8. Suppose that Xi,..., Xn is a sample from a normal population having unknown pa- rameters...
8. Suppose that Xi,..., Xn is a sample from a normal population having unknown pa- rameters μ and σ2 a) Devise a significance level α test of the null hypothesis 0 versus the alternative hypothesis for a given positive value σ1. b) Explain how the test would be modified if the population mean μ were known in advance 8. Suppose that Xi,..., Xn is a sample from a normal population having unknown pa- rameters μ and σ2 a) Devise a...
---------------------------------------------------------------------------------------------------------------------------------------------------------- Reference: 11.2.3 Suppose that X. , X are iid MI, σ2) where σ (E 9t*) is the unknown parameter but μ(€ 9) is assumed known. With preassigned α ε (0. 1), derive a level α LR test for a null hypothesis Ho : σ.-a> 0) against an alternative hypothesis H, : σ2 σ1 in the implementable form. {Note: Recall from the Exercise 8.5.5 that no UMP level a test exists for testing Ho versus 8.5.5 Let X, X, be...
Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n). Let X1,X2, , Xn be a random sample from a normal distribution with a known mean μ (xi-A)2 and variance σ unknown. Let ơ-- Show that a (1-α) 100% confidence interval for σ2 is (nơ2/X2/2,n, nơ2A-a/2,n).
N.B: Solve only for (i), but draw the power function for (i) as well. X, are iid random variables from the M11, σ2) unknown, μ 01 8.3.1 Suppose that X1, population where μ is assumed known but σ is fix a number α E (0, 1) and to positive numbers σο, σ1. g, () ( : M". Wi Derive the MP level a test for Ho : σ in the simplest implementable form: (i) σ0 versus H1 : σ-01 (>...
6. Let Xi 1,... ,Xn be a random sample from a normal distribution with mean u and variance ơ2 which are both unknown. (a) Given observations xi, ,Xn, one would like to obtain a (1-a) x 100% one-sided confidence interval for u as a form of L E (-00, u) the expression of u for any a and n. (b) Based on part (a), use the duality between confidence interval and hypothesis testing problem, find a critical region of size...
, Xn iid. N 5. Let Xi, (μ, σ2), μ E R and σ2 > 0 are both unknown. Find an asymp- totically likelihood ratio test (LRT) of approximate size α for testing μ-σ 2 H1:ťtơ2 Ho : versus , Xn iid. N 5. Let Xi, (μ, σ2), μ E R and σ2 > 0 are both unknown. Find an asymp- totically likelihood ratio test (LRT) of approximate size α for testing μ-σ 2 H1:ťtơ2 Ho : versus
We discuss population mien problen here. Usually, we use μ to denote the unknown mean of a population. Given a hypothesis is not difficult. From last homework, we have the following table: .R. Corresponding formula 。 Alternative hypothesis-RR Now, what we should do is just find proper test statistic and solve above equations. The choice of test statistics depends on different assumption. a. Normal population with known variance That is the case we have sample from a population with normal...
Suppose you have a sample of n independent observations X1,X2,...,Xn from a normal population with mean μ (known) and variance σ2 (unknown). (a) Find the ML estimator of σ2 . (b) Show that the ML estimator in (a) is a consistent estimator of θ. (c) Find a sufficient statistic for σ2. (d) Give a MVUE for θ based on the sufficient statistic.
1. The mean of a sample of 25 measurements of the diameter of a camshafts on a production line was 6.7 cm. Manufacturer specifications call for a mean diameter of 7 cm. Assume the diameters are known to have a normal distribution with unknown mean, μ, and known variance, σ2 = .2 (cm)2. a. Test H0: μ = 7 versus Ha: μ < 7 at level of significance α =.01. Find the p-value and state whether to reject the null...
please answer with full soultion. with explantion. (4 points) Let Xi, , Xn denote a randon sample from a Normal N(μ, 1) distribution, with 11 as the unknown parameter. Let X denote the sample mean. (Note that the mean and the variance of a normal N(μ, σ2) distribution is μ and σ2, respectively.) Is X2 an unbiased estimator for 112? Explain your answer. (Hint: Recall the fornula E(X2) (E(X)Var(X) and apply this formula for X - be careful on the...