# ISS (Statistical Services) Statistics Paper II (Old Subjective Pattern): Questions 33 - 37 of 39

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## Question number: 33

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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

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## Question number: 34

» Estimation » Optimal Properties » Minimum Variance Bound Estimators

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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

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## Question number: 35

» Estimation » Optimal Properties » Complete Statistics

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Appeared in Year: 2015

Essay Question▾

### Describe in Detail

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

Examine whether the statistic is complete for this distribution.

### Explanation

We know that is sufficient statistic for this distribution. Then,

The definition of completene

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## Question number: 36

» Multivariate Analysis » Multivariate Normal Distribution » Mutliple Correlation Coefficient

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Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Let X 1, X 2, …, X n be a random sample from an population with , a positive definite matrix. Derive 100 (1-α) % simultaneous confidence interval for for all .

### Explanation

Let X 1, X 2, …, X n be a random sample from an and assume the linear combination of the random sample is

From, the theorem of linear combinations of multivariate normal distrib

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## Question number: 37

» Multivariate Analysis » Principal Components » Correlations

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Appeared in Year: 2015

Essay Question▾

### Describe in Detail

Let

Determine

(i) The principal components y 1, y 2 and y 3.

(ii) The proportion of variance explained each one of them.

(iii) Correlation between the first principal components y 1 and the third original random variable.

### Explanation

First find the eigen value and eigen vector pairs

To solve this equation, we get the eigen values is

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