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

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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

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

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

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

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## Question number: 35

» Estimation » Optimal Properties » Complete Statistics

Appeared in Year: 2015

### 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

Appeared in Year: 2015

### 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

Appeared in Year: 2015

### 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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