Homework Answers
a). The size of a test is the maximum probability of incorrectly rejecting the null hypothesis when it is true, i.e. the maximum probability of committing a Type 1 error.
The size of this test = 
The size of this test is 10 % or probabilty = 0.1
b). The power of this test (at 
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Suppose that in the following hypothesis testing situation about a parameter θ (θ 1) Ho : θ-1 aga...
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho : θ 1 versus H1 :...
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho : θ 1 versus H1 : θ 1 . We have two tests: where 0<c<1 (a) Show that the power functions of the two tests are A(0)-1-(0.9)θ and β2(0)-1 + d|θ Inc-1), respectively. (b) Calculate the size of the φι test. Then, find the value of c that gives the same size for the φ2 test. (c) Is фг a most powerful test of...
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho...
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho : θ 1 versus H1 : θ 1 . We have two tests: where 0<c<1 (a) Show that the power functions of the two tests are A(0)-1-(0.9)θ and β2(0)-1 + d|θ Inc-1), respectively. (b) Calculate the size of the φι test. Then, find the value of c that gives the same size for the φ2 test. (c) Is фг a most powerful test of...
5. For X follows Exp(6) (exponential distribution with parameter θ), a hypothesis test rejects th...
5. For X follows Exp(6) (exponential distribution with parameter θ), a hypothesis test rejects the null hypothesis Ho : θ-1 when X k versus H1 : θ > 1. (a) Show that for any k greater than -log(0.05), the test has the probability of type I error less than 0.05 (b) Show that the power of the test at θ-10 is larger when k-1 than k-2. (c) Let k-_ log(0.05), calculate the power function in terms of θ when θ...
QUESTION 30 "In hypothesis testing, the hypothetical assumption about the population parameter is" the alternative hypothesis....
QUESTION 30 "In hypothesis testing, the hypothetical assumption about the population parameter is" the alternative hypothesis. the null hypothesis. either the null or the alternative. neither the null nor the alternative. QUESTION 31 "When the following hypotheses are being tested at a level of significance of a Ho: 1500 H1: <500 the null hypothesis will be rejected, if the p-value is" Sa. a. = a /2 < 1 -a/2.
N(0,02). We wish to use a 1. [18 marks] Suppose X hypothesis single value X = x to test the null Ho : 0 = 1 against the...
N(0,02). We wish to use a 1. [18 marks] Suppose X hypothesis single value X = x to test the null Ho : 0 = 1 against the alternative hypothesis H1 0 2 Denote by C aat the critical region of a test at the significance level of : α-0.05. (f [2 marks] Show that the test is also the uniformly most powerful (UMP) test when the alternative hypothesis is replaced with H1 0 > 1 (g) [2 marks Show...
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0....
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0. Consider testing Ho : θ-Bo versus H1: θ (a) Derive a size α likelihood ratio test (LRT). (b) Derive the power function P(0) of the LRT. θο, where θο is known. (c) Now consider putting an inverse gamma prior distribution on θ, namely, 1 00), a 4a where a and b are known. Show how to carry out the Bayesian test (d) Is the...
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS θ for the purpose of testing Ho : θ-2 against H, : θ < 2 at α : 0.10 level of significance. Suppose the decision rule is to be...
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS θ for the purpose of testing Ho : θ-2 against H, : θ < 2 at α : 0.10 level of significance. Suppose the decision rule is to be based on Xmax, the largest order statistic. What would be the probability of committing a Type II error when θ 1.7.
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS...
Suppose that X1, X2,..., Xn are iid from where a 1 is a known constant and...
Suppose that X1, X2,..., Xn are iid from where a 1 is a known constant and θ > 0 is an unknown parameter. (a) Show that the likelihood ratio rejection region for testing Ho : θ θο versus H : θ > θο can be written in terms of X(n), the maximum order statistic. (b) Derive the power function of the test in part (a). (c) Derive the most powerful (MP) level α test of Ho : θ-5 versus H1...
