# 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

### 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

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## 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

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## Question number: 33

» Hypotheses Testing » Unbiased Test

Appeared in Year: 2015

### 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

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## Question number: 34

» Estimation » Optimal Properties » Minimum Variance Bound Estimators

Appeared in Year: 2015

### 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

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