Sma2110:Real Analysis Question Paper
Sma2110:Real Analysis
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2014
QUESTION ONE - (30 MARKS)
a) Define the following; i. Type I and type II errors. (2 Marks) ii. Most powerful level ? test. (3 Marks) iii. Power of a test. (2 Marks) b) Consider a population with pdf ??(??,1) where ?? ? R, is unknown. An experimenter postulates two possible hypotheses. ??0:?? = 5.5 and ??1:?? = 8 . a random sample ?? = ??1 ….??9 is collected and denote
?? =
1 9
????
9
??-1
Consider the following tests. Test I: reject ??0 ?????? ??1 > 7 Test 2: reject ??0 ?????? 1 2 ??1 + ??2 > 7 Test 3: reject ??0 ?????? ?? > 6 Test 4: reject ??0 ?????? ?? > 7.5
i. Compute the probabilities of Type I and Type II errors for each test. (5 Marks) ii. Deduce the power function for the Test 4 ??? ? R. (3 Marks)
2
c) Suppose that x is an observable random variable with its pdf given by ?? ?? ,?? ? R+. Consider two function defined as
??0(??) =
3 64
??2 ???? 0 < ?? < 4 0 ??????????h??????
??1(??) =
3 16
?? ???? 0 < ?? < 4 0 ??????????h??????
Determine the MP test for ??0:?? ?? = ??0(??) ??1:?? ?? = ??1(??) (6 Marks)
Calculate the associated power for this test. (3 Marks) d) Given a random sample of size n from a ??(??,4) distribution and we wish to test ??0:?? = 10 vs ??1:?? = 8. The decision rule is to reject ??0 if ?? < ?? find n and c so that ?= 0.05 and ?? = 0.1. (5 Marks)
QUESTION TWO (20 MARKS)
a) Suppose that ??1 ….???? are ?????? ??(??,??2) where ?? ? R and ??2 ? R+. Assume that ?? is unknown. Consider choosing between a null hypotheses. ??0:?? = ??0vs ??1:?? ? ??0 with level ? where ??0 ? R . Determine the L.R level ? test for the mean when i. Variance is known. (8 Marks) ii. Variance is unknown (7 Marks) b) Let ??1~??(??,4). To test the hypothesis that ?? = 1 against the alternative that ?? = 2, based on a sample size 25, the following decision rule is adopted. If ?? > 1.6 claim that ?? = 2 If ?? = 1.6 claim that ?? = 1
Compute the probabilities of type I and type II errors. (5 Marks)
QUESTION THREE (20 MARKS
a) State and prove the Neyman Pearson Lemma. (12 Marks) b) Let ??1,…….???? be iid with probability density fucntion ?? ?? = ??-1?????? -?? ?? for ?? ? R+ with unknown ?? ? R+ and ?? (0,1). Use the Neyman Pearson Lemma, obtain the most powerful level ? test for
3
??0:?? = ??0 vs ??1:?? = ??1 (> ??0) where ??0,??1,? R+are two positive numbers. (8Marks)
QUESTION FOUR (20 MARKS)
a) Suppose that the random variables ????1,…….?????? are iid ?? ??,??2 ?? = 1,2 and that the observations are independent. Assume that all the three parameters are unknown and ?? = ??1,??2,?? ? R × R × R+. Given ?? 0,1 . Determine the likelihood Ration test for choosing between ??0:?? = ??2 vs ??1:?? ? ??2 with ??:= 2. (8 Marks) b) Given the following paired observations Before ??1 After ??2 70 80 85 83 67 75 74 80 89 94 78 86 63 69 82 78
Assume a bivariate normal distribution for ??1,??2 decide between the following i. ??0:??1 = ??2Vs ??1:??1 < ??2 at 1% level where ??1and ??2 are means for ??1 and ??2 respectively. (4 Marks) ii. ??1:?? = 0 Vs ??1:?? ? 0 at 10% level. Where e is the correlation coefficient of ??1 and ??2. (4 Marks) iii. ??0:??1 = ??2 vs. ??1:??1 > ??2 at 1% level. Where ??1, ??2 are standard deviations for ??1 and ??2 respectively. (4 Marks)
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