# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 8 - 12 of 165

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## Question number: 8

» Statistical Methods » Correlation Coefficient » Multiple Correlation

Appeared in Year: 2009

### Describe in Detail

(i) Show that the arithmetic mean of (positive) regression coefficients is greater than the correlation coefficient.

(ii) What is the value of the product of geometric mean of variances and the geometric mean of regression coefficients?

### Explanation

(i) The arithmetic mean of regression coefficients is and the correlation coefficient is r.

The square of the two real numbers cannot be negative. So, the arithmetic mean of (positive) regression coefficients is greater than the correlation coefficient.

(ii) Geometric mean of variances is

## Question number: 9

» Statistical Methods » Regression » Polynomial

Appeared in Year: 2013

### Describe in Detail

Given the two regression lines between X and Y, 2Y -X - 50 = 0, 3Y-2X - 10 = 0, compute the means of X and Y and the correlation coefficient between X and Y.

### Explanation

Given that

We know that the mean value of the given series satisfies the regression line that is

The means of X and Y is calculate by multiply equation (1) by 2 to subtract equation (2), we get

The correlation coefficient is defined by

## Question number: 10

» Statistical Methods » Standard Errors and Large Sample Tests

Appeared in Year: 2013

### Describe in Detail

The proportion of defective telephone instruments from the All-India basis was observed to be at 0.01. A new manufacturer of telephone instruments wants to estimate the proportion of defectives in his production lot. How many telephones should be sampled in order to estimate the fraction defective within 0.01 with 90 % confidence?

### Explanation

Let the proportion of defective telephone instruments from the All-India basis was observed to be at 0.01. The number of telephones should be sampled where the estimate fraction defective is p-p _{0} =0.01 and the true value of proportion of defective is 0.01. so, the test statistic for 90 %

## Question number: 11

» Statistical Methods » Dispersion and Skewness

Appeared in Year: 2012

### Describe in Detail

Show, by proving all intermediate results, that for a set of n unsorted data, the measure of skewness based on mean, median and standard deviation, necessarily lies between -3 and + 3.

### Explanation

The formula of measure of skewness given by Karl Pearson’s is

Also, we known the relation of mean, median and mode is

Mean-Mode = 3 (Mean-Median)

So,

Using Chebyshev’s inequality, X be a random variable with mean µ and variance σ ^{2} then any k > 0,

## Question number: 12

Appeared in Year: 2014

### Describe in Detail

In a lottery 1000 tickets are sold and the cost of a ticket is if 10. The lottery offers a first prize of if 1, 000, two second prizes of if 500 each, and three third prizes of if 100 each. A person purchases a ticket. If X denotes the amount he may get, find E (X) and V (X).

### Explanation

The probability of purchases a ticket is

First prize amount is 1000

Two second prizes amount is 500 each

Three third prizes amount is 100 each

X denotes the amount he may get

Then, the expected value of X is

The variance of X is