9-stratified sampling - Motivation...

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6/3/2013 1 9. Stratified Sampling 1 Motivation: A population of nine elements with measurements, 3, 3, 3, 6, 6, 6, 9, 9, 9. The mean and variance are, respectively, 6 and 6. A sample of 3 is selected with simple random sampling without replacement. Sample mean is an unbiased estimator of the population mean. The variance of the sample mean is 3/2. Question: Can we do better? 2 Motivation: Divide the population into three groups: G 1 ={3, 3, 3}, G 2 ={6, 6, 6}, and G 3 ={9, 9, 9}. New sampling scheme: Select one unit from each groups with simple random sampling. Sample size is still 3. Under this sampling scheme, the sample means of all possible samples have the value 6, which is exactly equal to the population mean. No error. 3 Definition A stratified random sample is one obtained by separating the population elements into nonoverlapping groups call strata and then nonoverlapping groups, call strata, and then selecting a simple random sample from each stratum. 4 Why use stratified sampling? We want to be protected from the possibility of obtaining a really bad sample We may want data of known precision for subgroups. These subgroups should be the strata, which then coincide with the domains of study A stratified sample may be more convenient to administer and may result in a lower cost for the survey Stratified sampling, if done correctly, will give more precise estimates for the whole population 5 Notations: N : Total number of elements (sampling units) in the population L : Number of strata N N b f l i h h h h : Number of elements in the th stratum Y hi : Measurement of i th element in the h th stratum 6
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6/3/2013 2 Notations and results: 1. 2. Total of h th stratum N N L h h = = 1 = h N hi h Y T 3. Mean of h th stratum = i 1 = = = h N i hi h h h h Y N N T Y 1 1 7 Notations and results: 4. Variance of h th stratum 5 Population total 2 1 2 ) ( 1 = = h N i h hi h h Y Y N σ L L L N Y N T Y T h ∑∑ = = = 5. Population total 6. Population mean h h h h h h i hi = = = = 1 1 1 1 = = = L h h h Y N N N T Y 1 1 8 Notations: For the sample, we define n : Sample size n h : Number of elements in the sample that are f h h from th stratum y hi : Measurement of the i th selected element from h th stratum 9 Then, 1. 2 Sample mean of the h th stratum = = L h h n n 1 = h n y y 1 2. Sample mean of the 3. Sample variance of the h th stratum = i hi h h n 1 2 1 2 ) ( 1 1 h n i hi h h y y n s h = = 10 How to draw a stratified random sample 1. Divide the N population elements into L subpopulation of N 1 , …, N L elements, respectively.
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