# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 78 - 83 of 165

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## Question number: 78

» Statistical Methods » Correlation Coefficient » Partial Correlation

Appeared in Year: 2014

### Describe in Detail

Given the following correlation matrix of order 3 x 3

Calculate (i) r _{12.3} (ii) r _{1.23}

### Explanation

In general, the correlation matrix is in these notations

(i)

(ii)

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## Question number: 79

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2013

### Describe in Detail

Find the distribution of the ratio of two iid random variables with density function:

### Explanation

Let consider x and y are two iid random variable with density function is same. Then find the distribution of their ratio that is x/y. The joint pdf of x and y is

To find this we use Jacobian transformation technique, assume

X/Y = u, Y = v

X = uv and Y = v

So, the joint density functi

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## Question number: 80

» Probability » Standard Probability Distributions » Normal

Appeared in Year: 2012

### Describe in Detail

12·3 % of the candidates in a public examination score at least 70%, while another 6·3 % score at most 30%. Assuming the underlying distribution to be normal, estimate the percentage of candidates scoring 80 % or more.

### Explanation

Let total marks obtain is 100. Assuming the underlying distribution to be normal, the mean µ and variance σ ^{2}. It is given that

The value of z corresponding to an area

0.500 - 0.123 = 0.377

We can write

Similarly, . It is given that

The value of z correspo

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## Question number: 81

» Probability » Probability of M Events Out of N

Appeared in Year: 2013

### Describe in Detail

A fair die is rolled twice. Let A be the event that the first throw shows a number ≤ 2, and B be the event that the second throw shows at least 5. Show that P (AUB) =5/9.

### Explanation

A fair die is rolled twice, the sample space consists of thirty six outcomes. The sample space is

(1,1), (1,2), (1,3), (1,4), (1,5), (1,6)

(2,1), (2,2), (2,3), (2,4), (2,5), (2,6)

(3,1), (3,2), (3,3), (3,4), (3,5), (3,6)

(4,1), (4,2), (4,3), (4,4), (4,5), (4,6)

(5,1), (5,2), (5,3), (5,4), (5,5), (5,6)

(6,1), (6,2), (6,3), (6,4), (5,5)

… (67 more words) …

## 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 |

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## 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

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