Chapter 18 Additional Topics in Sampling ©

Steps in Sampling Study Step 1: Information Required? Step 2: Relevant Population? Step 3: Sample Selection? Step 4: Obtaining Information? Step 5: Inferences From Step 6: Conclusions?

Examples of Nonsampling Errors The population actually sampled is not the relevant one. Survey subjects may give inaccurate or dishonest answers. Nonresponse to survey questions.

Simple Random Sampling simple random sample Suppose that it is required to select a sample of n objects from a population of N objects. A simple random sample procedure is one in which every possible sample of n objects is equally likely to be chosen.

Estimation of the Population Mean, Simple Random Sample Let X 1, X 2,..., X n denote the values observed from a simple random sample of size n, taken from a population of N members with mean  unbiased estimator of the population mean (i)The sample mean is an unbiased estimator of the population mean . The point estimate is: unbiased estimation procedure for the variance of the sample mean (ii)An unbiased estimation procedure for the variance of the sample mean yields the point estimate confidence intervals for the population mean (iii)Provided the sample size is large, 100(1 -  )% confidence intervals for the population mean are given by

Estimation of the Population Total, Simple Random Sample Suppose a simple random sample of size n from a population of N is selected and that the quantity to be estimated is the population total N  (i)An unbiased estimation procedure for the population total N  yields the point estimate NX. (ii)An unbiased estimation procedure for the variance of our estimator of the population total yields the point estimate (iii)Provided the sample size is large, 100(1 -  )% confidence intervals for the population mean are obtained from

Estimation of the Population Proportion, Simple Random Sample Let p be the proportion possessing a particular characteristic in a random sample n observations from a population, a proportion, , of whose members possess that characteristic. (i) The sample proportion, p, is an unbiased estimator of the population proportion, . (ii)An unbiased estimation procedure for the variance of our estimator of the population total yields the point estimate (iii)Provided the sample size is large, 100(1 -  )% confidence intervals for the population proportion are given by

Stratified Random Sampling Stratified random sampling Suppose that a population of N individuals can be subdivided into K mutually exclusive and collectively exhaustive groups, or strata. Stratified random sampling is the selection of independent simple random samples from each stratum of the population. If the K strata in the population contain N 1, N 2,..., N K members, then There is no need to take the same number of sample members from every stratum. Denote the numbers in the sample by n 1, n 2,..., n K. Then the total number of sample members is

Estimation of the Population Mean, Stratified Random Sample Suppose that random samples of n j individuals are taken from strata containing N j individuals (j = 1, 2,..., K). Let Denote the sample means and variances in the strata by X j and s j 2 and the overall population mean by . (i)An unbiased estimation procedure for the overall population mean  yields the point estimate:

Estimation of the Population Mean, Stratified Random Sample (continued) (ii)An unbiased estimation procedure for the variance of our estimator of the overall population mean yields the point estimate where confidence intervals for the population mean for stratified random samples (iii)Provided the sample size is large, 100(1 -  )% confidence intervals for the population mean for stratified random samples are obtained from

Estimation of the Population Total, Stratified Random Sample Suppose that random samples of n j individuals from strata containing N j individuals (j = 1, 2,..., K) are selected and that the quantity to be estimated is the population total, N . (i)An unbiased estimation procedure for the population total N  yields the point estimate (ii)An unbiased estimation procedure for the variance of our estimator of the population total yields the point estimate confidence intervals for the population total for stratified random samples (iii)Provided the sample size is large, 100(1 -  )% confidence intervals for the population total for stratified random samples are obtained from

Estimation of the Population Proportion, Stratified Random Sample Suppose that random samples of n j individuals from strata containing N j individuals (j = 1, 2,..., K) are obtained. Let  j be the population proportion, and p j the sample proportion, in the j th stratum, of those possessing a particular characteristic. If  is the overall population proportion: (i)An unbiased estimation procedure for  yields (ii)An unbiased estimation procedure for the variance of our estimator of the overall population proportion is

Estimation of the Population Proportion, Stratified Random Sample (continued) where is the estimate of the variance of the sample proportion in the j th stratum. 4Provided the sample size is large, 100(1 -  )% confidence intervals for the population proportion for stratified random samples are obtained from

