Statistics UPSC PYQ 2018

Let S_n be equal to the total obtained in n tosses of a symmetric die. Estimate an upper bound for P [ |S_n / n - 3.5| ≥ ε ] for ε > 0.

Let X_1, X_2, ..., X_n be a random sample from N(0, Θ). The variance Θ is unknown. Derive the most powerful test for H_0 : Θ = Θ_0 against H_1 : Θ = Θ_1, where Θ_1 is a fixed positive number. Is it a UMP test?

Stating the underlying assumptions, find the expression of the probability of happening n events in the time interval (0, t) on Poisson process. Hence obtain the distribution of inter-arrival times.

Control charts for x-bar and R are maintained for an important quality characteristic. The sample size is n = 7; x-bar and R are computed for each sample. After 35 samples, we have found that summation from i=1 to 35 of x-bar_i = 7805 and summation from i=1 to 35 of R_i = 1200. (i) Compute the central line and control limits for x-bar and R charts. (ii) Assuming both charts exhibit control, estimate the process mean and standard deviation. (iii) If the quality characteristic is normally distributed and if the specification limits are 220 +- 35, estimate C_p and C_pk. [Given : D_3 = 0.076, D_4 = 1.924, d_2 = 2.704, A_2 = 0.419]

Using central limit theorem, show that lim (n→∞) e^(-n) ∑_{k=0}^n n^k / (k! √2) = 1

Define principal components. Is the principal component affected by standardization? Establish your claim by illustrating the same with the covariance matrix Σ = [ 4 1 ; 1 100 ]

What do you understand by Balanced Design? Suppose four treatments 1, 2, 3, 4 are laid out in five blocks I, II, III, IV, V as given below : I : 1 2 II : 1 2 III : 3 4 IV : 3 4 V : 3 4 Establish that the above design is unbalanced.

Let X : i -> 0, 1, where X_1: 2 3 4 5 f_nx: 1/3 1/3 1/3 1 Find the best critical region of size α. What is its power?

Describe the behaviour of the hazard rate in a bathtub curve.

Let X_1, X_2, ..., X_n be iid binomial(1, p) random variables. Find the UMVUE for p(1-p) based on the sample.

Let {X_n} be a sequence of random variables and Y_n = 1/n ∑_{i=1}^n X_i. Prove that a necessary and sufficient condition for the sequence {X_n} to satisfy the weak law of large numbers is that E [ X_n^2 / (1 + Y_n^2) ] → 0 as n → ∞

For a random sample of size n from N_p(μ, Σ), we obtain Σ = [ 18, 17, 18, 14 ; ... ]

For testing a simple hypothesis H_0 : θ = θ_0 against a simple alternative H_1 : θ = θ_1, prove that the Sequential Probability Ratio Test (SPRT) terminates with probability one. Let X_1, X_2, ... be random variables from the Bernoulli distribution with P(X = 1) = p and P(X = 0) = 1 - p. Obtain a sequential probability ratio test for H_0 : p = p_0 against H_1 : p = p_1.

The period of oscillation, t, of a pendulum is 2π√(l/g), where l is the length of the pendulum and g is the gravitational constant. The periods observed are t_i (i = 1, 2, ..., k) and lengths l_i (i = 1, 2, ..., k) of the pendulum, in an experiment. Assuming the errors of observations to be uncorrelated with zero means and variances σ^2, show that the best linear estimator of θ = 2π/√(g) is θ = ∑_{i=1}^k t_i / ∑_{i=1}^k √(l_i)

What is the importance of sensitivity analysis? Mention what type of problems can be resolved using it. Using the following linear programming problem, show that the value 6 of the resource vector b. col. [6 8 1 2] can be increased to 7 : Maximize Z = 3x_1 + 2x_2 subject to x_1 + 2x_2 <= 6, 2x_1 - x_2 <= 8, -x_1 + x_2 <= 1, x_2 <= 2, x_1, x_2 >= 0

The following figures give the number of defectives in 20 samples each containing 2000 items: 305, 425, 430, 216, 341, 225, 322, 280, 306, 337, 356, 402, 216, 264, 126, 409, 193, 326, 280, 390. Can we conclude that the process is in control by setting up an appropriate control chart in a graph sheet?

In a randomized block design with t treatments and r replicates, one observation was found missing. Explain the method of estimating the missing value and perform the analysis of variance.

Suppose there are four items A, B, C and D respectively passed by 90%, 80%, 70% and 60% of individuals. Compare the differences in difficulty between A and B with the differences in difficulty between C and D.

