Exam Details
| Subject | statistics | |
| Paper | paper 2 | |
| Exam / Course | indian economic service and indian statistical service examination (ies/iss) | |
| Department | ||
| Organization | union public service commission | |
| Position | ||
| Exam Date | 2014 | |
| City, State | central government, |
Question Paper
x1,x2,... xn be a random sample from obtain the moment estimator of 0. Also find its variance.
1.(b) Obtain a g-inverse of A given below and verify that AA-A A
1.(c) Define completeness. Verify whether Bin is complete.
1.(d) Find the sum of squares due to
(E to E i=1 to p ai bj Ei=1 to p ai^2 E i=1 to p bj^2
under suitable assumptions on Yij's where ai and bj's are constants. State its use in Analysis of variance.
1.(e) Let X2, X3 and X4 be four random variables such that
E(X1) 0l 03, 0l 82-03,
E(X3) 01 01 -02 03
where 01, 02, 03 are unknown parameters. Assume var i 4. Check if 82 is estimable. If so obtain its BLUE.
1.(f) x1, x2, ... xn is a random sample from the following distribution
f(x, x
=0 otherwise.
Find MLE of a.
1.(g) Let x1, x2, ... xn be a random sample from 8]. Let T . Is T an unbiased 1=1 estimator of 8 If not suggest an unbiased estimator of 0 which is a function of T.
1.(h) Let X be exponentially distributed with parameter (J. Obtain MLE of based on a sample of size from the above distribution.
2.(a) Define estimability of a linear parametric function in· a Gauss Markoff model. State and prove a necessary and sufficient condition for estintabiIiry.
2.(b) For the Pareto distribution with pdf
f(x, 1 A 0
Show that method of moments fails if 0 1. State the method of moments estimator when it A 1. Is it consistent? Justify your answer.
2.(c) Consider the model with Donna] assumption on error variables.
E(Y1) 81-82 E1
E(Y2) 81-83 E2
.
.
E(Yn) 8n-8t En
Find the error function(s) and the BLUE of 81- 82•
3.(a) A manufacturer of television sets is interested in the effect of tube conductivity of five different types of coating for colour picture tubes. Sample means are
Y1'. 51, 53 Y5'. =58.
Error Mean sum of squares 54·0 Is there a difference in coating for colour picture tubes? F value 3·09.
(ii) Determine which pairs differ significantly using Bonferroni t-interval. Comment on the same.
[t15(0.05/20) 3.286]
3.(b) Xl, X2, ... Xn are i.i.d. random variables from where 9 is an integer. Obtain MLE of 8.
3.(c) Suppose
E(Yij) ai BiXij 1 n· 1 j K
where Yij are independent homoscedastic normal variables, Xij's are non stochastic and ai and Bi are unknown parameters.
Find a suitable test of HO1 B1 B2 BK.
(ii) Assuming B1 B2 BK derive a test for H02 a1 a2
4.(a) Obtain unbiased estimators of the variance components in two way classification model with interaction, with r observations per cell when 'there are 81 levels of factor A and S2 levels of factor B.
4.(b) Xl, X2, ... xn be a random sample from a population having pmf 1/N if ... N
0 otherwise
Derive UMVUE of N.
4.(c) Show that Z (X1 2X2 3X3 is not a sufficient estimator of the Bernaulli parameter 8.
5.(a) Let X be r. v. with pmf under Ho and HI" given below. Find M.P. test with a 03
x 1 2 3 4 5 6
fo(x) 0·01 0·01 0.01 0·01 0·01 0·95
f1(x) 0·05 0·04 0·03 0·02 0·01 0·85
5.(b) A single observation of a r.v. having a geometric disilibution with pmf
f(x, e(1 x .....
=0 otherwise.
The null hypothesis Ho: 8 0·5 against the alternative hypothesis H1 e 0.6 is rejected if the observed value of the r.v. is greater than or equal to 5. Find probabilities of type I error and type II error.
5.(c) The pdf of 5 variate normal distribution is given by -30 exp x3^2/5 (x4 x5^2/5
Obtain mean vector and variance covariance matrix of the distribution.
5.(d) Let E=
1 P 0
p 1 p
0 p 1
Find p such that X1 X2 X3 and X1 X2 X3 are independent.
