# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 8 - 11 of 164

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## Question 8

Appeared in Year: *2009*

### Describe in Detail

Essay▾(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 and geometric mean of regression coefficients is

The value o…

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## Question 9

Appeared in Year: *2013*

### Describe in Detail

Essay▾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

The first equation is written in terms of Y is

The second equation is written in terms of X is

Take r = 0.866 becaus…

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## Question 10

Appeared in Year: *2013*

### Describe in Detail

Essay▾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 % confidence is

Where ME = margin of error = 0.01

## Question 11

Appeared in Year: *2012*

### Describe in Detail

Essay▾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,

I am consider first the positive sign of

Assume k = = median

So, the median is less than equal to and when…

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