Simple random sampling

In many situations, we are interested in taking a sample of the data. There could be multiple reasons for doing this, but in most practical considerations this happens due to budget constraints. For example, what if we have a certain amount people in a neighborhood, and we want to estimate what proportion of them is supporting Candidates A or B in an election? Visiting each one of them would be prohibitively expensive, so we might decide to find a smaller subset that we can visit and ask them who they are going to vote for.

The sample function in R can be used to take a sample with or without the replacement of any arbitrary size. Once a sample is defined, we can sample those units. An interesting question is estimating the variability of that sample with respect to a given sample size. In order to do that, we will build a thousand replications of our sampling exercise, and we will get the upper and lower boundaries that enclose 95% of the cases. Nevertheless, we could also compute approximate confidence intervals for the proportion of people voting for Candidate A, using the well known Gaussian approximation to an estimated proportion.

When the sample size is not very small, the estimated proportion is distributed approximately as a Gaussian distribution:

This can be used to compute an approximate confidence interval, where we need to choose  in order to achieve an interval of _, as implemented in the following formula:

Both methods will be in agreement for reasonably large samples (>100), but will differ when the sample sizes are very small. Still, the method that uses the sample function can be used for more complex situations, such as cases when we use sample weights.