Chapter 9 Exercises Solutions

# Chapter 9 Exercises Solutions - Stat 332 R.J MacKay...

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Unformatted text preview: Stat 332 R.J. MacKay, University of Waterloo, 2005 Chapter 9 Solutions -1 Chapter 9 Exercises Solutions 1. In many surveys, there is interest in estimating strata averages or differences in strata averages. a) In general, for SRS, write down the distribution for the estimators ~ μ h and ~ ~ μ μ h k- . Assuming relatively large sample sizes within the strata we have approximately ~ ~ ( ,( ) ) ~ ~ ~ ( , ( ) ( ) ) / μ μ σ μ μ μ μ σ σ h h h h h h k h k h h h k k k G f n G f n f n 1 1 1 1 2 2 2---- +- b) In the well survey, find a 95% confidence interval for the proportion of wells in farms with animals that are contaminated The estimate of the proportion contaminated is \$ . π 1 172 = with associated estimated standard deviation 1 1 0 030 1 1-- = f n \$ ( \$ ) . π π so the 95% confidence interval is 0172 0 058 . . ± c) In the well survey, find a 95% confidence interval for the average Na difference between the two types of farm wells. For farms with animals, we have \$ . μ 1 237 3 = with associated estimated standard deviation ( ) \$ . 1 3 275 1 1 2 1- = f n σ . For farms without animals, we have \$ . μ 2 245 6 = with associated estimated standard deviation ( ) \$ . 1 3 614 2 2 2 2- = f n σ and hence we have \$ \$ . μ μ 2 1 8 30- = with associated estimated standard deviation 3 275 3 614 4 877 2 2 . . . + = . Hence a 95% confidence interval for μ μ 2 1- is 8 30 9 60 . . ± . There is no evidence of a difference in average Na levels between the two groups of farms. 2. Suppose that the purpose of the survey is to estimate a population proportion π . If there are H strata, a) Write down the stratified estimate of π and the variance of the corresponding estimator. Since π π π = + + W W H H 1 1 ... we have \$ \$ ... \$ π π π strat H H W W = + + 1 1 and Var W Var W Var W f n W f n strat H H H H H H H ( ~ ) ( ~ ) ... ( ~ ) ( ) ( ) ... ( ) ( ) π π π π π π π = + + =-- + +-- 1 2 1 2 1 2 1 1 1 1 2 1 1 1 1 (ignoring the factors n n h h / ( )- 1 ) b) What is the variance of ~ π strat for proportional allocation? If n W n h h = , we have Stat 332 R.J. MacKay, University of Waterloo, 2005 Chapter 9 Solutions -2 Var W n N W n W n N W n f n W W strat H H H H H H H H H ( ~ ) ( / ) ( ) ......
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Chapter 9 Exercises Solutions - Stat 332 R.J MacKay...

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