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b. Suppose of = 2, o% = 2.5, and m = n. Find the sample size n so that X - Ý will be within one unit of µ1 – µ2 with probability 0.95.

b. Suppose of = 2, o% = 2.5, and m = n. Find the sample size n so that X - Ý will be within one unit of µ1 – µ2 with probability 0.95.

MATLAB: An Introduction with Applications
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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**Exercise 5** Suppose that \( X_1, \cdots, X_m \) and \( Y_1, \cdots, Y_n \) are two samples, with \( X_i \sim N(\mu_1, \sigma_1) \) and \( Y_i \sim N(\mu_2, \sigma_2) \). The difference between the sample means, \( \bar{X} - \bar{Y} \), is then a linear combination of \( m + n \) normal random variables. All the random variables are independent. Answer the following questions: a. Use moment generating functions to show that \( \bar{X} - \bar{Y} \) follows a normal distribution. Find the mean and variance of this distribution. b. Suppose \( \sigma_1^2 = 2 \), \( \sigma_2^2 = 2.5 \), and \( m = n \). Find the sample size \( n \) so that \( \bar{X} - \bar{Y} \) will be within one unit of \( \mu_1 - \mu_2 \) with probability 0.95. (The text here is obscured by redaction.) (Note: The redacted part in the image obscures some information that might be relevant for solving part b.)
Transcribed Image Text:**Exercise 5** Suppose that \( X_1, \cdots, X_m \) and \( Y_1, \cdots, Y_n \) are two samples, with \( X_i \sim N(\mu_1, \sigma_1) \) and \( Y_i \sim N(\mu_2, \sigma_2) \). The difference between the sample means, \( \bar{X} - \bar{Y} \), is then a linear combination of \( m + n \) normal random variables. All the random variables are independent. Answer the following questions: a. Use moment generating functions to show that \( \bar{X} - \bar{Y} \) follows a normal distribution. Find the mean and variance of this distribution. b. Suppose \( \sigma_1^2 = 2 \), \( \sigma_2^2 = 2.5 \), and \( m = n \). Find the sample size \( n \) so that \( \bar{X} - \bar{Y} \) will be within one unit of \( \mu_1 - \mu_2 \) with probability 0.95. (The text here is obscured by redaction.) (Note: The redacted part in the image obscures some information that might be relevant for solving part b.)
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