# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 31 - 33 of 39

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## Question number: 31

» Estimation » Optimal Properties » Rao-Blackwell Theorem

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Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Explain how the Rao-Blackwell theorem helps one to find a uniformly minimum variance unbiased estimator (UMVUE) of an unknown parameter. What is the relevance of the Lehman-Scheffe theorem in this scenario? If X 1, X 2, …, X n are Bin (1, p) variates, find the UMVUE of p.

### Explanation

Let U be an unbiased estimator of θ and T be a sufficient statistic for θ, then E (U|T) is free from θ and it is an estimation. Using the identity , we have

True for all θ. Then is an unbiased for θ.

Next, we find that is has minimum variance.

Using th

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## Question number: 32

» Statistical Quality Control » Control Charts » Variable

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Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Sample of sizes n = 5 are taken from a manufacturing process every hour. A quality characteristic is measured, and and R are computed for each sample. After 25 samples have been analyzed, we have and . Assume that the quality characteristic is normally distributed.

(i) Find the control limit for the and R charts.

(ii) Assume that both chart exhibit control, if specifications are 26.40±0.50, estimate the fraction nonconforming. Express your answers in terms of CDF of N (0,1) random variable.

[For n = 5, A 2 =0.577, A = 1.342, A 3 =1.427, D 1 =0, D 2 =4.918, D 3 =0, D 4 =2.115 and d 2 =2.326]

### Explanation

Given that and

(i) For chart, the control limits are

For R chart, the control limits are

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## Question number: 33

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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 test based on it is never l

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