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

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## Question 31

Rao-Blackwell Theorem
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Appeared in Year: 2015

### Describe in Detail

Essay▾

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 X1 , X2 , … , Xn 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 the identity

Here X = U, Y = T

Since V (U|T) ⩾ 0, then

This says that is have uniformly minimum varia&#8230;

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## Question 32

Appeared in Year: 2015

### Describe in Detail

Essay▾

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, A2 = 0.577, A = 1.342, A3 = 1.427, D1 = 0, D2 = 4.918, D3 = 0, D4 = 2.115 and d2 = 2.326]

### Explanation

Given that and

(i) For chart, the control limits are

For R chart, the control limits are

(ii) UCL = 26.40 + 0.50 = 26.90

LCL = 26.40 - 0.50 = 25.90

Fraction nonconforming = P (X > 26.90) + P (X < 25.90)

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## Question 33

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Appeared in Year: 2015

### Describe in Detail

Essay▾

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 H0 when it is false is at least as much as the probability of rejecting H0 when it is true.

where . The power of a test based on it is never less than its size is called unbiased critical region and test id unbiased test. In terms of probability, the con&#8230;

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