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.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter10: Statistics
Section10.5: Comparing Sets Of Data
Problem 14PPS
Related questions
Question
answer question3
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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