A population has seven units consisting of 30, 10, 25, 5, 20, 6 and 4 fields. Select a PPS sample of units of size 3 using random numbers 72, 48 and 94.

Find the range of values for p and q that will render the cell (2, 2) a saddle point in the game with the following pay-off matrix: Player B (1, 4, 6) Player A (p, 5, 10), (6, 2, 3)

In a single sampling plan (1000, 89, 2), compute the probability of acceptance of the lot if the lot fraction defective is 0.01. Also compute Average Outgoing Quality (AOQ) of the lot.

Discuss the importance of scaling in psychometry. Describe any two scaling methods.

Two dentists A and B make a survey of the state of the teeth of 200 children in a village. Dr. A selects a simple random sample of 20 children and counts the number of decayed teeth for each child with the following results: No. of decayed teeth: 0 1 2 3 4 5 6 7 8 9 10 No. of children: 8 1 2 2 1 1 1 0 0 1 1 Dr. B using the same dental technique examines all 200 children counting merely those who have no decayed teeth. He finds 60 children with no decayed teeth. Estimate the total number of decayed teeth using (i) Dr. A's results only (ii) Both Dr. A's and Dr. B's results Which estimate do you prefer and why?

Discuss the concept of reliability of a test. Define reliability ratio of a test and show that it is always between 0 and 1. What is the index of reliability? Is it also bounded by 0 and 1?

Explain the theoretical basis for control charts.

Distinguish between OLS and GLS estimators. Show that OLS estimators are BLUEs.

Discuss a transportation problem (TP) and show that it can be formulated as a linear programming problem. Obtain a necessary and sufficient condition under which a TP has always a feasible solution. Find the first feasible solution of the following TP using Vogel's approximation method (VAM) Destination (5, 3, 7, 3, 8, 5) Requirement 3, Port (5, 6, 12, 5, 7, 11) 4, (2, 8, 3, 4, 8, 2) 2, (9, 5, 10, 5, 10, 9) 8, Availability 3, 3, 6, 2, 1, 2. Discuss whether the solution is degenerate or non-degenerate.

Consider two independent and identically distributed uniform U(0, 1) random variables. Denote the distribution of U = X / Y. Comment on E(U).

Show that Hotelling's T^2 is a generalization of the Student's t. Prove the invariance property of T^2 under the non-singular linear transformations of the variables. Briefly mention different uses of T^2.

What is a life table? Distinguish between a complete life table and an abridged life table. Complete the following life table : (Table with x, l_x, d_x, q_x, L_x, T_x, e^0_x for x=83 and 84)

Shots aimed at a target reach the plane of the target at a distance R from the centre, where R has the probability density function f(r) = r/σ^2 exp(-r^2 / 2σ^2), r ≥ 0 A hit is scored if R ≤ a. If n shots are fired and there are exactly m hits, these being at distances r_1, r_2, ..., r_m from the centre, prove that the maximum likelihood estimator of σ^2 is given by σ^2 = 1/(2m) [∑_{i=1}^m r_i^2 + (n-m) a^2]

(i) Distinguish between total and partial confounding in a 2^k factorial experiment. (ii) If a 2^3 factorial experiment is laid in a block of size 4 with four replicates, write down the allocations of degrees of freedom for analyzing the results of such a design, when the highest order interaction is totally confounded.

Given the following demand and supply model Demand : Q_i = alpha_0 + alpha_1 P_t + alpha_2 I_t + u_{1t}, alpha_1 < 0, alpha_2 > 0 Supply : Q_i = beta_0 + beta_1 P_t + beta_2 I_{t-1} + u_{2t}, beta_1 > 0, beta_2 > 0 construct the reduced-form model and justify whether the parameters can be estimated completely.

If X is a p-component vector of random variables X ~ N_p(0, Σ), Σ being positive definite matrix, then show that X' Σ^(-1) X is distributed as chi-square with p degrees of freedom.

Let {X_n, n > 0} be a Markov chain having a sample space S = {1, 2, 3, 4} and transition probability matrix P = [[1/3, 2/3, 0, 0], [1, 0, 0, 0], [1/2, 0, 1/2, 0], [0, 0, 1/2, 1/2]]. Show that states 3 and 4 are transient states.

The following data represents lifetime in hours of batteries for two different brands: Brand A : 40 30 40 15 55 30 Brand B : 50 50 45 55 60 40 Use the Median test to conclude if the two samples come from the same population.

Distinguish between wholesale price index number and consumer price index number. How are they calculated? The following data relate to the wholesale prices of six different pulses in June and December of a year : (Table of 6 pulses, weights 1.26, 0.35, 1.79, 1.88, 0.98, 1.12; June prices: 120, 135, 148, 132, 146, 128; December prices: 135, 156, 162, 148, 158, 150). Calculate the index number for December with June as the base by weighted aggregative method.

