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

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.

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

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

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

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

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