Proportional Allocation: Sample Size The proportion of sample members in any stratum is the same as the proportion of population members in the stratum. Thus, for the j th stratum, sample size for the j th stratum using proportional allocation So that the sample size for the j th stratum using proportional allocation is

Optimal Allocation: Sample Size for j th Stratum, Overall Population Mean or Total sample size for the j th stratum using optimal allocation If it is required to estimate an overall population mean or total and if the population variances in the individual strata are denoted,  j 2, it can be shown that the most precise estimators are obtained with optimal allocation. The sample size for the j th stratum using optimal allocation is:

Optimal Allocation: Sample Size for j th Stratum, Population Proportion sample size for the j th stratum for population proportion using optimal allocation For estimating the overall population proportion, estimators with the smallest possible variance are obtained by optimal allocation. The sample size for the j th stratum for population proportion using optimal allocation is:

Sample Size: Population Mean or Total, Simple Random Sampling Consider estimating the mean of a population of N members, which has variance  2. If the desired variance, of the sample mean is specified, the required sample size to estimate population mean through simple random sampling is (i)Often it is more convenient to specify directly the width of confidence intervals for the population mean rather than. This is easily accomplished since, for example, a 95% confidence interval for the population mean will extend an approximate amount 1.96 on each side of the sample mean, X. (ii)If the object of interest is the population total, the variance of the sample estimator os this quantity is N 2 and that confidence intervals for it extend an approximate amount of 1.96N on each side of NX. I

Sample Size: Population Proportion, Simple Random Sampling Consider estimation of the proportion  of individuals in a population of size N who possess a certain attribute. If the desired variance,, of the sample proportion is specified, the required sample size to estimate the population proportion through simple random sampling is The largest possible value for this expression, whatever the value of , is A 95% confidence interval for the population proportion will extend an approximate amount 1.96 on each side of the sample proportion.

Sample Size: Overall Mean (Stratum Population variances Specified), Stratified Sampling Suppose that a population of N members is subdivided in K strata containing N 1, N 2,...,N K members. Let  j 2 denote the population variance in the j th stratum, and suppose that an estimate of the overall population mean is desired. If the desired variance,, of the sample estimator is specified, the required total sample size, n, is as follows (i) Proportional allocation:

Sample Size: Overall Mean (Stratum Population variances Specified), Stratified Sampling (continued) (ii) Optimal allocation:

Estimators for Cluster Sampling A population is subdivided into M clusters and a simple random sample of m of these clusters is selected and information is obtained from every member of the sampled clusters. Let n 1, n 2,..., n m denote the numbers of population members in the m sampled clusters. Denote the means of these clusters by and the proportions of cluster members possessing an attribute of interest by  1,  2,...,  m. The objective is to estimate the overall population mean  and proportion . (i) Unbiased estimation procedures give and

Estimators for Cluster Sampling (continued) (ii) Estimates of the variance of these estimators, following from unbiased estimation procedures, are, and

Estimation of the Population Mean, Cluster Sampling confidence intervals for the population mean using cluster sampling Provided the sample size is large, 100(1 -  )% confidence intervals for the population mean using cluster sampling are given by

Estimation of the Population Proportion, Cluster Sampling confidence intervals for the population proportion using cluster sampling Provided the sample size is large, 100(1 -  )% confidence intervals for the population proportion using cluster sampling are given by

Key Words 4Cluster Sampling 4 Estimation 4Population Mean, Simple Random 4Population Mean, Stratified 4Population Mean, Cluster 4Population Total, Simple Random 4Population Total, Stratified 4Population Proportion, Simple Random 4Population Proportion, Stratified 4Population Proportion, Cluster 4 Finite Population Correction factor 4 Nonprobabilistic Methods 4 Nonsampling Error 4 Optimal Allocation 4 Proportional Allocation 4 Quota Sampling 4 Sampling Study 4 Sampling Error 4 Simple Random Sample 4 Stratified Random Sample

Key Words (continued) 4Sample Size 4Optimal Allocation, j th Stratum 4Optimal Allocation, Total Sample 4Population Mean, Simple Random & Stratified 4Population Total, Simple Random 4Population Proportion, Simple Random & Stratified 4Proportional Allocation, j th Stratum and Total Sample 4Systematic Sampling 4Two-Phase Sampling