = V₀₁

where V₀₁ is the value index number.

Hence or otherwise show that L(p)/L(q) = P(p)/P(q)

Also, show that if A(p) = ½ [L(p) + P(q)], then A(p) is greater than the Fisher’s Ideal Index Number. 8

Q6. (a) For an autoregressive series of order two, namely

y_{t+2} + a y_{t+1} + b y_t = ε_{t+2}, |b| < 1

show that the correlogram ρ_k of order k is given by

ρ_k = p^k sin(kθ + ψ)/sin ψ, p = +√b

where tan ψ = (1 + p²)/(1 − p²) tan θ, cos θ = −a/(2p)

(b) Explain an ARMA (p, q) process and ARIMA (p, d, q) process. How does Box-Jenkins (BJ) method facilitate the identification of ARMA or ARIMA model? 10

(c) Consider the general linear model given by

Y = Xβ + U

where Y is an (n × 1) vector observation on the explained variable
X is an (n × k) matrix of observations on the explanatory variable
U is an (n × 1) vector of error
β is (k × 1) vector of parameters to be estimated

Assume that U ∼ N(0, σ²I_n) is independently and identically distributed and rank(X) = k < n

Show that β̂ is the least square estimator of β and is Best Linear Unbiased Estimator (BLUE) of β. Test the significance of the Null Hypothesis H₀ : β = β₀. 15

Q7. (a) Define the force of mortality and show that its approximate value is

μ_x = (d_{x−1} + d_x)/(2l_x), x ≥ 1.

If l(x) = 100√(100 − x), find μ(84) exactly and by using approximate method. 15

(b) State the scaling procedures developed for psychological research. Explain z-score and linear deviate score in brief.

The following table gives the mean and standard deviations of scores and the raw scores of two students in a number of examinations. Compare the overall performance of the two students. Give your conclusions. 15

Examinations | Mean | Standard Deviation | Raw Score of Students
I | II | III
A | 155.7 | 26.4 | 195 | 162
B | 33.7 | 8.2 | 20 | 54
C | 54.5 | 9.3 | 39 | 72
D | 87.1 | 25.8 | 139 | 84
E | 24.8 | 6.8 | 41 | 25

(c) What do you mean by projection of population? Explain how you would do adjustment of migratory disturbances while projecting population. 10

8. (a) Consider the following extended Keynesian model of income determination :

Consumption function : C_t = β₁ + β₂Y_t − β₃T_t + U_{1t}
Investment function : I_t = α₀ + α₁Y_{t−1} + U_{2t}
Taxation function : T_t = γ₀ + γ₁Y_t + U_{3t}
Income identity : Y_t = C_t + I_t + G_t

where C = Consumption expenditure
Y = Income
I = Investment
T = Taxes
G = Government expenditure
U’s = the disturbance terms

In the model, the endogeneous variables are C, I, T and Y and the predetermined variables are G and Y_{t−1}

By applying the order condition, check the identifiability of each of the equations in the system and of the system as a whole? Write your conclusion. 15

(b) Define Gross reproduction rate and Net reproduction rate. Distinguish between these two reproduction rates. Which one attains greater value? Why? Which one can be used for forecasting future population changes? Why? 15

(c) Explain quasi-stable population. Stating the underlying assumptions, derive the age distribution function for quasi-stable population with changing mortality. 10

Indian Forest Service (Main) Examination, 2024-STATISTICS II

UPSC Indian Forest Service (IFoS) Main Previous Year Question Paper 2024 Statistics-I

UPSC Indian Forest Service (IFoS) Main Previous Year Question Paper 2024 PHYSICS-II

UPSC Indian Forest Service (IFoS) Main Previous Year Question Paper 2024 PHYSICS-I