# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 30 - 32 of 39

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## Question number: 30

» Estimation » Optimal Properties » Cramer-Raoinequality

Appeared in Year: 2015

### Describe in Detail

Stating the regularity conditions, give the Cramer-Rao lower bound for the variance of an unbiased estimator of a parameter. Give an example, each, of a situation where the regularity conditions (i) does not hold (ii) holds

### Explanation

Suppose that X _{1}, …, X _{n} is a sample from a distribution with joint pdf f _{n} (__x__, θ) and T (__X__) is an estimator. Also assume that f _{n} () satisfies the conditions that allow

(i) Interchange of differentiation and integration operations i.

## Question number: 31

» Estimation » Optimal Properties » Rao-Blackwell Theorem

Appeared in Year: 2015

### 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

## Question number: 32

» Statistical Quality Control » Control Charts » Variable

Appeared in Year: 2015

### 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