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

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## Question 31

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 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 the identity

Here X = U, Y = T

Since V (U|T) ⩾ 0, then

This says that is have uniformly minimum varia…

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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, 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

(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

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 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 less than its size is called unbiased critical region and test id unbiased test. In terms of probability, the con…

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