Let Y, represent the ith normal population with unknown mean 4, and unknown varianceof for i=1,2. Consider independent random samples, Ya, Y₁2, Yin, of size n,, fromthe ith population with sample mean Y, and sample variance S?=1 Σ₁-1 (₁-₁)².(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: #₁ = μio versusHa Hi Ho 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.

Consider the independent random variables, of size from the ith population with sample mean and…

Answered: Let Y, represent the ith normal…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

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Question

Let Y, represent the ith normal population with unknown mean 4, and unknown variance
of for i=1,2. Consider independent random samples, Ya, Y₁2, Yin, of size n,, from
the ith population with sample mean Y, and sample variance S?=1 Σ₁-1 (₁-₁)².
(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: #₁ = μio versus
Ha Hi Ho 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 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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