Get premium membership and access revision papers, questions with answers as well as video lessons.
Sma2110:Real Analysis Question Paper
Sma2110:Real Analysis
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2013
SECTION A (COMPULSORY)
QUESTION ONE (30 MARKS)
a) Define the following; i. Power function of a test. (2 Marks) ii. Uniformly most powerful level test. (2 Marks) b) Suppose that random variable x has a normal distribution with mean ?? and variance 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 i. Compute the probability of type I and type II errors. (4 Marks) ii. Write statements in R-software on how and can be computed. (2 Marks) iii. What is the power of this test? (2 Marks) c) Let ??1,.?? ?? ???? (??,2) with unknown ?? , assume that 2 + is known with (0.1). derive the most powerful level test for 0:?? = ??0 Vs 1:?? = ??1 Where ??1 > ??0 are unknown real numbers. (5 Marks) d) Suppose that ??1 .??15 are ???? (??,2)with unknown ?? but 2 = 9. Compute the associated P-value and test the hypothesis 0:?? = 3.1 Vs 1:?? = 3.1 at 5% level. (5 Marks)
2
e) i) Suppose that ??1,.?? are iid observations from (??,2) population where ?? ,2 +. Assume that both ?? and are unknown. Given (0,1) find a level ?? test for choosing between a null hypothesis. 0: = 0 Vs 1: 0 Where 0 is a fixed positive real number? (5 Marks)
ii) A preliminary mathematics screening test was given to a group of twenty applicants for the position of actuary. his roup’s test scores x) ave ?? = 81.26 and = 15.39. Assume a normal distribution for X. The administrator in charge wish to test 0: = 12 vs 1: 12 ???? 15% level. Based on the given information, advice him accordingly. (4 Marks)
SECTION B
QUESTION TWO (20 MARKS)
a) State and prove the Neyman –Pearson Lemma (N-p Lemma). (12 Marks) b) Suppose that ??1,.?? iid with the common pdf ??-1?? -?? ?? for ?? + with unknown ?? + and (0,1). Using the Neyman –Pearson Lemma, obtain the most powerful (MP) level test for 0: = ??0 vs 1:?? = ??1 (> ??0) where ??0,??1 are two positive numbers. (6 Marks)
c) Under what circumstances does the Neyman-pearson Lemma hold in developing MP test? (2 Marks)
QUESTION THREE (20 MARKS)
a) When do we use the likelihood ratio test to develop an ump level test?(2 Marks) b) Suppose that ??1,.?? are iid observations from (??,2) where ?? and 2 +. Assume that ?? is unknown. Consider choosing between a null hypothesis. 0:?? = ??0 vs a two sided alternative Hypothesis 1:?? ??0 with level where ??0 . Develop a likelihood ratio test for the mean when i. Variance is known (9 Marks) ii. Variance is unknown. (9 Marks)
QUESTION FOUR (20 MARKS)
a) Suppose that the random variables ????1,.???? are iid ??,2 ?? = 1,2 and that the ??1?? are independent of the ??2??. Assume that all the three parameters are unknown and = ??1,??2, × × +. Given (0,1) compute the LR test for choosing between 0:?? = ??2
3
vs 1:?? ??2 with : 2. (8 Marks) b) In a large establishment, suppose that ??1??, ??2?? respectively denote the job performance score before and after going through a week long job training program for the ith employee ?? = 1.( 2). Assume that employees are picked randomly and independently of each other. We wish to compose the average job performance scores in the population, before and after the training. Eiht employee’s ob performance scores out of points) were recorded as follows Y ID ??1 ??2 1 70 80 2 85 83 3 67 75 4 74 80 5 89 94 6 78 86 7 63 69 8 82 78 Assume a bivariate normal distribution for ??1,??2 i. Was the training program effective at 1% level? (4 Marks) ii. Test whether the job performance scores before and after the training are correlated at 10% level. (4 Marks) iii. Test whether the job performance scores after the training are less variable than those taken before the training at 1% level. (4 Marks)
More Question Papers
Popular Exams
Return to Question Papers