3. Suppose £1,..., n are a simple random sample from an exp(A) distribution. It can be shown thatthe UMP a-level tests of Ho: A = Ao against HA:A> Ao and HA A(x) =1 if në > X², 2n/(20)0 otherwise== {1{}Y<(x) = { !Determine the Uniformly Most Accurate (1-a) x 100% lower bound for X.

In the context of hypothesis testing for the exponential distribution with parameter , you have the…

Answered: 3. Suppose ₁,... In are a simple random…

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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1. Based on a random sample of 3 observations, consider two competing estimators of the population mean u: în =(X1 + X2 + Xa), ia = ;X1 +x2 + X3 4 (a) Are they unbiased? (b) Which estimator is more efficient? How much more efficient? 2. Let {X1, X2, ..., Xm} be an independent random sample from Poisson(A), (a) Show that both X and S² are unbiased estimator of ). (b) Calculate the Cramer-Rao Lower Bound with respective to A. (c) Which estimator should be preferred and why?
Let Y₁, Y₂, ..., Y₁ denote a random sample from a normal distribution with known mean μ and unknown variance o². fl Find the most powerful a-level test of H₁: ² = of vs. Hå: o² = 02, where 0² > 08. Show that this test is equivalent to a x² test. [Hint: Recall that for Z₁, Z₂, Zn independent standard normal random variables, ₁Z² has a x2² distribution with n df.] 9 Is the test from part (b) uniformly most powerful (UMP) for Hå: 0² > of?
Suppose the rbinom() function is used to generate 20 sets of data (each with sample size of n=10 from the distribution Bin(1, 0.8). For each of the 20 datasets, I conduct a sign test. The null hypothesis for the sign test is that the population probability of getting 1 is 0.5; the alternative hypothesis is that the population probability is not 0.5. Then I calculate the proportion of rejection across the 20 datasets and found out that 6 of the 20 datasets had a significant result at α=0.5; in other words, the rejection rate is 6/20 =0.3. In addition, I also recorded the confidence intervals for estimating the population proportion for the 20 datasets. Given the rejection rate, what is the proportion of the CI that contains 0.5? A. 0.05 B. 0.3 C. 0.5 D. 0.7 E. 1 F. not enough information
A large population (consisting of measurements of diameters of ball bearings) has mean u and standard deviation o, neither of which is perfectly known. A random sample of size n = 25 observations will be taken, namely the diameters U1, U2, ·.., U25 will be observed. In other words, U1, U2, · .., U25 are independent and identically distributed random variables with u = E(U1) and Var(U1) = o². No further knowledge about the shape of the distribution is known. Define, 25 1 X 25 Ui 25 i=D1 (i) If someone says E(X 25) = 13, what information does he give you about u, if any? (ii) If someone says o = 5, what is Var(X25)? • (iii) What is the chance, approximately, that a sample of size n = 25 will have its mean, X25, smaller than the population mean u? In other words, find an approximation of P(X 25 < u). [Hint: see if the value of o is relevant or not to answer the question.] (iv) In part (iii) while approximating what theorem did you use, if any? • (v) What is the chance, approximately, that…
Say that we have a null hypothesis that X1,..., X16 that are Normally distributed with mean 0 and variance 1. Call the sample mean X. Consider the test where p-value = 20 (4 X) - 24 (4|X) – 1 We will reject this if p-value < a. a) Show that the type I error rate of this test is a. b) Say that X1,..., X16 that are Normally distributed with mean 1 and variance 1. What is the type II error rate of this test when a = 0.05? •..
show that if X11, X12,..., X1, X21, X22,..., X2n₂ are independent random variables, with the first n constitut- ing a random sample from an infinite population with the mean μ₁ and the variance of and the other n2 constitut- ing a random sample from an infinite population with the mean μ2 and the variance o2, then 1 M2 (a) E(X₁-X₂) = μ₁ −μ2; 07 0 22 (b) var(X₁-X₂) = + n₁ n₂
b. Let X1, X2, X3, and X4 represent the weight of shipment packages at a certain shipment facility. Suppose they are independent normal random variables with means H1 = 3.6, 42=0.9, 3= 1.8, 4 = 7.4 pounds and variances o = = = Find P (2X₁ + 1X2 + 3X3 + 1X4 ≤ 17.6). = 1.
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]
2. If X is a normal random variable with mean u 3 and variance o2 16, find P{X-1 }
3. If x1, X2, X3, ..., Xn is a random sample from a normal 1 distribution N (µ, 1) show that t= E xí is an unbiased estimator of n i = 1 H² + 1.
1. Six batteries taken at random from a large lot were subjected to life tests with the following results: 19, 22, 18, 16, 25, and 20 hours. Find the probability that the sample mean of x= 20 hours, as observed here, or one less than 20 hours could arise if the true mean of the lot of batteries is 21.5 hours, assuming a known variance of o = 10 hours, for the individual batteries in the lot. %3D
3.3.24. Let X1, X2 be two independent random variables having gamma distribu- tions with parameters a₁ = 3, B₁ = 3 and a2 = 5, B2 = 1, respectively. (a) Find the mgf of Y = 2X₁ +6X2. (b) What is the distribution of Y?

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