# 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 . The power of a

## 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

The Fisher information is

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

The Lower bound is

## 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 completeness is

Assume is free from and

This equation is true iff

This is also true if and only if this is a polynomial of n ^{th}

## 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 distribution is that every linear combination of X follows an univariate normal distribution.

## 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

The corresponding eigen vector for each eigen value is by using normalize the eigen vector by the equation

For λ _{1} =6,