5.(e) Find 3-sigma control limits for a C chart with process average equal to 4 nonconfotmities.
(ii) U chart with process average c 4 and n 4.
5.(f) Justify the statement:
P chart is equivalent to Chi-square test of homogenity.
5.(g) A sample of size n from nonnal distribution with 0.2 4 was observed. 95% confidence interval for 8 was computed from the above sample. Find the value of n if the confidence interval is (9·02, 10.98).
5.(h) Bring out the difference between a randomized test and a nonrandomized test. Explain how the decision based on a randomized test can be taken in the discrete set up.
6.(a) Consider the Hypothesis Ho against H1 P 1 for a binomial X for which n =2. List all possible critical regions for which a 1/2. Which of these regions minimizes a B
6.(b) Consider an i.i.d. sample of size n 5 from bivariate normal a a [0.05]
For what values of a would the hypothesis Ho: M Or be rejected in favour of H1 M at level of significance. (Chi-square table value 5·99).
6.(c) Obtain expressions for OC and ASN function under Single sampling plan (iO Double, sampling plan.
7.(a) Let X E=
1 2 -1
2 6 0
-1 0 4
Set Y1 =Xj Xj,. Y2 2XI and Y3 2X] -X2• Find the conditional distribution of lJ given Y1 Y2 1.
7.(b) X1, X2) ... xn is a random sample frOin e(12 not specified). Derive likelihood ratio test of testing Ho 90 against HI e 00,
7.(c) A X chart is used to control the mean of a normally distributed quality characteristic. It is known that 6 and n 4. The center line is 200. If the process mean shifts to 188) find the probability that this shift is detected on the first subsequent sample.
8.(a) Let X be a r.v. following exponential distribution
8e^-8x x>0 =
0 otherwise
Obtain SPRT of strength a,B for testing Ho 8 8 against H1 8 8
8.(b) X2, X3) N3
0 1 p12 p13
1 p23
1
Show that 1 2P12P13P23 P12^2 P13^2 P23^2
8.(c) A proposed triple sampling plan is as follows
Take a first sample of 2 articles. If both good accept, if both bad reject. If 1 good and I bad reject, take a second sample of 2. If both good accept. If both bad reject. If I good and 1 bad take a third sample of 2. If both articles in the third sample are good, accept; otherwise reject. Find the OC. of this plan assuming that a large lot of defective is submitted for inspection.
1.(b) Obtain a g-inverse of A given below and verify that AA-A A
1.(c) Define completeness. Verify whether Bin is complete.
1.(d) Find the sum of squares due to
(E to E i=1 to p ai bj Ei=1 to p ai^2 E i=1 to p bj^2
under suitable assumptions on Yij's where ai and bj's are constants. State its use in Analysis of variance.
1.(e) Let X2, X3 and X4 be four random variables such that
E(X1) 0l 03, 0l 82-03,
E(X3) 01 01 -02 03
where 01, 02, 03 are unknown parameters. Assume var i 4. Check if 82 is estimable. If so obtain its BLUE.
1.(f) x1, x2, ... xn is a random sample from the following distribution
f(x, x
=0 otherwise.
Find MLE of a.
1.(g) Let x1, x2, ... xn be a random sample from 8]. Let T . Is T an unbiased 1=1 estimator of 8 If not suggest an unbiased estimator of 0 which is a function of T.
1.(h) Let X be exponentially distributed with parameter (J. Obtain MLE of based on a sample of size from the above distribution.
2.(a) Define estimability of a linear parametric function in· a Gauss Markoff model. State and prove a necessary and sufficient condition for estintabiIiry.
2.(b) For the Pareto distribution with pdf
f(x, 1 A 0
Show that method of moments fails if 0 1. State the method of moments estimator when it A 1. Is it consistent? Justify your answer.
2.(c) Consider the model with Donna] assumption on error variables.
E(Y1) 81-82 E1
E(Y2) 81-83 E2
.
.
E(Yn) 8n-8t En
Find the error function(s) and the BLUE of 81- 82•
3.(a) A manufacturer of television sets is interested in the effect of tube conductivity of five different types of coating for colour picture tubes. Sample means are
Y1'. 51, 53 Y5'. =58.
Error Mean sum of squares 54·0 Is there a difference in coating for colour picture tubes? F value 3·09.
