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3. Now take the same setting again and continue with the two-sided test in this case as specified above. (a) HARDER: Recall the likelihood ratio test statistic. Show that in this case the likelihood ratio statistic is given by: where 2 = -n/2 - μο ~X(20) - ( 1 + ( ² = 2^)²") ² = 1/2 Σ²-1(x₁ - x) ². n i=1 (b) HARDER: Thus, by appropriate rearrangement, show that the rejection region obtained by using the likelihood ratio statistic as the test statistic for the specified two-sided test is the same as the region derived in Question 2. Note this is not generally the case e.g. in Qu 1 the LRT gives a different rejection region from using the MLE as the test statistic.

3. Now take the same setting again and continue with the two-sided test in this case as specified above. (a) HARDER: Recall the likelihood ratio test statistic. Show that in this case the likelihood ratio statistic is given by: where 2 = -n/2 - μο ~X(20) - ( 1 + ( ² = 2^)²") ² = 1/2 Σ²-1(x₁ - x) ². n i=1 (b) HARDER: Thus, by appropriate rearrangement, show that the rejection region obtained by using the likelihood ratio statistic as the test statistic for the specified two-sided test is the same as the region derived in Question 2. Note this is not generally the case e.g. in Qu 1 the LRT gives a different rejection region from using the MLE as the test statistic.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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2. Take the setting of Question 2 from Tutorial sheet 4. However, now suppose that instead of a one-sided test we are performing a two sided test of size a such that: Ho Ho vs H₁ μμo In Tutorial 4 we derived that under the null hypothesis the test statistic â(X) – Ho ~tn-1 T(X) : = where (X) is the mean of the X;'s and S² is the sample variance has a tñ−1 is the student t distribution on n - 1 degrees of freedom. Derive an expression for the rejection region for this two-sided test of size a that uses T(X) as the test statistic. 3. Now take the same setting again and continue with the two-sided test in this case as specified above. (a) HARDER: Recall the likelihood ratio test statistic. Show that in this case the likelihood ratio statistic is given by: n X(x) = =) = (1 + (ª = μ10) ²) 2 -n/2 where 2 = ¹₁(x₁ - x)². (b) HARDER: Thus, by appropriate rearrangement, show that the rejection region obtained by using the likelihood ratio statistic as the test statistic for the specified two-sided test is the same as the region derived in Question 2. Note this is not generally the case e.g. in Qu 1 the LRT gives a different rejection region from using the MLE as the test statistic.
Transcribed Image Text:2. Take the setting of Question 2 from Tutorial sheet 4. However, now suppose that instead of a one-sided test we are performing a two sided test of size a such that: Ho Ho vs H₁ μμo In Tutorial 4 we derived that under the null hypothesis the test statistic â(X) – Ho ~tn-1 T(X) : = where (X) is the mean of the X;'s and S² is the sample variance has a tñ−1 is the student t distribution on n - 1 degrees of freedom. Derive an expression for the rejection region for this two-sided test of size a that uses T(X) as the test statistic. 3. Now take the same setting again and continue with the two-sided test in this case as specified above. (a) HARDER: Recall the likelihood ratio test statistic. Show that in this case the likelihood ratio statistic is given by: n X(x) = =) = (1 + (ª = μ10) ²) 2 -n/2 where 2 = ¹₁(x₁ - x)². (b) HARDER: Thus, by appropriate rearrangement, show that the rejection region obtained by using the likelihood ratio statistic as the test statistic for the specified two-sided test is the same as the region derived in Question 2. Note this is not generally the case e.g. in Qu 1 the LRT gives a different rejection region from using the MLE as the test statistic.
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