# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 31 - 34 of 39

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## Question number: 31

» Estimation » Optimal Properties » Rao-Blackwell Theorem

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 the identity

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## Question number: 32

» Statistical Quality Control » Control Charts » Variable

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

(ii) UCL = 26.40 + 0.50 = 26.90

LCL = 26.40 - 0.50 = 25.90

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## Question number: 33

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 less than its size is called unbiased critical r

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## Question number: 34

» Estimation » Optimal Properties » Minimum Variance Bound Estimators

Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Let X 1, X 2, …, X n be a random sample from the probability distribution with density

= 0; otherwise

where 0 < θ < ∞. Show that is a minimum variance bound estimator and has variance .

### Explanation

By using Cramer-Rao lower bound we find the minimum variance

The Fisher information is

Here E (X) =θ. The unbiased estimator of exponential distribution is .

The Lower b

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