We are looking to calculate the power of a one-sided test from n independent observations xi from a N (µ, σ2 ) distribution with a null hypothesis of H0 : µ = µ0 and an alternative H1 : µ > µ0. Supposing that we know σ2, we can form a test statistic T = (x¯ − µ0)/(σ/√n) and reject the null hypothesis when T > 1.645. This test has level α = 0.05. We want a formula for the power of this test against the alternative that µ = µ1. This power formula can be written as equal to P{Z > a} where Z is a standard normal. You need to find the a that is a function of n, σ2 , µ0, and µ1.
We are looking to calculate the power of a one-sided test from n independent observations xi...
We are looking to calculate the power of a one-sided test from n independent observations Xi from a N(μ, σ2) distribution with a null hypothesis of Ho : μ-μ0 and an alternative H, : μ 〉 μ0. Supposing that we know σ2, we can form a test statistic T= and reject the null hypothesis when T 〉 1.645. This test has level α 0.05. We want a formula for the power of this test against the alternative that μ-74-This power...
We are looking to calculate the power of a one-sided test from n independent observations from a N(μ, σ2) distribution with a null hypothesis of Ho : μ-μο and an alternative H1 : μ > μο. Supposing that we know σ2, we can form a test statistic o/Vn and reject the null hypothesis when T > 1.645. This test has level α = 0.05. We want a formula for the power of this test against the alternative that μ-A-This power...
The one-sample t statistic from a sample of n = 21 observations for the two-sided test of H0: μ = 60, Ha: μ ≠ 60 has the value t = –1.98. Based on this information: we would reject the null hypothesis at α = 0.05. All of the answers are correct. 0.025 < P-value < 0.05. we would reject the null hypothesis at α = 0.10
235 independent normal observations were collected from a population with known variance σ 2 = 15. Suppose that we want to perform a test of H0 : µ = 65 HA : µ > 65. Calculate the size of the test that rejects the null hypothesis when ¯x > 65.33. Calculate the power of the test from part (a) against the alternative that µ = 65.6.
4. It is desired to test H0 : µ = 20 Vs H1 : µ < 20, on the basis of a random sample of size 64 from normal distribution with population standard deviation σ = 2.4. The sample mean and sample standard deviation are found to be 19.5 and 2.5, respectively. (a) Test the hypothesis at α = 0.05. Compute the test statistics, critial regions, and perform the test. Will the result be difference if α is changed to...
2. A randon sample XI, X. is drawn frotn Normal(μ, σ2), where-oo < μ < oo and 0 < σ2 < x. To test the null hypothesis Ho : σ2-1 against the alternative H1: σ2 > 1, we have designed the following test Reject Ho if S>k where S2 = "LE:-1(x,-X)2, k ís a constant. Noticed that (n-1) distribution with degree of freedom 1 has a (a) Determine k so that the test will have size a. (b) Use k...
2. Suppose that we have n independent observations x1, ,Tn from a normal distribution with mean μ and variance σ2, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 7-2 log max L(μ, σ*) )-2 log ( max L(u, i) μισ (c) Explain as clearly as you can what happens to T, when our estimate of σ2 is less than 1. (d) Show...
Problem 3. Consider two independent samples, X1, . . . , Xm from a N(µ1, σ12 ) distribution and Y1, . . . , Yn from a N(µ2, σ22 ) distribution. Here µ1, µ2, σ12 and σ2 are unknown. Consider testing the null hypothesis that the two population variance are equal, H0 : σ12 = σ22 , against the alternative that these variances are different, H1 : σ12 ≠ σ12 . (a) Derive the LR test statistic Λ
Suppose that X1, X2, . . . , Xn is an iid sample of N (0, σ2 ) observations, where σ 2 > 0 is unknown. Consider testing H0 : σ 2 = σ 2 0 versus H1 : σ 2 6= σ 2 0 ; where σ 2 0 is known. (a) Derive a size α likelihood ratio test of H0 versus H1. Your rejection region should be written in terms of a sufficient statistic. (b) When the null...
2. Suppose that we have n independent observations x1,..., xn from a normal distribution with mean μ and variance σ, and we want to test (a) Find the maximum likelihood estimator of μ when the null hypothesis is true. (b) Calculate the Likelihood Ratio Test Statistic 2 lo g max L(μ, σ log | max L( 1) (c) Explain as clearly as you can what happens to T when our estimate of σ2 is less than 1. (d) Show that...