Chapter 12 - ECMT1020 Chapter 12 Survey Sampling Dr Boris...

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ECMT1020: Chapter 12 Dr Boris Choy 1 ECMT1020 Chapter 12 Survey Sampling Dr Boris Choy for ECMT1020
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ECMT1020: Chapter 12 Dr Boris Choy 2 Topics covered 1. Simple Random Sampling 2. Stratified Random Sampling 3. Allocation Under Stratified Random Sampling References Lecture notes Pre-requisite: Black 7.1, 7.2, 7.3 (Chapters covered by ECMT1010)
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ECMT1020: Chapter 12 Dr Boris Choy 3 Learning Objectives Understand the simple random sampling and make adjustment on the variance of the estimator (of population mean, total or proportion) for finite population. Understand the stratified random sampling Able to calculate sample mean, standard deviation and variance from the stratified samples.
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ECMT1020: Chapter 12 Dr Boris Choy 4 Simple Random Sampling
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ECMT1020: Chapter 12 Dr Boris Choy 5 Simple Random Sampling A population has N elements. Each element is alike. n elements are selected randomly and independently (= a random sample of size n ). Let X denote a characteristic to be measured. Let and σ be the (unknown) population mean and (unknown) population standard deviation of X.
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ECMT1020: Chapter 12 Dr Boris Choy 6 Simple Random Sampling The sample of size of n produces the observations X 1 , X 2 ,..., X n with sample mean Case 1: [Well known results] If N ∞, the sample mean has If X i takes either “1” or “0”, then the sample proportion (of “1”) has where p is the population proportion. n i i X n X 1 1     ) 1 ( ˆ and ˆ 1 ˆ 1 n p p p E p p E X n p n i i     n X V X E X n X n i i 2 1 and 1
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ECMT1020: Chapter 12 Dr Boris Choy 7 Simple Random Sampling Using Central Limit Theorem, we have The confidence intervals can be constructed: The hypothesis testings can be constructed: Test statistics are ) 1 ( , ˆ and , ~ 2 n p p p ~ N p n N X n p p z p n s t X n ) ˆ 1 ( ˆ ˆ and 2 / 2 / , 1 ) 1 , 0 ( ~ ) 1 ( ˆ and ~ 0 0 0 1 0 N n p p p p z t n s X t obs n obs
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ECMT1020: Chapter 12 Dr Boris Choy 8 Simple Random Sampling Case 2: If N < ∞, i.e. a finite population, the variance of an estimate will be adjusted by a factor For an infinite population, an unbiased estimate of σ 2 is (denoted by lower case s ) For a finite population, an unbiased estimate of σ 2 is (denoted by upper case S ) n i i X X n s 1 2 2 ) ( 1 1 1 N n N N i i X X N S 1 2 2 ) ( 1 1
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