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  • 1. Midterm Two (STAT 5432 - Spring 2010) * * * Show your work to receive credit * * * 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, <r 2 ). 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. a.) . U***. 1(5, '/ ; a
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  • 5. 2. (30%) Let Xi,X2, • • • ,XQ 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. J f. - ^L X ~ i H. >Sx; £k t'- p,; t«>-: ~k< < k s- I '^ Ve.
  • 6. 1 U i s - o — 4 o- J H ; 3 "4 •D ° J |) . ^ -tf i •£ H 1 «4- -5 <L O a ' '«> 1 ^> ^ J r 5 i ^ -» t> i * 4-t- 3
  • 7. Q $ i) -h I -v J 'r? "* o C <£ IP 5 i 6 t> * X A 1 h H ^v H J U d ^rtr ^ o j Lo Jf j :
  • 8. 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 rejection region for your test. _. . - i , UK] 1- t' 4
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  • 10. /a'VI 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. ** -1- f. --L o e U 0 o
  • 11. ~ Q