Last updated: 2019-07-15

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File Version Author Date Message
Rmd 4f254dd John Blischak 2019-07-15 Add some statistics examples from Mastering ’Metrics

Reproducing the demonstration of the Central Limit Theorem in Master ’Metrics (p. 39-42).

Consider a Bernoulli random variable with probability of success of \(p = 0.8\). The sampling distribution of the mean of this distribution approximates a normal distribution, especially with increasing sample sizes. This is an example of the Central Limit Theorem.

# n - number of draws from a Bernoulli random variable with p = 0.8
randvar <- function(n) rbinom(n, size = 1, prob = 0.8)

# Calculate the t-statistic
tstat <- function(x) (mean(x) - 0.8) / (sd(x) / sqrt(length(x)))

# Visualize distribution compared to standard normal
viz <- function(x, ...) {
  hist(x, freq = FALSE, xlab = "t-statistic", ylab = "Fraction",
       xlim = c(-4, 4), ...)
  x <- seq(-4, 4, by = 0.25)
  y <- dnorm(x)
  lines(x, y, col = "red", lty = "dashed")
}

Sample size of 10

draws <- replicate(500000, randvar(10))
tstats <- apply(draws, 2, tstat)
viz(tstats, main = "n = 10")

Sample size of 40

draws <- replicate(500000, randvar(40))
tstats <- apply(draws, 2, tstat)
viz(tstats, main = "n = 40")

Sample size of 100

draws <- replicate(500000, randvar(100))
tstats <- apply(draws, 2, tstat)
viz(tstats, main = "n = 100")


sessionInfo()
R version 3.6.1 (2019-07-05)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 18.04.2 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/atlas/libblas.so.3.10.3
LAPACK: /usr/lib/x86_64-linux-gnu/atlas/liblapack.so.3.10.3

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8    
 [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8   
 [7] LC_PAPER=en_US.UTF-8       LC_NAME=C                 
 [9] LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] workflowr_1.4.0.9000 Rcpp_1.0.1           digest_0.6.20       
 [4] rprojroot_1.2        backports_1.1.4      git2r_0.26.1        
 [7] magrittr_1.5         evaluate_0.14        highr_0.8           
[10] stringi_1.4.3        fs_1.3.1             whisker_0.3-2       
[13] rmarkdown_1.13.4     tools_3.6.1          stringr_1.4.0       
[16] glue_1.3.1           xfun_0.8             yaml_2.2.0          
[19] compiler_3.6.1       htmltools_0.3.6      knitr_1.23