The Central Limit Theorem

Investigate one of the following statements about the central limit theorem for simple random sampling. To do this, create a population and consider various values for the sample size. You may want to consider a few different populations before settling on one to present. Support (or refute) the statement by writing a one paragraph summary of your simulation findings. Include as evidence your population distribution and 3 normal quantile plots.

The statement you investigate is determined by the last character in your stat login:

• a-g : the first statement
• h-o : the second statement
• p-z : the third statement

For example, if your account is s135as then you will address the third statement below.

1. For the CLT to hold, n must be large in absolute terms and small relative to the population size.
2. The SD of a sample is a random variable, and it too follows the CLT. Do all sample statistics follow the normal distribution eventually?
3. When the population size is really large, there is no difference between sampling with and without replacement.

Rules of thumb for 0-1 populations

Consider one of the following statements for populations that consist of only 0,1 values. Here p is the proportion of 1s in the population, n is the sample size, and draws are made with replacement.

1. The normal approximation works well when square-root(np(1-p)) is at least 3.
2. The continuity correction, which takes into account the fact that the sum can only be integer-valued, is important even when sample sizes are large.

Make a table of various values of n and p that you considered (at least 8) and write a brief summary of your findings. Include one normal-quantile plot to support your conclusions.