Calculate (i) crude death rate and (ii) standardised death rates of Town A from the following data: (Table with Age Group 0-5, 5-10, 10-25, 25-45, 45 and above; Population of Town A: 2000, 1600, 2400, 6000, 8000; Deaths in Town A: 100, 48, 24, 30, 104; Standard Population: 20000, 16000, 24000, 60000, 80000). Are the two rates equal? If so, why?

Define a series system and a parallel system Compute the reliability of the following configuration whose components work independently : It is given that the components c_1, c_2, c_3, c_4 and c_5 have component reliabilities 0.6, 0.4, 0.7, 0.8 and 0.5 respectively.

For a model Y_i = β_0 + α_1 X_i + ε_i, i = 1, 2, 3 where X_1 = -1, X_2 = 0, X_3 = 1, find BLUE's of β_1, β_0. Suppose the model is not correct and the true model is Y_i = β_0 + α_1 X_i + β_2 X_i^2 - ε_i Find the bias of the BLUE's obtained.

What is time series? Discuss various components of a time series and their significances in contributing variation in the model.

What are the importances of index numbers? Define the different tests an index number has to satisfy. Which index numbers satisfy these tests?

Compare ratio and regression methods of estimation. Show that ratio estimator gives a biased estimate of population total. Derive an approximate expression of the bias.

Describe Monte Carlo simulation technique and mention its different steps. Also describe how you would generate random numbers from a Weibull density with parameters (theta, nu).

The joint density of X, Y is f(x, y) = e^(-y), 0 < x < y < ∞ Find the marginal distributions of X, Y and the conditional distribution of Y given X.

If a linear programming problem has an optimal solution, then prove that it attains its optimum solution at an extreme point of the convex set generated by the set of all the feasible solutions to the problem.

Consider a two variable regression model Y_t = beta_1 + beta_2 X_t + u_t. Suppose u_t is autocorrelated as u_t = rho u_{t-1} + e_t, -1 < rho < 1, where rho is the autocovariance coefficient and e_t is the stochastic disturbance term satisfying OLS assumptions. Obtain an estimate for beta_2 and its corresponding variance. Also discuss the Durbin-Watson statistic for detecting autocorrelation in the model, and its interpretation.

A 2^3 experiment involving factors A, B, C, D and E is conducted in blocks of size 8. The key tracks of the replications 1 and 2 are successively given by 1 : abd, abe, acd, bce, abc, cde 2 : acd, abde, bcd, cde, abce Identify the confounded effects in each replication.

Describe registration and census methods of demographic data collection.

We have a bag with 3 coins in it. One of them is a fair coin, but the others are biased trick coins. When flipped, the three coins come up 'heads' with probability 0.5, 0.6, 0.7 respectively. One of these coins is picked at random and flipped three times. (i) What is P(H)? (ii) Assuming that HHH occurred, what is the probability of having chosen a fair coin? Here H, T denote Heads and Tails respectively.

Let X_1, X_2, ..., X_n be a sample from a distribution with density function f(x) = (1 + θ) x^ heta, 0 < x < 1, θ > 0 Obtain an estimate of Θ by method of moments.

Describe the Box Jenkins methodology to identify a particular time series model.

Define failure rate and reliability function of a random variable T. Obtain the failure rate and reliability function of the Weibull failure model. State the conditions under which the failure rate is increasing, decreasing and constant.

If X_n P-> X, Y_n P-> Y, then show that (i) X_n + Y_n P-> X + Y (ii) X_n - Y_n P-> X - Y (iii) X_n / Y_n P-> X / Y, provided C ≠ 0 If X_n a.s-> X, Y_n a.s-> Y, then show that (i) X_n + Y_n a.s-> X + Y (ii) X_n - Y_n a.s-> X - Y (iii) X_n Y_n a.s-> XY (iv) X_n / Y_n a.s-> X / C, provided C ≠ 0

Explain stratified random sampling and give reasons for its preference to simple random sampling. With two strata, a sampler would like to have n_1 = n_2 for administrative convenience instead of using the values given by the Neyman allocation. If V(y_st) and V_{opt}(y_st) denote the variances given by the n_1 = n_2 and Neyman allocations respectively, show that the fractional increase in the variance is (V(y_st) - V_{opt}(y_st)) / V_{opt}(y_st) = (r - 1)^2 / r + 1 where r = n_1 / n_2 as given by Neyman allocation.

Write a brief note on National Statistical Commission stating the components, functions and objectives of each.