# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 84 - 88 of 165

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## 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 get

the estimated linear regression of X on Y i…

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## 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 pairwise independent because

Similarly the events…

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## 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

This two chi-square random variable are independent. So, H _{0} can be test b…

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## 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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## Question number: 88

» Probability » Elements of Measure Theory

Appeared in Year: 2012

### Describe in Detail

Of three independent events A, Band C, A only happens with probability ¼, B only happens with probability 1/8 and C only happens with probability 1/12. Find the probability that at least one of these three events happens.

### Explanation

Given that P (A) =1/4, P (B) =1/8, P (C) =1/12

then probability that at least one event of these three events happens is

The events are independent because only one event happens

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