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Simulation study: CLT

The central limit theorem is an important computational short-cut for generating and making inference from the sampling distribution of the mean. You’ll recall that the central limit theorem short-cut relies on a number of conditions, specifically:

  1. Independent observations
  2. Identically distributed observations
  3. Mean and variance exist
  4. Sample size large enough for convergence

In this simulation study, you are going to compare the sampling distribution of the mean generated by simulation to the sampling distribution implied by the central limit theorem. You will compare the distributions graphically in QQ-plots.

This will be a 4 × 4 factorial experiment. The first factor will be the sample size, with N = 5, 10, 20, and 40. The second factor will be the degree of skewness in the underlying distribution. The underlying distribution will be the Skew-Normal distribution. The Skew-Normal distribution has three parameters: location, scale, and slant. When the slant parameter is 0, the distribution reverts to the normal distribution. As the slant parameter increases, the distribution becomes increasingly skewed. In this simulation, slant will be set to 0, 2, 10, 100. Set location and scale to 0 and 1, respectively, for all simulation settings.

Use the rsn function in the sn package.

The output of each combination of factors will be a QQ-plot. Generate a sampling distribution of 5000 draws from both the CLT approximation and the simulation approximation. When analyzing data, the parameters for the mean and variance (in the case of the CLT shortcut) or the parameters for the distribution (in the case of MLE, MM, etc) are replaced with sample estimates. (This is often called the plug-in approach.) For the purposes of this simulation, treat the mean and variance as known values and use the actual population parameters and the population mean and variance instead of sample estimates.

Here is a template that may help for generating the QQ-plot for a single simulation setting

require(magrittr)
# Parameters that do not change
R <- 5000; location <- 0; scale <- 1

# Parameters that change
N <- 5
slant <- 10

# Quantites to calculate/generate
pop_mean <- ... code here ...#
pop_sd <- ... code here ...#
sample_dist_clt <- ... code here ...#
sample_dist_sim <- ... code here ...#

# QQ plot
qqplot(sample_dist_sim, sample_dist_clt, asp = 1)
abline(0,1)

# Optional (Dislay the 95% CI from the clt and the sim)
... code here ...

The table of figures below is a possible (but not required) template for displaying the results.

You will write a blog post to explain your simulation study and results. The audience of your blog post is a Senior Data Scientist who you hope to work with in the future. Think about how you will communicate the results. Comment on any patterns you observe.

Advanced (optional)

Overlay in each figure the QQ-plot that results when the mean and variance parameters are estimated from data, instead of using the population parameters.

Submission instructions

  1. Within the repo Probability and Inference Portfolio Lastname Firstname, create a folder called 10-CLT-approximation
  2. Within the folder, create an .html for your blog post
  3. The name of the blog post file must be writeup.html
  4. Within the folder, include code scripts or .rmd or some other document that will successfully generate the output of the simulation when executed from within the folder. (Do not use absolute file paths.)
  5. Edit the README to be an index for the portfolio.
  6. Be prepared to share your blog post with the class when the deliverable is due.

Other instructions

  1. The deliverable should be your own work. You may discuss concepts with classmates, but you may not share code or text.