# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 38 - 39 of 39

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## Question number: 38

» Hypotheses Testing » Uniformly Most Powerful Test

Appeared in Year: 2015

### Describe in Detail

Fins a Most powerful test for testing the simple hypothesis against the simple alternative hypothesis based on n random observations from N (µ, σ ^{2}) where µ is know. Show that this MP test is UMP (Uniformly most powerful).

### Explanation

Let X _{i} ’s are n random observations from N (µ, σ ^{2}) and the pdf is

The hypothesis is

To find the most powerful test, we use following step

The N-P test is

Given α, we can find k by solving if

## Question number: 39

» Multivariate Analysis » Wishart's Distribution » Reproductive Properties

Appeared in Year: 2015

### Describe in Detail

(i) Let A _{i} be distributed as Wishart , i = 1, 2 and A _{1}, A _{2} be independent. Show that A _{1} +A _{2} is distributed as .

(ii) If A is distributed as , then CAC’ is distributed as where C is a nonsingular matrix of order m.

### Explanation

(i) We have that is

where ) and Y _{i} ’s are independent.

Similarly, that is

where ) and Z _{i} ’s are independent.

We know that A _{1}, A _{2} are independent. So, Y _{i} ’s are independent of Z _{i} ’s