Bl. (a) (b) In a test about a real parameter 0, the null and alternative hypotheses...

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Bl. (a) (b) In a test about a real parameter 0, the null and alternative hypotheses are: H:0≤0, and H₁:0>0. Explain what is meant by a Uniformly Most Powerful (UMP) test at significance level a. (3 Marks) A random sample X₁, X.....X, is drawn from a population with probability density function (pdf) f(x,0)= (i) P(-1) Aexp 20² Aexp . *so x>0 where a>0 is an unknown parameter and A= all x values with x > 0. √√2 (1+0)√ What is the well-known distribution when a = 1? (2 marks) A test is required of the null hypothesis Ho=1 against the alternative hypothesis H₁:=o', where o'>1 is a fixed value. Show that the Neyman-Pearson's approach leads to the rejection of the null hypothesis when >k for some suitably chosen k, where x is the sum of for 0<x<1. (8 marks) n=30, x=80.0 and =2.0. Cont/... 1 CS CamScam (iii) Deduce that this test is UMP against the alternative hypothesis H₂:0>1. (4 marks) (iv) Perform the Generalised Likelihood Ratio Test (GLRT) of H₂:0=1 against the alternative H₁:01 when Here is the unrestricted Maximum Likelihood Estimate (MLE) of a, and you should use an approximate 5% significance level. (8 marks) Bl. (a) (b) In a test about a real parameter 0, the null and alternative hypotheses are: H:0≤0, and H₁:0>0. Explain what is meant by a Uniformly Most Powerful (UMP) test at significance level a. (3 Marks) A random sample X₁, X.....X, is drawn from a population with probability density function (pdf) f(x,0)= (i) P(-1) Aexp 20² Aexp . *so x>0 where a>0 is an unknown parameter and A= all x values with x > 0. √√2 (1+0)√ What is the well-known distribution when a = 1? (2 marks) A test is required of the null hypothesis Ho=1 against the alternative hypothesis H₁:=o', where o'>1 is a fixed value. Show that the Neyman-Pearson's approach leads to the rejection of the null hypothesis when >k for some suitably chosen k, where x is the sum of for 0<x<1. (8 marks) n=30, x=80.0 and =2.0. Cont/... 1 CS CamScam (iii) Deduce that this test is UMP against the alternative hypothesis H₂:0>1. (4 marks) (iv) Perform the Generalised Likelihood Ratio Test (GLRT) of H₂:0=1 against the alternative H₁:01 when Here is the unrestricted Maximum Likelihood Estimate (MLE) of a, and you should use an approximate 5% significance level. (8 marks) Bl. (a) (b) In a test about a real parameter 0, the null and alternative hypotheses are: H:0≤0, and H₁:0>0. Explain what is meant by a Uniformly Most Powerful (UMP) test at significance level a. (3 Marks) A random sample X₁, X.....X, is drawn from a population with probability density function (pdf) f(x,0)= (i) P(-1) Aexp 20² Aexp . *so x>0 where a>0 is an unknown parameter and A= all x values with x > 0. √√2 (1+0)√ What is the well-known distribution when a = 1? (2 marks) A test is required of the null hypothesis Ho=1 against the alternative hypothesis H₁:=o', where o'>1 is a fixed value. Show that the Neyman-Pearson's approach leads to the rejection of the null hypothesis when >k for some suitably chosen k, where x is the sum of for 0<x<1. (8 marks) n=30, x=80.0 and =2.0. Cont/... 1 CS CamScam (iii) Deduce that this test is UMP against the alternative hypothesis H₂:0>1. (4 marks) (iv) Perform the Generalised Likelihood Ratio Test (GLRT) of H₂:0=1 against the alternative H₁:01 when Here is the unrestricted Maximum Likelihood Estimate (MLE) of a, and you should use an approximate 5% significance level. (8 marks) Bl. (a) (b) In a test about a real parameter 0, the null and alternative hypotheses are: H:0≤0, and H₁:0>0. Explain what is meant by a Uniformly Most Powerful (UMP) test at significance level a. (3 Marks) A random sample X₁, X.....X, is drawn from a population with probability density function (pdf) f(x,0)= (i) P(-1) Aexp 20² Aexp . *so x>0 where a>0 is an unknown parameter and A= all x values with x > 0. √√2 (1+0)√ What is the well-known distribution when a = 1? (2 marks) A test is required of the null hypothesis Ho=1 against the alternative hypothesis H₁:=o', where o'>1 is a fixed value. Show that the Neyman-Pearson's approach leads to the rejection of the null hypothesis when >k for some suitably chosen k, where x is the sum of for 0<x<1. (8 marks) n=30, x=80.0 and =2.0. Cont/... 1 CS CamScam (iii) Deduce that this test is UMP against the alternative hypothesis H₂:0>1. (4 marks) (iv) Perform the Generalised Likelihood Ratio Test (GLRT) of H₂:0=1 against the alternative H₁:01 when Here is the unrestricted Maximum Likelihood Estimate (MLE) of a, and you should use an approximate 5% significance level. (8 marks)