1. Consider the following hypothesis test for a Poisson(a) population Ho : α = 1 H1 :a = 2 a) Fin...
1. Consider the following hypothesis test for a Poisson(a) population Ho : α = 1 H1 :a = 2 a) Find the rejection region for a likelihood ratio test with k-4 and sample size n. (b) Find the level of the rejection region found in the previous part with n 15 (c) Find the power of a 05-level test with n 100.
1. Consider the following hypothesis test for a Poisson(a) population Ho : α = 1 H1 :a =...
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho : θ 1 versus H1 : θ 1 . We have two tests: where 0<c<1 (a) Show that the power functions of the two tests are A(0)-1-(0.9)θ and β2(0)-1 + d|θ Inc-1), respectively. (b) Calculate the size of the φι test. Then, find the value of c that gives the same size for the φ2 test. (c) Is фг a most powerful test of...
Suppose Xi and X2 are iid from 0, otherwise, where θ 0, and consider testing Ho : θ 1 versus H1 : θ 1 . We have two tests: where 0<c<1 (a) Show that the power functions of the two tests are A(0)-1-(0.9)θ and β2(0)-1 + d|θ Inc-1), respectively. (b) Calculate the size of the φι test. Then, find the value of c that gives the same size for the φ2 test. (c) Is фг a most powerful test of...
5. For X follows Exp(6) (exponential distribution with parameter θ), a hypothesis test rejects the null hypothesis Ho : θ-1 when X k versus H1 : θ > 1. (a) Show that for any k greater than -log(0.05), the test has the probability of type I error less than 0.05 (b) Show that the power of the test at θ-10 is larger when k-1 than k-2. (c) Let k-_ log(0.05), calculate the power function in terms of θ when θ...
QUESTION 30 "In hypothesis testing, the hypothetical assumption about the population parameter is" the alternative hypothesis. the null hypothesis. either the null or the alternative. neither the null nor the alternative. QUESTION 31 "When the following hypotheses are being tested at a level of significance of a Ho: 1500 H1: <500 the null hypothesis will be rejected, if the p-value is" Sa. a. = a /2 < 1 -a/2.
N(0,02). We wish to use a 1. [18 marks] Suppose X hypothesis single value X = x to test the null Ho : 0 = 1 against the alternative hypothesis H1 0 2 Denote by C aat the critical region of a test at the significance level of : α-0.05. (f [2 marks] Show that the test is also the uniformly most powerful (UMP) test when the alternative hypothesis is replaced with H1 0 > 1 (g) [2 marks Show...
Suppose that Xi, X2,..., Xn is an iid sample from r > 0 where θ 0. Consider testing Ho : θ-Bo versus H1: θ (a) Derive a size α likelihood ratio test (LRT). (b) Derive the power function P(0) of the LRT. θο, where θο is known. (c) Now consider putting an inverse gamma prior distribution on θ, namely, 1 00), a 4a where a and b are known. Show how to carry out the Bayesian test (d) Is the...
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS θ for the purpose of testing Ho : θ-2 against H, : θ < 2 at α : 0.10 level of significance. Suppose the decision rule is to be based on Xmax, the largest order statistic. What would be the probability of committing a Type II error when θ 1.7.
A random sample of size n -8 is drawn from uniform pdf f(x,θ)- , 0-XS...
Suppose that X1, X2,..., Xn are iid from where a 1 is a known constant and θ > 0 is an unknown parameter. (a) Show that the likelihood ratio rejection region for testing Ho : θ θο versus H : θ > θο can be written in terms of X(n), the maximum order statistic. (b) Derive the power function of the test in part (a). (c) Derive the most powerful (MP) level α test of Ho : θ-5 versus H1...
1. Consider the following hypothesis test for a Poisson(a) population Ho : α = 1 H1 :a = 2 a) Find the rejection region for a likelihood ratio test with k-4 and sample size n. (b) Find the level of the rejection region found in the previous part with n 15 (c) Find the power of a 05-level test with n 100.
1. Consider the following hypothesis test for a Poisson(a) population Ho : α = 1 H1 :a =...
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