You are employed by a local health district to provide advice on estimating the proportion of
the population who are currently vaccinated against COVID. The local health district is
divided into 4 areas. You know the population size in each district and from a small pilot
study you have estimates of the proportion of people who have been vaccinated in each
district. This information is summarised in the table below.
Estimated proportion
District Population vaccinated
A 25000 0.45
B 15000 0.40
C 30000 0.05
D 25000 0.10
All districts 95000 0.22
You now plan to conduct a survey in the local health district to estimate the proportion who
have been vaccinated. Based on resources the total sample size will be 300. Use the data from
the table to answer the following questions
a) If you took a simple random sample of the population what would be the standard
error of the proportion who have been vaccinated? (Note: assuming you find a similar
proportion to that from the pilot study.)
(1 mark)
b) How many people would you sample from each stratum if you allocate the sample
size of 300 using

1. Proportional allocation (1 mark)
2. Optimal allocation (1 mark)
3. Comment on the difference in sample sizes. (1 mark)
   c) If you took a stratified random sample of the population what would be the standard
   error of the proportion who have been vaccinated under
4. Proportional allocation (1 mark)
5. Optimal allocation (1 mark)
   Note: assume the estimated proportions from the table are those obtained in your
   stratified sample.
   d) Comment on your results from (a) and (c). (1 mark)

Sample Solution