Midterm Two (STAT 5432 - Spring 2010)

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Due by 7:00pm on Tuesday April 13—

(Please Print Name) Last First

1. (30%) Let Xi, Xi, • • • , .Xie be iid random sample from a normal distribution JV(0, <r2).

Find the UMP size a = 0.05 test for the following hypotheses and give explicit rejection

region for each test.

(a) HQ : a = 4 versus HI : a = 1.

(b) H0 : a < 1 versus #1 : a > 1.

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2. (30%) Let Xi,X2, • • • ,X\Q be iid random sample from a Bernoulli(p). Find the UMP

test for the following hypotheses and give explicit rejection region for each test.

(a) HQ : p = I versus H\ p = |, and the size a is close to 0.05.

(b) H0 : p < | versus HI : p > |, and the size a is close to 0.01.

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3. (20%) Let Xi,X-2, • • • ,X-2§ be iid random sample from a normal distribution N(n, 1).

Find the UMP size a — 0.05 test of H0 : //, < 10 versus HI : // > 10 and give explicit

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4. (20%) Let Xi,X2,--• ,X-iQ be iid random sample from an exponential distribution

f ( x \ 0 ) = \e--ex>x>Q. Find the UMP size a = 0.05 test of H0 : 0 > 100 versus

HI : 9 < 100 and give explicit rejection region for your test.

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