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48 SECTION 4 STRATIFIED RANDOM SAMPLING 4.1 What is Stratification? 4.1.1 Definition Stratification --- The process of dividing a population of elements into distinct subpopulations called strata. Strata are formed so that each population element is assigned to only one stratum. 4.1.2 Some Examples A. We wish to stratify a list of employees of a large company by age. Since the age of all employees is available on our sampling frame, we form the following strata:49 h Stratum Composition: ---- -------------------------------- 1 Less than 25 years old 2 25 to 34 years old 3 35 to 44 years old 4 45 to 54 years old 5 55 years of age or older B. To draw a sample of United States counties we wish to stratify by region. To this end we form the following strata: h Stratum Composition All Counties Located in the --- -------------------------------------- 1 Northeast census region 2 South census region 3 North Central census region 4 West census region50 4.1.3 How is Stratification Used in Sample Surveys? A. The population is divided into strata so that each population element is a member of only one stratum. We use the letter H to represent the number of strata that are formed and Nh to denote the number of population elements which fall in the h-th stratum. The total number of elements in the population is therefore N N N N NH hhH= + + + ==1 21... (4.1) B. A sample size is chosen for each stratum. We denote the sample size in the h-th stratum by the symbol, nh. The total sample size over all strata is then n n n n nH hhH= + + + ==1 21... (4.2) The corresponding sampling rate for the h-th stratum would be51 fnNhhh= (4.3) with the overall sampling rate being fnN= (4.4) C. A probability sample is separately chosen in each stratum so that the choice of elements in one stratum does not depend upon choices made in the other strata. Selection procedures among strata are often the same although different selection methods can be used, if needed. D. The population value to be estimated (e.g., mean, proportion, etc.) is estimated separately for each stratum. E. An estimate for the entire population is produced by appropriately combining the individual stratum estimates. (Skip to Section 4.7 on p. 75 )52 4.2 What is Stratified Simple Random Sampling: 4.2.1 Definition Stratified Simple Random Sampling: Stratified sampling in which simple random sampling is used in each stratum. NOTE: Stratified simple random sampling is a simple form of stratified sampling. There are many other types of stratified sampling, however. 4.2.2 How to Select a Stratified Random Sample A. Divide the population into strata. B. Determine sample sizes for each stratum. C. Select a separate simple random sample in each stratum.53 4.3 Estimating a Population Mean From a Stratified Random Sample 4.3.1 Setting A. A stratified random sample has been selected. B. Data from each element in the sample have been collected. C. We wish to estimate the population mean per element for some characteristic. This mean can be expressed as YYNN Y N Y N YNoH H= =+ + +1 1 2 2..... = + + + ==W Y W Y W Y W YH H h hhH1 1 2 21..... (4.5) where WNNhh= (4.6) is the proportion of all population elements which fall in the h-th stratum.54 4.3.2 Estimator of Y yNN y N y N yNN yst H H h hhH= + + + ==111 1 2 21... = W y W y W y W yH H h hhH1 1 2 21+ + + ==... (4.7) where yh is the estimator of Yh for the h-th stratum calculated as yynhhiinhh==1 (4.8) and yhi is the value of the observation made on the i-th sample element in the h-th stratum.55 4.3.3 Some Statistical Notes About yst A. yst is a random variable since different stratified random samples of the same size from the same population are likely to lead to different values of yst. B. yst is an unbiased estimator of Y. C. With sufficiently large stratified random samples (n>30), the sampling distribution will closely resemble the normal distribution. 4.3.4 Estimated Variance of yst 2 2 2st 1 1 2 2 H Hv(y ) W v(y ) W v(y ) ... W v(y )= + + + =− =WfnshhhhhH2 211 (4.9)56 where sh2 is the estimated element variance for the h-th stratum, which can be calculated by using the same formula as used to estimate element variance from a simple random sample, only applied for a specific stratum. 4.3.5 Some Statistical Notes About stv(y ) A. stv(y ) is a random variable (see Section 4.3.3). B. sh2 is an unbiased estimator of the true element variance in the h-th stratum. C. stv(y ) is an unbiased estimator of the true variance of yst. 4.3.6 Standard Error of yst H2 2hst st h hh 1h1 fse(y ) v(y ) W sn= −= =   (4.11)57 4.3.7 Confidence Intervals for Y (n>30) Lower Boundary: st sty {t}{se(y )}− (4.12) Upper Boundary: st sty {t}{se(y )}+ (4.13) The value of t which we use depends on the confidence level for our interval (see Section 2.2.7 for some commonly used values of t). Interpretation: We are 95 percent sure that Y is covered by the interval whose boundaries are (t=1.96) defined by formulas (4.12) and (4.13). 4.4 Estimating a Population Total from a Stratified Random Sample 4.4.1 Setting A. A stratified random sample has been selected. B. Data from each element in the sample have been collected.58 C. We wish to estimate the aggregate total of some characteristic for the population. This total can be expressed as Y N Y N Y N Y N YoH H h hhH= + + + ==1 1 2 21..... (4.14) 4.4.2 Estimator of Yo y Ny N yostst h hhH= ==1 (4.15) 4.4.3 Some Statistical Notes About yost A. yost is a random variable. B. yost is an unbiased estimator of Yo. C. When we have large sample sizes, the sampling distribution of yost will closely resemble the normal distribution.59 4.4.4 Estimated Variance of yost Ho2 2 2hst h hsth 1h1 fv y N v(y ) N sn= − = =     (4.16) 4.4.5 Some Statistical Notes About ostv(y ) A. ostv(y ) is a random variable B. ostv(y ) is an unbiased estimator of the true variance of yost.60 4.4.6 Standard Error of yost Ho o2 2hh hst sth 1h1 fse(y ) v(y ) N sn= −= =  


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UNC-Chapel Hill BIOS 662 - Stratification

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