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(a) We write (u') to be the probability of type II error given the true value of μ = μ'. Prove that Po - μ² o/√√n B(μ') = where Þ(z) = P(Z ≤ z) is the cdf of Z ~ N(0, 1). = 6 Za+

(a) We write (u') to be the probability of type II error given the true value of μ = μ'. Prove that Po - μ² o/√√n B(μ') = where Þ(z) = P(Z ≤ z) is the cdf of Z ~ N(0, 1). = 6 Za+

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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3. (Sample size determination for hypothesis testing) Suppose that X₁,..., Xn N(μ, o2) and o² is known. We would like to test We consider the test statistic Ho: μ = po, and the critical region at significance level a 1- {=; C = T = X x; x > µo + %α² H : μ > μο· B(μ') = = (a) We write ß(µ') to be the probability of type II error given the true value of µ = µ'. Prove that 0 (za + 10 = 1²). where Þ(z) = P(Z ≤ z) is the cdf of Z ~ N(0,1). 1 n = = O 2.2.d. (b) We would like to determine the sample size n so that the probability of type I error is a and the probability of type II error is ß. Using part (a), prove that 2 - ( 0¹ (²0 + zp)) ². Po - (c) Let u denote the true average tread life of a certain type of tire. Consider testing Ho : μ = 25,000 versus H₁ : μ> 25, 000 from a normal population distribution with o = 1300. Assuming the true value of μ is μ' = 24,000. What is the sample size n so that the probability of type I error is a = 0.05 and the probability of type II error is = 0.025?
Transcribed Image Text:3. (Sample size determination for hypothesis testing) Suppose that X₁,..., Xn N(μ, o2) and o² is known. We would like to test We consider the test statistic Ho: μ = po, and the critical region at significance level a 1- {=; C = T = X x; x > µo + %α² H : μ > μο· B(μ') = = (a) We write ß(µ') to be the probability of type II error given the true value of µ = µ'. Prove that 0 (za + 10 = 1²). where Þ(z) = P(Z ≤ z) is the cdf of Z ~ N(0,1). 1 n = = O 2.2.d. (b) We would like to determine the sample size n so that the probability of type I error is a and the probability of type II error is ß. Using part (a), prove that 2 - ( 0¹ (²0 + zp)) ². Po - (c) Let u denote the true average tread life of a certain type of tire. Consider testing Ho : μ = 25,000 versus H₁ : μ> 25, 000 from a normal population distribution with o = 1300. Assuming the true value of μ is μ' = 24,000. What is the sample size n so that the probability of type I error is a = 0.05 and the probability of type II error is = 0.025?
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