(ii) Determine which pairs differ significantly using Bonferroni t-interval. Comment on the same.
[t15(0.05/20) 3.286]
3.(b) Xl, X2, ... Xn are i.i.d. random variables from where 9 is an integer. Obtain MLE of 8.
3.(c) Suppose
E(Yij) ai BiXij 1 n· 1 j K
where Yij are independent homoscedastic normal variables, Xij's are non stochastic and ai and Bi are unknown parameters.
Find a suitable test of HO1 B1 B2 BK.
(ii) Assuming B1 B2 BK derive a test for H02 a1 a2
4.(a) Obtain unbiased estimators of the variance components in two way classification model with interaction, with r observations per cell when 'there are 81 levels of factor A and S2 levels of factor B.
4.(b) Xl, X2, ... xn be a random sample from a population having pmf 1/N if ... N
0 otherwise
Derive UMVUE of N.
4.(c) Show that Z (X1 2X2 3X3 is not a sufficient estimator of the Bernaulli parameter 8.
5.(a) Let X be r. v. with pmf under Ho and HI" given below. Find M.P. test with a 03
x 1 2 3 4 5 6
fo(x) 0·01 0·01 0.01 0·01 0·01 0·95
f1(x) 0·05 0·04 0·03 0·02 0·01 0·85
5.(b) A single observation of a r.v. having a geometric disilibution with pmf
f(x, e(1 x .....
=0 otherwise.
The null hypothesis Ho: 8 0·5 against the alternative hypothesis H1 e 0.6 is rejected if the observed value of the r.v. is greater than or equal to 5. Find probabilities of type I error and type II error.
5.(c) The pdf of 5 variate normal distribution is given by -30 exp x3^2/5 (x4 x5^2/5
Obtain mean vector and variance covariance matrix of the distribution.
5.(d) Let E=
1 P 0
p 1 p
0 p 1
Find p such that X1 X2 X3 and X1 X2 X3 are independent.
5.(e) Find 3-sigma control limits for a C chart with process average equal to 4 nonconfotmities.
(ii) U chart with process average c 4 and n 4.
5.(f) Justify the statement:
P chart is equivalent to Chi-square test of homogenity.
5.(g) A sample of size n from nonnal distribution with 0.2 4 was observed. 95% confidence interval for 8 was computed from the above sample. Find the value of n if the confidence interval is (9·02, 10.98).
5.(h) Bring out the difference between a randomized test and a nonrandomized test. Explain how the decision based on a randomized test can be taken in the discrete set up.
6.(a) Consider the Hypothesis Ho against H1 P 1 for a binomial X for which n =2. List all possible critical regions for which a 1/2. Which of these regions minimizes a B
6.(b) Consider an i.i.d. sample of size n 5 from bivariate normal a a [0.05]
For what values of a would the hypothesis Ho: M Or be rejected in favour of H1 M at level of significance. (Chi-square table value 5·99).
6.(c) Obtain expressions for OC and ASN function under Single sampling plan (iO Double, sampling plan.
7.(a) Let X E=
1 2 -1
2 6 0
-1 0 4
Set Y1 =Xj Xj,. Y2 2XI and Y3 2X] -X2• Find the conditional distribution of lJ given Y1 Y2 1.
7.(b) X1, X2) ... xn is a random sample frOin e(12 not specified). Derive likelihood ratio test of testing Ho 90 against HI e 00,
7.(c) A X chart is used to control the mean of a normally distributed quality characteristic. It is known that 6 and n 4. The center line is 200. If the process mean shifts to 188) find the probability that this shift is detected on the first subsequent sample.
8.(a) Let X be a r.v. following exponential distribution
8e^-8x x>0 =
0 otherwise
Obtain SPRT of strength a,B for testing Ho 8 8 against H1 8 8
8.(b) X2, X3) N3
0 1 p12 p13
1 p23
1
Show that 1 2P12P13P23 P12^2 P13^2 P23^2
8.(c) A proposed triple sampling plan is as follows
Take a first sample of 2 articles. If both good accept, if both bad reject. If 1 good and I bad reject, take a second sample of 2. If both good accept. If both bad reject. If I good and 1 bad take a third sample of 2. If both articles in the third sample are good, accept; otherwise reject. Find the OC. of this plan assuming that a large lot of defective is submitted for inspection.