Presentation on theme: "Introduction Simple Random Sampling Stratified Random Sampling"— Presentation transcript:
1Introduction Simple Random Sampling Stratified Random Sampling Chap 7: Survey SamplingIntroductionSimple Random SamplingStratified Random Sampling
27.1: IntroductionFor small pop’n, a census study are used because data can be gathered on all.For large pop’n, sample surveys will be used to obtain information from a small (but carefully chosen) sample of the pop’n. The sample should reflect the characteristics of the pop’n from which it is drawn.Sampling methods are classified as either probabilistic or non-probabilistic in nature.
3Sampling methods: Probability Sampling: Random Sampling Systematic SamplingStratified SamplingNonProbabilitySamplingConvenienceSamplingJudgment SamplingQuota SamplingSnowball Sampling
4The winner is: Probability Sampling. In non-probability sampling, members are selected from the pop’n in some non-random manner and the sampling error (=degree to which a sample might differ from the pop’n) is unknown.In probability sampling, each member of the pop’n has a specified probability of being included in the sample. Its advantage is that sampling error can be calculated.
57.2: Pop’n parametersDefinition: Parameters are those numerical characteristics of the pop’n that we will estimate from a sample.Notations:
6Pop’n mean, total, variance: Pop’n total:Pop’n variance: its square root is the StdDev
77.3: Simple Random Sampling The most elementary form of sampling is s.r.s. where each member of the pop’n has an equal and known chance of being selected at most once.There are possible samples of size n taken without replacement.In this section, we will derive some statistical properties of the sample mean.
87.3.0: The Sample Mean: The sample mean estimates the pop’n mean where so that will estimate the pop’n total
97.3.1: Expectation & Variance of the Sample Mean: Theorem A (UNBIASEDNESS) :under s.r.s. ,Theorem B: under s.r.s. ,Recall: The variance of in sampling without replacement differs from that in sampling with replacement by the factor which is called the finite population correction.The ratio is called the sampling fraction.
107.3.2: Estimation of the Population Variance: Theorem A: under s.r.s.,whereCorollary A: An unbiased estimator of isgiven by
117.3.3: Normal approximation to the sampling dist’n of the sample mean We will be using the CLT (Central Limit Theorem, see Section 5.3) in order to find the probabilistic bounds for the estimation error.Application 1:the probability that the error made in estimating by is using the CLT.Application 2:a CI (Confidence Interval) for the pop’n mean is given by using the CLT.
127.4: Estimating a ratio: Ratio arises frequently in Survey Sampling. If a bivariate sample is drawn, then the ratiois estimated byWe wish to derive E(R) and Var(R) using approximation methods seen in Section 4.6 because R is a nonlinear function.
137.4: Estimating a ratio (cont’d) Theorem A:With s.r.s., the approximate variance of R isSince the population correlation pop’n is then
147.4: Estimating a ratio (cont’d) Theorem B:With s.r.s., the approximate expectation of R is
15Standard Error estimate of R: The estimate variance of R iswhere and the pop’n covarianceis estimated by
16Confidence Interval for r: An approximate CI (Confidence Interval) for the ratio of interest r is given by
177.5: Stratified Random Sampling: 7.5.1: Introduction (S.R.S.)The pop’n is partitioned into sub-pop’s or strata (stratum, singular) that are then independently sampled and are combined to estimate pop’n parameters.A stratum is a subset of the population that shares at least one common characteristicExample: males & females; age groups;…
18Why is Stratified Sampling superior to Simple Random Sampling? S.R.S. reduces the sampling errorS.R.S. guarantees a prescribed number of observations from each stratum while s.r.s. can’tThe mean of a S.R.S. can be considerably more precise than the mean of a s.r.s., if the pop’n members within each stratum are relatively homogeneous and if there is enough variation between strata.
197.5.2: Properties of Stratified Estimates: Notation: Let be the total pop’n size if denote the pop’n sizes in the L strata.The overall pop’n meanis a weighted average of the pop’n means of the L strata, where denotes the fraction of the pop’n in the stratum.
207.5.2: Properties of Stratified Estimates: (cont’d) Stratified sampling requires two steps:Identify the relevant strata in the pop’nUse s.r.s. to get subject from each stratumWithin each stratum, a s.r.s. of size is taken to obtain the sample mean in the stratum will be denoted bywhere denotes the observation in the stratum.
217.5.2: Properties of Stratified Estimates: (cont’d) Theorem A: The stratified estimate, , of the overall pop’n mean is UNBIASED.Since we assume that the samples from different strata are independent of one another and that within each stratum a s.r.s. is taken, then the variance of can be easily calculated in:Theorem B: The variance of the stratified sample is
22Neglecting / Ignoring the finite population correction: Approximation:If the sampling fractions within all stratawere small, Theorem B will then reduce to:
23Expectation and Variance of the stratified estimate of the pop’n total: This is a corollary of Theorems A & B.Practice with examples A & B in the textbook on pages to get healthy with these calculations.
247.5.3: Methods of allocation: For small sampling fractions within strata i.e. when neglecting/ignoring the finite pop’n correction,Question: How to choose to minimize subject to the constraint when resources of a survey allowed only a total of n units to be sampled?Note: We could include finite pop’n corrections but the results will be more complicated. Try it!
257.5.3 a: Neyman allocationTheorem A: The samples sizes that minimize subject to the constraintare given byCorollary A: stratified estimate & optimal allocations
267.5.3 b: Proportional allocation If a survey measures several attributes for each pop’n member, it will be difficult to find an allocation that is simultaneously optimal for each of those variables. Using the same sampling fractionin each stratum will provide a simple and popular alternative method of allocation.
277.5.3b:Proportional allocation (cont’ Theorem B: With stratified sampling based on proportional allocation, ignoring the finite pop’n correction,Theorem C: With stratified sampling based on both allocation methods, ignoring the finite pop’n correction,
287.6:ConclusionsA mathematical model for Survey Sampling was built using s.r.s. and probabilistic error bounds for the estimates derived.The theory and techniques of survey probability sampling include Systematic Sampling, Cluster Sampling, etc…as well as non-probability sampling methods such as Quota Sampling, Snowball Sampling,…