# ISS (Statistical Services) 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

### 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

x | 1 | 2 | 3 | 4 | 5 | 6 |

f | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.95 |

f | 0.05 | 0.04 | 0.03 | 0.02 | 0.01 | 0.85 |

### Explanation

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

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

### 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

### 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

## Question number: 22

» Estimation » Optimal Properties » Sufficient Estimator

Appeared in Year: 2015

### 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

## Question number: 23

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

Appeared in Year: 2015

### 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,

## Question number: 24

» Statistical Quality Control » Concepts of ATI

Appeared in Year: 2015

### 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