# ISS (Statistical Services) Statistics Paper III: Questions 34 - 39 of 109

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## Question number: 34

» Sampling Techniques » Cluster Sampling with SRS - Intra-Class Correlation Coefficient

Appeared in Year: 2010

### Describe in Detail

What is cluster sampling? In the case of cluster sampling for proportions, obtain an unbiased

Estimator of population proportion. Also, compute variance of your estimator.

### Explanation

**Cluster sampling:**

Cluster sampling is a type of sampling method. In this method, the population is divided into separate groups, called clusters. Then, a simple random sample of clusters is selected from the population. Then analysis is done on the data from the obtained sampled clusters.

The samplings lead to several difficulties if the population s

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## Question number: 35

» Sampling Techniques » Simple Random Sampling with and Without Replacement

Appeared in Year: 2011

### Write in Short

**Explain the concept of simple random/sampling with and without replacement. In a simple Random sampling, show that is an unbiased estimator of .**

## Question number: 36

» Applied Statistics » Index Numbers » Price Relatives and Quantity or Volume Relatives

Appeared in Year: 2011

### Describe in Detail

Explain the errors in the measurement of price and quantity index number.

### Explanation

All index numbers are affected by mainly three types of errors.

Formula error

Sampling error

Homogeneity error

Formula error:

There is no universally accepted index number formula that will measure the price or quantity changes excatly. Each formula in common use has its own defeat and consequently some error is natural in all index number formula. This

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## Question number: 37

» Sampling Techniques » Stratified Random Sampling

Appeared in Year: 2011

### Describe in Detail

Let for a stratified random sampling where is constant, is the-cost per unit-in i^{th} stratum**,** is the mean square based on ** ** units.

Estimate the sample sizen under optimum allocation for fixed cost** .**

### Explanation

Under optimum allocation, are obtained such that

(i) minimize the variance of the estimate for (a) fixed sample size n;

(i) var ( is minimum for fixed n

(ii) var ( is minimum for fixed total cost C

(iii) Total cost C is minimum for fixed value of var (

And the cost function C in stratified sampling is given b

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## Question number: 38

» Econometrics » Ordinary Least Squares (OLS)

Appeared in Year: 2011

### Describe in Detail

If is an individual’s utility function of two goods, show that the demand for the goods is and

Where and are the fixed price and µ be the individual fixed income.

### Explanation

For consumer’s equilibrium, we have

and hence, we get

→

→

→

→

→

→

→ ………………. (i)

since, µ the indivi

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## Question number: 39

» Econometrics » Auto-Correlation

Appeared in Year: 2011

### Describe in Detail

Discuss the effect of imperfect multicollinearity on tests and errors. Consider a model

with

and with

, ,

and

Do you think multicollinearity is present in the model? If yes, give reasons. Further show that as

increases, also increases.

### Explanation

Effect of imperfect multicollinearity on tests and errors:

OLS estimators will be BLUE, but they have large variances and covariances. this makes the estimation difficult.

The confidence intervals will be wider. and hence, we may not reject the ‘zero null hypothesis i. e. , We may conclude that B’s are not significantly different from zero even thoug

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