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21 Jul 2022: Logic. The following multiple choice self-test questions are intended to give an indication whether you haveprerequisites for the Part III courses in logic.

3 Aug 2022: Suppose thereexists a likelihood ratio test of size α for H0 against H1. ... Specify the test statistic and thecritical region. Is this test uniformly most powerful for the alternative H1 : π > 0?

21 Jul 2022: Set Theory. The following multiple choice self-test questions are intended to give an indication whether you haveprerequisites for the Part III courses in set theory.

17 Mar 2022: To test this enter. >> amatrix=printsquares(1,10);. >> amatrix. Note that we have used amatrix as the name of the matrix in which to store the output fromthe function printsquares; we ... 4.4 Exercises. 1. Test your function printsquares with m60 and m>n,

4 Aug 2022: a) Let Tn =1n. ni=1(θ1 θ0). (xi θ0θ12. ). Show that a likelihood ratio test should. ... Assuming thisfact, show that a hypothesis test based on Sn corresponds to a likelihood ratiotest of.

3 Aug 2022: Part IA, Paper 1. 3. 3D Analysis IState the alternating series test. ... You may use the comparison test, provided that you accurately state it.].

4 Aug 2022: Explainwhy a test for the hypothesis of part (a) based on the first K eigenfunctions ofC, as defined in part (b), might be preferable to a test based on the first ... Define a finite-dimensional statistic, based on the first K eigenfunctions of

3 Aug 2022: d) Describe a test of size α for the null hypothesis H0 : σ21 = σ. ... 22 against the. alternative H1 : σ21 < σ. 22, using the test statistic.

4 Aug 2022: Show that the smallest. possible total error probability that can be achieved by any hypothesis test is equalto:. ... b) Consider a simple-versus-simple hypothesis test between two distributions P andQ 6= P having full support on a finite alphabet A.

3 Aug 2022: d) Describe a test of size α for the null hypothesis H0 : σ21 = σ. ... Suppose thereexists a likelihood ratio test of size α for H0 against H1.