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Unformatted text preview: Stat 332 R.J. MacKay, University of Waterloo, 2005 Chapter 7 Solutions 1 Chapter 7 Exercise Solutions 1. Consider the sampling protocols defined in Example 1. a) Show that the inclusion probability for each unit in the frame is 1/100 for every protocol. For each protocol, the model is uniform. That is, the chance of any possible sample is equal. To find the inclusion probability, we need only count the number of samples that contain a particular unit. Once the unit is in the sample, we count the ways of selecting the remaining 99 units. SRS: there are 99 9999 ways to select the other units so 100 1 100 10000 99 9999 = = i p Stratified sampling: there are 9 10 1000 9 999 ways to select the other units so 10 1 10 1000 10 1000 9 999 10 9 = = i p Cluster sampling: there are 9 999 ways to choose the remaining clusters so 100 1 10 1000 9 999 = = i p Systematic sampling: there is only one way to select the sample so p i = 1 100 / Two stage sampling: there are 1 9 ways to select the second primary unit. Then the other 99 secondary units can be selected in 50 1000 49 999 ways so 10 1 50 1000 2 10 50 1000 49 999 1 9 2 = = i p b) On a final examination, a student once defined simple random sampling as follows: simple random sampling is a method of selecting units from a population so that every unit has the same chance of selection. Is this a correct answer? No because there are many sampling protocols that satisfy this definition as shown in a) Stat 332 R.J. MacKay, University of Waterloo, 2005 Chapter 7 Solutions 2 c) Show that the estimator corresponding to the sample average $ = y n i i s is unbiased for for each of the protocols. Let I i i N i = R S T = 1 1 if unit is in thesample 0 otherwise ,..., so that E I p i i ( ) = . Then we can write ~ = y I n i i i U and E y E I n y p n i i i U i i i U ( ~ ) ( ) = = = since p n N i = / for all five protocols. 2. Consider the estimate $ ( ) = y y n i i s 2 1 and the corresponding estimator ~ . a) For SRS, show that ~ 2 is an unbiased estimator for 2 . [Hint: Use the fact that ( ) y y y ny i i s i i s = 2 2 2 ]....
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This note was uploaded on 06/27/2009 for the course STAT 332 taught by Professor Xu(sunny)wang during the Spring '09 term at Waterloo.
 Spring '09
 Xu(Sunny)Wang
 Probability

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