2 Sample designThe focus of the design for a sample must be on the magnitude of the standard errors of sampling not than on an arbitrary percentage of the target population.The standard errors are used to calculate confidence intervals around the sample data.
3 Standard errorsThe next sequence aims to explain standard errors, and how they relate to the underlying target population and a sample drawn from this population.
4 Graph: Target population Population: mean = , standard deviation =
5 Graph: SampleSample: mean = x, standard deviation = s
6 Graph: Means from many samples However we could get many different samples with different sample means from the population.
7 Graph: Distribution of sample means This gives us a sampling distribution of sample means:
8 Sampling distribution of sample means normal distributionmean = = mean of underlying population distributionstandard deviation = / n
9 Standard error of a population mean The standard deviation of the sampling distribution of sample means is termed the standard error of a mean.standard error of population mean = / n
10 Central limit theoremThe central limit theorem states that the link between the mean of one sample and the population mean is given by:x = z. se(popn mean)If z = 1.96 we produce a confidence interval where we find 95% of the sample means, relative to
11 Estimated population mean However, we are not interested in finding the sample means given the population mean.Our aim is to locate the population mean given what we know about the sample.
12 Estimated population mean We start with a simple random sample and assume:s is a good estimate of se(sample mean) is a good estimate of se(popn mean)
13 Estimated population mean Then instead ofx = z. se(popn mean)we can write = x z. se(sample mean)wherese(sample mean) = s / n
14 Standard error of a proportion (srs) The standard error of a percentage (proportion) is:se(prop) = [p(1-p)/n]
15 Standard error of a proportion = 0.50 (srs) For p=0.50se(p50) = [0.50(1-0.50)/n]The standard error may be multiplied by a finite population correction (FPC) of (N-n)/N
16 Confidence intervalsConfidence intervals are usually expressed at the 95 per cent level (1.96 standard errors of sampling for a proportion)
18 Two stage samplesThe most efficient method is usually sampling at the first stage with probability proportional to size (pps).This produces a self-weighting sample.Easier logistics for administration.
19 Two stage samples Stage 1 Primary sampling units (psu) are selected with a probability proportional to the size of the target population in the psu.Example of psu: schools
20 Two stage samples Stage 2 A random cluster of secondary sampling units (ssu) is selected at random from each of the psu.Example of ssu: students in schools
21 DeffTwo-stage sampling is less efficient than a simple random sample (srs) of the same size.deff = (standard error of sampling for complex sample)2 / (standard error of sampling for srs)2
22 DeftThe square root of deff is deft, which gives the ratio of the standard errors of sampling.deft = (standard error of sampling for complex sample) / (standard error of sampling for srs)
23 Simple equivalent sample The simple equivalent sample (ses) is the size of a simple random sample which has the same standard error as the complex sample.We sometime use the term effective sample (neff)
25 Compare srs and sesFor a simple random sample of n=1000, the 95 per cent confidence interval is given by:= 1.96 se(p50)= 1.96 [0.50(1-0.50)/1000]= = 3.1%
26 Compare srs and sesFor a simple equivalent sample of n=345 (corresponding to a complex sample of n=1000), the 95 per cent confidence interval is given by:= 1.96 se(p50)= 1.96 [0.50(1-0.50)/345]= = 5.3%
27 Table: Values for deff and simple equivalent sample
28 Random clustersPPS sampling assumes a random cluster which is the same size for each ssu.In practice, we often draw an intact group at random.This usually increase the intraclass correlation for the sample.
29 WeightingAchieved samples are unlikely to properly represent the proportions of persons in the target populations for the strata.Weights are applied so that the achieved sample for each stratum represents its proportion in the total target population.
30 Weighting wh = Nh/nh where nh = the size of the achieved sample for the stratumNh = the size of the target population for the stratum