# ISS (Statistical Services) Statistics Paper III: Questions 62 - 68 of 109

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## Question number: 62

» Applied Statistics » Index Numbers » Income Distribution-Pareto and Engel Curves

Appeared in Year: 2012

### Write in Short

Discuss Engel’s law and Engel’s curve. Explain Engel ‘s curve for constant price.

## Question number: 63

» Sampling Techniques » Stratified Random Sampling

Appeared in Year: 2012

### Describe in Detail

Discuss the allocation of a sample to different strata in stratified sampling. Explain proportional allocation and hence obtain the variance of mean of a stratified sample under proportional allocation.

### Explanation

For a population consisting of N sampling units it can be divided into k relatively homogeneous sub-groups of sizes, , …, (called as strata) such that N=.

Then a simple random sample of size (i = 1,2, …, k) is drawn from each of the stratum such that n=, and this sample is the stratified random sample of size n.

**Allocati**

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## Question number: 64

» Sampling Techniques » Simple Random Sampling with and Without Replacement

Appeared in Year: 2012

### Write in Short

Show that for a simple random sample unbiased estimate of

## Question number: 65

» Econometrics » Ordinary Least Squares (OLS)

Appeared in Year: 2012

### Write in Short

If the demand curve is of the form , where *p* is the price and *x* is the demand, prove that the elasticity of demand is · Hence deduce the elasticity of demand for

## Question number: 66

» Applied Statistics » Index Numbers » Laspeyre

Appeared in Year: 2013

### Describe in Detail

Why do the errors occur in the measurement of price and quantity index numbers? Give the names of those errors.

### Explanation

**Errors in the measurement of price and quantity index numbers: **

Generally, index numbers for prices are constructed from *m* commodities, which are common to both the base year and the given year. But in practical calculations, it may not be possible to consider all common commodities say, M.

Therefore, value index can be either in terms of

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## Question number: 67

» Sampling Techniques » Probability Sampling or Random Sampling

Appeared in Year: 2012

### Write in Short

Let be the *y*-value of the i^{th} unit and be corresponding selection probability,

i = 1,2, …, N. Then, based on a sample of size n drawn with replacement, show that an unbiased estimator for population total is

## Question number: 68

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

Appeared in Year: 2013

### Describe in Detail

Explain the following:

(i) Weighted Aggregates method

(ii) Criteria of a good index number

### Explanation

** (i) Weighted Aggregates Method: **

In this method, an index number of prices for any given year is calculated after assigning the appropriate weights to the different items included in that index number.

Usually weights are assigned on the basis of quantities, values or sale price of the commodities consumed during the base year or in some typical

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