Sma2110:Real Analysis Ii Question Paper
Sma2110:Real Analysis Ii
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2013
QUESTION ONE – (30 MARKS)
(a) Using sampling theory, prove that the sample mean is an unbiased estimator of the population mean. (6 Marks) (b) Given that N is the population size, n the sample size. Prove that ?? = ??2 - . (c) Show that an unbiased estimate of the variance of the sample proportion is given by 1 1 n pp pVar where p is the population proportion p is the sample proportion n the sample size. (5 Marks) (d) To estimate the proportion of diseased citrus plants in a certain farm, a random sample of = 75 plants selected by SRWS was examined and 55 % of them were found to be diseased. Obtain an estimate of the proportion of the diseased plants and an estimate of its variance. (4 Marks) (e) In a population = 6, = 2, the value of ?? are 0 1 2 in stratum 1 and 4 6 11 in stratum 2. A sample with = 4 is to be taken. (i) Show that the optimum under a Neymann allocation when rounded to integer are = 1 in stratum I and = 3 in stratum 2. (3 Marks)
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(ii) Compute the estimator of ?? for every possible sample that can Le draw under optimum allocation and under proportional allocation. Verify that the estimators are unbiased. (4 Marks)
QUESTION TWO – (20 MARKS)
(a) A simple random sample of size n = 9 hospital records is drawn to estimate the average amount of money due on N = 484 open accounts. The sample values for their nine records are listed below. Estimate , the average amount outstanding and place abound on your error of estimation Amount of Money Owed. 1 = 23.5 6 = 41.00 2 = 32.00 7 = 45.00 3 = 52.00 8 = 42.00 4 = 43.00 9 = 39.00 5 = 40.00 (10 Marks) (b) Discuss five aspects one should consider when designing a questionnaire. (10 Marks)
QUESTION THREE – (20 MARKS)
(a) The following table gives the results of stratified random sample. Stratum i Ni ni ??2 1 20 5 1.6 3.3 2 9 3 2.8 4.0 3 12 4 0.6 2.2
Find (i) ?? (3 Marks) (ii) y (4 Marks) (iii) yVar (3 Marks)
(b) A national park has been divided into 80 Zones. A survey is taken with the aim of obtaining the number of zebras in the par. Suppose that 35% of the 80 zones are assumed to be inhabited by zebras and each of the zone is assumed to be large. (i) How large a sample of the 80 zones should be selected in order to obtain an estimate of the population proportion of occupied zones to within 5% of the true proportion with 95% confidence? (5 Marks) (ii) How large a sample is covered if the estimate is to be 5% of the true value with 95% confidence. (5 Marks)
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QUESTION FOUR – (20 MARKS)
(a) Discuss how linear systematic sampling schemes can be used to generate samples of size n. (4 Marks) (b) Show that in systematic sampling, the estimator for the population mean given by k i iy N k y 1 is unbiased. (6 Marks) (c) Given the following data Replicate 1 2 3 4 5 6 7 8 9 10 ti 14 10 9 8 20 15 9 16 12 10 compute the mean and its variance for these replicate. (5 Marks) (d) State five factors to be considered when developing a national sample. (5 Marks)
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