Last updated: 2024-03-14

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Knit directory: for-future-reference/

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File Version Author Date Message
html 1155510 John Blischak 2019-07-15 Build site.
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")

Version Author Date
1155510 John Blischak 2019-07-15

Sample size of 40

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

Version Author Date
1155510 John Blischak 2019-07-15

Sample size of 100

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

Version Author Date
1155510 John Blischak 2019-07-15 

sessionInfo()
R version 4.3.3 (2024-02-29 ucrt)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 11 x64 (build 22631)

Matrix products: default


locale:
[1] LC_COLLATE=English_United States.utf8 
[2] LC_CTYPE=English_United States.utf8   
[3] LC_MONETARY=English_United States.utf8
[4] LC_NUMERIC=C                          
[5] LC_TIME=English_United States.utf8    

time zone: America/New_York
tzcode source: internal

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

other attached packages:
[1] workflowr_1.7.1

loaded via a namespace (and not attached):
 [1] jsonlite_1.8.8    compiler_4.3.3    highr_0.10        promises_1.2.1   
 [5] Rcpp_1.0.12       stringr_1.5.1     git2r_0.33.0      callr_3.7.5      
 [9] later_1.3.2       jquerylib_0.1.4   yaml_2.3.8        fastmap_1.1.1    
[13] R6_2.5.1          knitr_1.45        tibble_3.2.1      rprojroot_2.0.4  
[17] bslib_0.6.1       pillar_1.9.0      rlang_1.1.3       utf8_1.2.4       
[21] cachem_1.0.8      stringi_1.8.3     httpuv_1.6.14     xfun_0.42        
[25] getPass_0.2-4     fs_1.6.3          sass_0.4.8        cli_3.6.2        
[29] magrittr_2.0.3    ps_1.7.6          digest_0.6.34     processx_3.8.3   
[33] rstudioapi_0.15.0 lifecycle_1.0.4   vctrs_0.6.5       evaluate_0.23    
[37] glue_1.7.0        whisker_0.4.1     fansi_1.0.6       rmarkdown_2.26   
[41] httr_1.4.7        tools_4.3.3       pkgconfig_2.0.3   htmltools_0.5.7