# ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 86 - 91 of 165

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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.

… (133 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

… (68 more words) …

## 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

… (53 more words) …

## Question number: 89

» Probability » Standard Probability Distributions » Binomial

Appeared in Year: 2010

### Describe in Detail

Let X _{1}, X _{2}, …, X _{m} be i. i. d. random variables with common p. m. f.

obtain the p. m. f. of S _{m} = X _{1} + X _{2} + …. + X _{m}.

### Explanation

Let X _{1}, X _{2}, …, X _{m} i. i. d. random variables with common p. m. f. is P (X = k) which is a binomail random variables with common parameters n and p respectively. Then, the p. m. f. of S _{m} = X _{1} + X _{2} + …. + X _{m}, sum of random variables are found by moment generating function. The moment generating function of binomial distribution is

… (95 more words) …

## Question number: 90

» Statistical Methods » Tests of Significance » Chi-Square

Appeared in Year: 2011

### Describe in Detail

Explain how to carry out the chi-squared test for on the basis of a random sample

X _{1}, X _{2}, …, X _{n} from N (µ, σ ^{2}) population.

### Explanation

Given that a random sample X _{1}, X _{2}, …, X _{n} from N (µ, σ ^{2}) population. The hypothesis is

The null hypothesis is test by chi-square test only assume the sample size is less than 30.

For this we use the likelihood ratio test

Under the model, the parameter space is

… (130 more words) …

## Question number: 91

» Probability » Standard Probability Distributions » Gamma

Appeared in Year: 2011

### Describe in Detail

Let

Show that f (x) is a probability density function. Obtain V (X).

### Explanation

if X is a continuous random variable and f (x) is a continuous function of X, then f (x) is a probability density function if

… (128 more words) …