usually unimportant in social surveys: n =10,000 and N = 5,000,000: 1- f = 0.998 n =1000 and N = 400,000: 1- f = 0.9975 n =1000 and N = 5,000,000: 1-f = 0.9998 effect of changing n much more important than effect of changing n/N

The estimated variance Usually we report the standard error of the estimate: Confidence intervals for m is based on the Central Limit Theorem:

Example n N = 341 residential blocks in Ames, Iowa yi = number of dwellings in block i 1000 independent SRS for different values of n n Proportion of samples with |Z| <1.64 Proportion of samples with |Z| <1.96 30 0.88 0.93 50 70 0.94 90 0.90 0.95

For one SRS with n = 90:

Absolute value of sampling error is not informative when not related to value of the estimate For example, SE =2 is small if estimate is 1000, but very large if estimate is 3 The coefficient of variation for the estimate: A measure of the relative variability of an estimate. It does not depend on the unit of measurement. More stable over repeated surveys, can be used for planning, for example determining sample size More meaningful when estimating proportions

Estimation of a population proportion p with a certain characteristic A p = (number of units in the population with A)/N Let yi = 1 if unit i has characteristic A, 0 otherwise Then p is the population mean of the yi’s. Let X be the number of units in the sample with characteristic A. Then the sample mean can be expressed as

So the unbiased estimate of the variance of the estimator:

Examples A political poll: Suppose we have a random sample of 1000 eligible voters in Norway with 280 saying they will vote for the Labor party. Then the estimated proportion of Labor votes in Norway is given by: Confidence interval requires normal approximation. Can use the guideline from binomial distribution, when N-n is large:

In this example : n = 1000 and N = 4,000,000 Ex: Psychiatric Morbidity Survey 1993 from Great Britain p = proportion with psychiatric problems n = 9792 (partial nonresponse on this question: 316) N @ 40,000,000

General probability sampling Sampling design: p(s) - known probability of selection for each subset s of the population U Actually: The sampling design is the probability distribution p(.) over all subsets of U Typically, for most s: p(s) = 0 . In SRS of size n, all s with size different from n has p(s) = 0. The inclusion probability:

Illustration U = {1,2,3,4} Sample of size 2; 6 possible samples Sampling design: p({1,2}) = ½, p({2,3}) = 1/4, p({3,4}) = 1/8, p({1,4}) = 1/8 The inclusion probabilities:

Some results

Estimation theory probability sampling in general Problem: Estimate a population quantity for the variable y For the sake of illustration: The population total

CV is a useful measure of uncertainty, especially when standard error increases as the estimate increases Because, typically we have that

Some peculiarities in the estimation theory Example: N=3, n=2, simple random sample

For this set of values of the yi’s:

Let y be the population vector of the y-values. This example shows that is not uniformly best ( minimum variance for all y) among linear design-unbiased estimators Example shows that the ”usual” basic estimators do not have the same properties in design-based survey sampling as they do in ordinary statistical models In fact, we have the following much stronger result: Theorem: Let p(.) be any sampling design. Assume each yi can take at least two values. Then there exists no uniformly best design-unbiased estimator of the total t

Proof: This implies that a uniformly best unbiased estimator must have variance equal to 0 for all values of y, which is impossible