8.3.11 Let X₁,..., Xn and Y₁,..., Ym be independent random samples from thedistributions N(01, 03) and N(02,04), respectively.=1(a) Show that the likelihood ratio for testing Ho : 01alternatives is given by02, 03 04 against all=n/2m/2mΣ(xi(T₁ - T)²/nΣ(yi - g)2/m57² /m]m[(x₂ −u)² + Σ (Yi- 1)²] / (m−u)² (m + n)where u = (na+my)/(n + m).(n+m)/2¹

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Answered: 8.3.11 Let X₁,..., Xn and Y₁,..., Ym be…

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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Suppose X₁,, Xn from a normal distribution N(μ, o²) where both μ and o are unknown. We wish to test the hypotheses Ho: o² = o vs. Ha: 0² # of at the level a. Show that the likelihood ratio test is equivalent to the x² test.
Let X1, X2, X3, and X4 be a random sample of observations from a population with mean μ and variance σ2. Consider the following estimator of μ: �"!= 0.15 X1 + 0.35 X2 + 0.20 X3 + 0.30 X4.What is the variance of � "!
a. Let Z ~ Normal(0, 1) be a continuous standard normal random variable with mean of 0 and standard deviation of 1. What is the probability that Z comes out to be less than -1.64? Normal(26.2, 5) is a continuous normal random variable 26.2 and standard deviation of o = 5. What is the probability that b. Suppose instead that X ~ with mean of µ = X comes out to be less than 18? c. Working still with X That is, find a value c such that the probability that X is bigger than c is 2.5%. Normal(26.2, 5), find a value c such that P(X > c) = 0.025.
Calculate the value of the unpooled test statistic t in each of the following situations. In each case, assume the null hypothesis is Ho: H, - H, = 0. (Ro (a) x, = 38, s, = 8, n, = 64; x2 = 36, s2 = 7, n, = 49 t = (b) The difference in sample means is 43, s.e.(x, - x2) = 27.
(e) Show that the sample mean of a simple random sample (drawn without replacement from a normal population) is dis- tributed normally with mean u and variance -² (1--=-1). n where N and n are population and sample sizes respectively. 4 In the 1 1 TI
620A city law-enforcement official has stated that 20% or the items sold by pawn shops within the city have been stolen. Ralph has just purchased 4 items from one of the city's pawn shops. Assuming that the official is correct, and for x = the number of Ralph's purchases that have been stolen, determine the following: a. P(x = 0) c. P(1 s x s 3) b. P(2 s x) d. P(x s 2) %3D
Suppose X is a random variable of uniform distribution between 1 and 7. Find E(X)
16. Let X, X2 X, be a random sample from a normal distribution, X, N(6, 25), and denote by X and S2 the sample mean and sample variance. Use tables from Appendix C to find each of the following: (a) P[3 < X < T. (b) P[L860< 3(X-6)/S]. (c) PIS <319375]
8.3
5. Suppose there are 2 populations, each with mean ₁ and 4₂ and standard deviation of ₁ and 2 respectively. The interest is to measure a random variable X by taking a random sample of size n from the first population. In a similar manner, another random variable Y is measured by independently taking a random sample of size m from the second population. Assuming n and m are sufficiently large, then demonstrate the following properties: a. X~N (μ₁, 1) b. 8~ N(H₂. /) c. (8-8)-№ (1₁ - 1₂ + 15t m
5.9. Show that Z is a standard normal random variable; then, for x > 0, (a) P(Z > x) = P(Z x) = 2P(Z > x}; (c) P{|Z| < x) = 2P(Z < x} 1.

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