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

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Appeared in Year: 2013

Essay Question▾

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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

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Appeared in Year: 2015

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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

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Appeared in Year: 2012

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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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Question number: 89

» Probability » Standard Probability Distributions » Binomial

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Appeared in Year: 2010

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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

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Appeared in Year: 2011

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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

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Appeared in Year: 2011

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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) …