# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 82 - 87 of 165

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## Question number: 82

» Probability » Probability of M Events Out of N

Appeared in Year: 2014

### Describe in Detail

A positive integer X is selected at random from the first 50 natural numbers.

Calculate P (X + 96/X > 50).

### Explanation

Total number of possible outcomes = number of ways in which the one random number can be chosen out of 50

Let Y be the event whose satisfied the relation, X + 96/X > 50

For which X we choose is

So, X takes values = 1,2, 48,49,50

Favorable | Others | Total | |

Avaible | 5 | 45 | 50 |

Choose number | 1 | 0 | 1 |

Choices |

Number

… (70 more words) …

## Question number: 83

» Statistical Methods » Tests of Significance » F-Test

Appeared in Year: 2012

### Describe in Detail

Indicate how you would test the hypothesis that the means of k independent normal populations are identical, clearly mentioning the null and the alternative hypotheses, the assumptions made, the test statistic used, and the critical region.

### Explanation

Let there are k independent normal population with different sample size n _{1}, n _{2}, …, n _{k}. The test of null hypothesis is that the means of k normal population is same and the alternative hypothesis is any two mean are not same.

Vs H _{1}: at least two mean are not equal

Under the assumptions that the po

… (93 more words) …

## Question number: 84

» Statistical Methods » Regression » Linear

Appeared in Year: 2011

### Describe in Detail

From a bivariate data set of 4995 observations, the following quantities have been calculated:

Obtain the estimated linear regression of X on Y.

### Explanation

Let the linear equation is

To solve this equation by least square method. In this approach, the residual sum of squares is minimized by partially differentiating with respect to and .

To differential this, we get the estimate of and

The given value is putting in these equation, we ge

… (75 more words) …

## Question number: 85

» Probability » Elements of Measure Theory

Appeared in Year: 2014

### Describe in Detail

Suppose that all the four outcomes 0 _{1}, 0 _{2}, 0 _{3} and 0 _{4} of an experiment are equally likely. Define A = (0 _{1}, 0 _{4}), B = (0 _{2}, 0 _{4}) and C = (0 _{3}, 0 _{4}). What can you say about the pairwise independence and mutually independence of the events A, B and C?

### Explanation

Given that there are four outcomes. Define A = (O _{1}, O _{4}), B = (O _{2}, O _{4}), C = (O _{3}, O _{4})

Their intersection is and Union is AUBUC = (O _{1}, O _{2}, O _{3}, O _{4})

P (A) = P (B) = P (C) = 1/2

P (AUBUC) =1

The events A, B and C are not pairwise independent because

But the events A and B are pa

… (144 more words) …

## Question number: 86

» Statistical Methods » Tests of Significance » F-Test

Appeared in Year: 2013

### Describe in Detail

Obtain 100 (1 -α) % confidence interval for the ratio of population variances by using two independent random samples from N (µ _{1}, σ _{1} ^{2}) and N (µ _{2}, σ _{2} ^{2}) under the assumption that the population means are (i) known and (ii) unknown.

### Explanation

Let X _{1}, X _{2}, …, X _{n} and Y _{1}, Y _{2}, …Y _{m} are the samples taken from independent N (µ _{1}, σ _{1} ^{2}) and N (µ _{2}, σ _{2} ^{2}).

In this question the hypothesis for testing is

Let s _{1} ^{2} and s _{2} ^{2} be the estimates variances of σ _{1} ^{2} and σ _{2} ^{2} based on sample sizes n and m.

and

Then

… (195 more words) …

## Question number: 87

» Probability » Tchebycheffs Inequality

Appeared in Year: 2015

### Describe in Detail

Let X be a random variable with E [X] = 3 and E [X ^{2}] = 13. Use Chebyshev’s inequality to obtain P [-2 < X < 8].

### Explanation

Let X be a random variable with mean µ and variance σ ^{2}. Then any k > 0, the Chebyshev’s inequality is

or

σ ^{2} = E [X ^{2}] - (E [X] ) ^{2} =4

Then, a lower bound for the probability

Using Chebyshev’s inequality

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