ISS Statistics Paper II (Old Subjective Pattern): Questions 19 - 24 of 39

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Question number: 19

» Hypotheses Testing » Most Powerful Test

Appeared in Year: 2014

Essay Question▾

Describe in Detail

Let X be r. v. with pmf under H ₀ and H ₁ given below. Find M. P. test with α=0.03

**Possible X Values given in table**
Finding a values of f 0 (x) and f 1 (x) to given a table.
x	1	2	3	4	5	6
f ₀ (x)	0.01	0.01	0.01	0.01	0.01	0.95
f ₁ (x)	0.05	0.04	0.03	0.02	0.01	0.85

Explanation

To find the M. P. test, first we compute

**Calculate Z in given table**
Finding a values of P (Z = Z) and Fz (z) to given a table.
Z	5	4	3	2	1	0.8947

The possible value of Z is

Question number: 20

» Estimation » Estimation Methods » Maximum Likelihood

Appeared in Year: 2014

Essay Question▾

Describe in Detail

Let X be exponentially distributed with parameter θ. Obtain MLE of θ based on a sample of size n, from the above distribution.

Explanation

Let X be exponentially distributed with parameter θ.

The likelihood function is

The log-likelihood function is

Differentiable with respect to θ, equating to zero

Question number: 21

» Estimation » Optimal Properties » Confidence Interval Estimation

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Let y ₁, y _{2, …,} y _n be a random sample from N (µ, σ ²) where µ and σ ² are both unknown. Obtain a confidence interval of µ with confidence coefficient (1-α)

Explanation

When population mean and population standard deviation in not know. If is the samplemean and replace σ by its estimate s and t _α/2 be the critical value of the student t-test such that have of the area on the left hand side and other half on the right… (25 more words) …

Question number: 22

» Estimation » Optimal Properties » Sufficient Estimator

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Obtain the sufficient statistics for the following distribution.

(i)

(ii)

Explanation

By using factorization theorem, the condition is that

where h (x) is free from θ and g (. ) depends on X only through T.

(i)

The joint pdf of random sample is

Let T= . By factorization theorem

h (x) =1,

So, h… (70 more words) …

Question number: 23

» Multivariate Analysis » Distribution of Hotelling T2 Statistic » Use for Testing

Appeared in Year: 2015

Essay Question▾

Describe in Detail

Show that T ² statistic is invariant under changes in the unit of measurements for a p×1 random vector X of the form Y =C X + d where C is a p×p nonsingular matrix, d is a p×1 vector.

Explanation

T ² statistic can be computed from X. Here we have to compute it form Y and both will be same. Here

Let , then and

So, T ² -statistic comes into existence for testing hypothesis of the form

Since is known vector,… (35 more words) …

Question number: 24

» Statistical Quality Control » Concepts of ATI

Appeared in Year: 2015

Essay Question▾

Describe in Detail

For a sequential probability ratio test of strength (α, β) and stopping bounds are A and B (B < A), show that A ≤ 1-β/α and B ≥ β/1-α

Explanation

Let X= (X ₁, …, X _k) and also let E _k be the set of all points in k dimensional Euclidean space R _k, for which we reject H ₀ using the sequential probability ratio test. Also, let F _k be the set of all points… (76 more words) …