Let Y, represent the ith normal population with unknown mean #, and unknown varianceof for i=1,2. Consider independent random samples, Ya,Ya. Yin, of size n,, fromthe ith population with sample mean Y, and sample variance S?=₁1(Yu - Y.².(g) For non-zero constants a,'s, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State allthe relevant parameters of the distribution.(h) Find the standard error of U₂ in part (g), assuming that of = o2 = 0².(i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the twopopulation means, #, and p2, when of=o=o² is known.(j) Define appropriate rejection regions, in terms of Y₁ - ₂, for testing Ho: #₁ = ₂against a two-sided alternative hypothesis at the a level of significance.

For the first variable :The sample size is .The sample mean is .The sample variance is .For the…

Answered: Let Y, represent the th normal…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

Let Y, represent the ith normal population with unknown mean #, and unknown variance
of for i=1,2. Consider independent random samples, Ya,Ya. Yin, of size n,, from
the ith population with sample mean Y, and sample variance S?=₁1(Yu - Y.².
(g) For non-zero constants a,'s, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all
the relevant parameters of the distribution.
(h) Find the standard error of U₂ in part (g), assuming that of = o2 = 0².
(i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the two
population means, #, and p2, when of=o=o² is known.
(j) Define appropriate rejection regions, in terms of Y₁ - ₂, for testing Ho: #₁ = ₂
against a two-sided alternative hypothesis at the a level of significance.

Expert Solution

Step 1: Write the given information.

For the first variable $Y subscript 1$ :

$Y subscript 1 tilde space N open parentheses mu subscript 1 comma space sigma subscript 1 superscript 2 close parentheses$

The sample size is $n subscript 1$ .

The sample mean is $top enclose Y subscript 1$ .

The sample variance is $S subscript 1 superscript 2 equals fraction numerator 1 over denominator n subscript 1 minus 1 end fraction sum from j equals 1 to n subscript 1 of open parentheses Y subscript j minus Y with bar on top subscript 1 close parentheses squared$ .

For the second variable $Y subscript 2$ :

$Y subscript 2 tilde N open parentheses mu subscript 2 comma space sigma subscript 2 superscript 2 close parentheses$

The sample size is $n subscript 1$ .

The sample mean is $Y with bar on top subscript 2$ .

The sample variance is $S subscript 2 superscript 2 equals fraction numerator 1 over denominator n subscript 2 minus 1 end fraction sum from j equals 1 to n subscript 2 of open parentheses Y subscript j minus Y with bar on top subscript 2 close parentheses squared$ .

The samples are independent.