# 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

### 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

## Question number: 7

» 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: 8

» Hypotheses Testing » Unbiased Test

Appeared in Year: 2015

### 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

## Question number: 9

» 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