Let X1, X2,., X25 be a simple random sample from a population having distri-bution N(0, o²), where o² is an unknown parameter. For the hypotheses...Ho : o < 1,H1 : o² > 1,consider the test that rejects Ho when 5 x? > c, where the constant c ischosen to have level of significance a =i=D10.05.Obtain the constant c. [Hint: The sampling distribution of z EE1 X7 isXi25): Now find c by solving max,2<1 P(E1 X? > c) = 0.05.]i=125• After we observed a simple-random sample and obtained , X? = 38 whatwill be your decision?25• Instead, if our simple random sample gave E1 X? = 36 what will be yourdecision?25i3D1

Let X1,X2,...,X25 be a simple random sample of population having distribution N(0,σ2). Here σ2 is…

Answered: Let X1, X2, . , X25 be 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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Let X₁,X₂,...,Xₙ denote a random sample from a distribution that is N(0,θ), where the variance θ is an unknown positive number. Show that there exists a uniformly most powerful test of size α for testing the simple hypothesis H₀ : θ = θ', where θ' is a fixed positive number.
Let X1 and X2 constitute a random sample from a nor-mal population with σ2 = 1. If the null hypothesis μ = μ0 is to be rejected in favor of the alternative hypothesis μ = μ1 > μ0 when x > μ0 + 1, what is the size of the criti-cal region?
If X1, X2, ... , Xn constitute a random sample from anormal population with μ = 0, show that ni=1X2inis an unbiased estimator of σ2.
Suppose that three random variables X1, X2, X3 form a random sample from the uniform distribution on interval [0, 1]. Determine the value of E[(X1-2X2+X3)2]
Consider a random sample X1,...,Xn,... ∼ iid Beta(θ,1) for n > 2. Prove that the MLE and UMVUE are both consistent estimators for θ
There are two traffic lights on a commuter's route to and from work. Let X1 be the number of lights at which the commuter must stop on his way to work, and X2 be the number of lights at which he must stop when returning from work. Suppose that these two variables are independent, each with the pmf given in the accompanying table (so X1, X2 is a random sample of size n = 2). x1 0 1 2 p(x1) 0.1 0.2 0.7 ? = 1.6, ?2 = 0.44 (a) Determine the pmf of To = X1 + X2. to 0 1 2 3 4 p(to) (b) Calculate ?To. ?To = How does it relate to ?, the population mean? ?To = · ? (c) Calculate ?To2. ?To2 = How does it relate to ?2, the population variance? ?To2 = · ?2
If X1, X2, ... , Xn constitute a random sample of size nfrom a geometric population, show that Y = X1 + X2 +···+ Xn is a sufficient estimator of the parameter θ.
Suppose that the random variables X1,...,Xn form a random sample of size n from the uniform distribution on the interval [0, 1]. Let Y1 = min{X1,. . .,Xn}, and let Yn = max{X1,...,Xn}. Find E(Y1) and E(Yn).
There are two traffic lights on a commuter's route to and from work. Let X1 be the number of lights at which the commuter must stop on his way to work, and X2 be the number of lights at which he must stop when returning from work. Suppose that these two variables are independent, each with the pmf given in the accompanying table (so X1, X2 is a random sample of size n = 2). Can you help me with 3 and 4?
For the z test, the critical region for rejection of H0: A. Is determined only by α B. Is determined by α and N C. Allows us to accept the null hypothesis D. Depends on N
Suppose X1, . . . , Xn be a random sample from the Beta(θ, 1) distribution. Find the P-value for the LRT test of the hypotheses H0 : θ ≥ 1 vs H1 : θ < 1
Suppose {Xi} are iid 〜Unif(0,θ). Consider the test Ho : θ=θo against H1 :θ<θo Use the likelihood ratio test to show that the critical region C has the form C={(x1,...xn)|yn≤cθo} Find c in term of the significant level a and θo. .

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