Let Y, represent the ith normal population with unknown mean 44, and unknown varianceof for i 1,2. Consider independent random samples, Ya, Y₁2,,Yin, of size ni, fromthe ith population with sample mean Y, and sample variance S?=1Σj=1(Y₁j - Y₁².(a) What is the distribution of Y;? State all the relevant parameters of the distribution.(b) Find a level a test (that is, the rejection region) for testing Ho : μi = μio versusHa: Pipio when of is unknown and n, is small.(c) In the context of the test in part (b), state the Type I error and give a probabilitystatement for the level of significance, a.

Answered: m-1 j=) t is the distribution of Y,?…

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 44, and unknown variance
of for i 1,2. Consider independent random samples, Ya, Y₁2,,Yin, of size ni, from
the ith population with sample mean Y, and sample variance S?=1Σj=1(Y₁j - Y₁².
(a) What is the distribution of Y;? State all the relevant parameters of the distribution.
(b) Find a level a test (that is, the rejection region) for testing Ho : μi = μio versus
Ha: Pipio when of is unknown and n, is small.
(c) In the context of the test in part (b), state the Type I error and give a probability
statement for the level of significance, a.

Expert Solution

Step 1: Write the distribution of Yi bar and state its all the relevant parametersWrite the distribution of

Write the distribution of Yi bar and state its all the relevant parametersWrite the distribution of Yi bar and state its all the relevant parameters

Consider the independent random variables, $Y subscript i 1 end subscript comma space Y subscript i 2 end subscript comma space. space. space. comma space Y subscript i n subscript i end subscript$ of size $n subscript i$ from the i^th population with sample mean $Y with bar on top subscript i$ and sample variance $S subscript i superscript 2 equals fraction numerator 1 over denominator n subscript i minus 1 end fraction sum from j equals 1 to n subscript i of open parentheses Y subscript i j end subscript minus Y with bar on top subscript i close parentheses squared$ .

(a)

The distribution of the sample mean $Y with bar on top subscript i$ will have the following parameters:

Mean ( $E open parentheses Y with bar on top subscript i close parentheses$ ): The expected value of $Y with bar on top subscript i$ will be equal to the population mean, which is $μ_{i}$ .
Variance ( $Var open parentheses Y with bar on top subscript i close parentheses$ ): The variance of will be equal to the population variance divided by the sample size, i.e., $Y with bar on top subscript i$ $fraction numerator sigma subscript i superscript 2 over denominator n subscript i end fraction.$
Standard deviation ( $S D open parentheses Y with bar on top subscript i close parentheses$ ): The standard deviation of $Y with bar on top subscript i$ will be the square root of its variance, i.e., $square root of fraction numerator sigma subscript i superscript 2 over denominator n subscript i end fraction end root.$

The distribution of $Y with bar on top subscript i$ is normal distribution since the samples are taken from a normal distribution.

$table row cell E open square brackets Y with bar on top subscript i close square brackets end cell equals cell E open square brackets fraction numerator sum from j equals 1 to n subscript i of Y subscript i j end subscript over denominator n subscript i end fraction close square brackets end cell row blank equals cell 1 over n subscript i cross times E open square brackets sum from j equals 1 to n subscript i of Y subscript i j end subscript close square brackets end cell row blank equals cell 1 over n subscript i cross times sum from j equals 1 to n subscript i of E open square brackets Y subscript i j end subscript close square brackets end cell row blank equals cell 1 over n subscript i cross times sum from j equals 1 to n subscript i of mu subscript i end cell row blank equals cell 1 over n subscript i cross times n subscript i mu subscript i end cell row blank equals cell mu subscript i end cell end table$

$table row cell V a r open square brackets Y with bar on top subscript i close square brackets end cell equals cell V a r open square brackets fraction numerator sum from j equals 1 to n subscript i of Y subscript i j end subscript over denominator n subscript i end fraction close square brackets end cell row blank equals cell 1 over open parentheses n subscript i close parentheses squared cross times V a r open square brackets sum from j equals 1 to n subscript i of Y subscript i j end subscript close square brackets end cell row blank equals cell 1 over open parentheses n subscript i close parentheses squared cross times sum from j equals 1 to n subscript i of V a r open square brackets Y subscript i j end subscript close square brackets end cell row blank equals cell 1 over open parentheses n subscript i close parentheses squared cross times sum from j equals 1 to n subscript i of sigma subscript i superscript 2 end cell row blank equals cell 1 over open parentheses n subscript i close parentheses squared cross times n subscript i sigma subscript i superscript 2 end cell row blank equals cell fraction numerator sigma subscript i superscript 2 over denominator n subscript i end fraction end cell end table$

The relevant parameters of the distribution are sample mean is $mu subscript i$ and sample variance is $fraction numerator sigma subscript i superscript 2 over denominator n subscript i end fraction$ .

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m-1 j=) t is the distribution of Y,? State all the relevant parameters of the evel a test (that is, the rejection region) for testing H₁ : µ; = o when of is unknown and n; is small. ntext of the test in part (b), state the Type I error and give a or the level of significance, a.