Consider the post stratification method introduced in Exercise 5. Show that the expected value for nj equals the sample size for proportional allocation.
It is not always possible to stratify a sample before taking it. In this case, when a simple random sample is taken, it can be stratified after the units are surveyed and it is known to which group each sampled unit belongs. That is, the sample is divided into strata after sampling and treated as a stratified random sample. This technique is called post stratification. For J strata, with strata sizes Nj and weights
the post stratified estimator is
Because we have post stratified, the number of units in the stratum, nj , is random. The variance of the post stratified estimator is approximately
Where Are the mean and standard deviation for stratum j. the first term in the variance is the same as the variance under proportional allocation,
TABLE 3.5. Numbers of female and male students in the sample of statistics students (Chapter 2) who did/did not play video games in the week prior to the survey, and who do/do not own a PC.
with the finite population correction factor included. The second term is due to there being a random number of units in each stratum. Post stratify the sample of statistics students (Chapter 2) by sex, and provide a 95% confidence interval for the proportion of students who played video games in the week prior to the survey (Table 3.5). Of the 314 students in the statistics class, 131 were women.