# Hypotheses Testing (ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern)): Questions 6 - 9 of 9

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## Question number: 6

» Hypotheses Testing » Randomised Test

Appeared in Year: 2014

Essay Question▾

### Describe in Detail

chart is used to control the mean of normally distributed characteristic. It is know that . 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.

### Explanation

Type II error is the probability of saying that the process is under control when it is not under control, that is, the probability of a point falling within control limits after the shift the mean. This is also known as the probability of not detecting the shift after shift

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## Question number: 7

» 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

 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

 Z 5 4 3 2 1 0.8947

The possible value of Z is

Possible Z Values given in table

Finding

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## Question number: 8

» Hypotheses Testing » Unbiased Test

Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Explain the notion of unbiasedness with regards to a test of a hypothesis. Examine the validity of the statement. A most powerful test is invariably unbiased.

### Explanation

A test T of the null hypothesis is said to be an unbiased test if the probability of rejection H 0 when it is false is at least as much as the probability of rejecting H 0 when it is true.

where . The power of a

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## Question number: 9

» Hypotheses Testing » Uniformly Most Powerful Test

Appeared in Year: 2015

Essay Question▾

### 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

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