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

Access detailed explanations (illustrated with images and videos) to **39** questions. Access all new questions we will add tracking exam-pattern and syllabus changes. *Unlimited Access for Unlimited Time*!

View Sample Explanation or View Features.

Rs. 200.00 or

How to register?

## 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

Z | 5 | 4 | 3 | 2 | 1 | 0.8947 |

P (Z = z) | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |

F | 1/6 | 2/6 | 3/6 | 4/6 | 5/6 | 1 |

The next ste

… (174 more words) …

## 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

… (19 more words) …

## 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 side that is

For first inequality we get

… (88 more words) …

## 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 fa

… (186 more words) …

## 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 know

… (342 more words) …

## 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 in R _{k} for which we accept H _{0}. Notice that (E _{k}, k = 1,2, …) are mutually disjoint and (F _{k}, k = 1,2, …) are also mutually disjoint. Assume

… (289 more words) …