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?
If , and 02 be two unbiased estimators of 0 with variances of and o, respectively. What is the value of k so that the unbiased estimator Ô3 = 3kô, + (1- 3k)@, of 0 has a minimum variance. af (c) 3(a+of) (a) (b) af 3(0+o) (d) (e) None
Ch.6. "A carpet manufacturing company is comparing the tensile strengths of two different types of synthetic fiber. Synthetic fiber A used in manufacturing carpet has tensile strength that is normally distributed with mean 75 psi and standard deviation 8 psi, and synthetic fiber B has tensile strength that is 70 psi and standard deviation 12 psi. Random sample of nA = 16 and nB = 9 of fiber specimens are selected. Find the probability that 3.5 < x̄a - x̄b < 6.5. Determine the value of zs
Let X and S? be the sample mean and sample variance calculated from nj number of Normal (u1, 01) random variables. Let X, and S; be the sample mean and sample variance calculated from n2 number of Normal (2, 02) random variables. (a). Show that X - X, is an unbiased estimator of u1 - µ2. (b). Find the standard deviation of X1 - X2. (C). Propose an estimator to estimate the standard deviation of X - X2. (d). Suppose that the two samples have the same variance o? = o; = o?. Show that S3 = (n -1)S+(n2-1)S n1+n2-2 is an unbiased estimator of o?. (e). In the setting of part (d), compute the relative efficiency of S? to S for estimating o2.
Let X is Normally (Gaussian) distributed with mean 3 and variance 25. Compute the following. P(X>7) P(3T) = 10,7 = ? P(X>T) = 0.9999, T = ?
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
Let X1,X2,..., X3 denote a random sample from a population having mean u and variance o?. Which of the following is/are unbiased estimator/s of u? X1+X2++X, 7 2X1-X6+X4 2 3X1-X3+X4 O3 2 2(X,+X,++X,) 7
Let X1, X2,..., X3 denote a random sample from a population having mean u and variance o?. 302 Which of the estimators have a variance of 2 X1+X2++X7 7 2X1-X6+X4 3X1-X3+X4 2 2(X1+X2+…+X7) 4 7
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…
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Suppose you have a sample of 9 observations Y1,...,Y, from a normal population with mean 2 and variance 4. Let Y and S denote the sample mean and sample variance, respectively. What is the distribution of E ( (a) (b) 653 (c) Suppose X1,X2, X3, X4, X5 are independent random variables from a normal population with mean 0 and variance 4. Define S3 = E(X, - X). What is the distribution of 253 + S3?

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