7.7

library(tidyverse)

Warning: package 'tidyverse' was built under R version 4.4.3

Warning: package 'ggplot2' was built under R version 4.4.3

Warning: package 'tidyr' was built under R version 4.4.3

Warning: package 'readr' was built under R version 4.4.3

Warning: package 'purrr' was built under R version 4.4.3

Warning: package 'dplyr' was built under R version 4.4.3

Warning: package 'stringr' was built under R version 4.4.3

Warning: package 'forcats' was built under R version 4.4.3

Warning: package 'lubridate' was built under R version 4.4.3

── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.4     ✔ readr     2.1.5
✔ forcats   1.0.0     ✔ stringr   1.5.1
✔ ggplot2   4.0.1     ✔ tibble    3.2.1
✔ lubridate 1.9.4     ✔ tidyr     1.3.1
✔ purrr     1.0.4     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors

library(purrrfect)

Attaching package: 'purrrfect'

The following objects are masked from 'package:base':

    replicate, tabulate

N <- 10000
clt_Unif_sims <-( parameters( ~n, c(5,10,20,40,80,160))
               
%>% add_trials(N) 
%>% mutate(Y_sample = map(n, ~ runif(.x,0,8)))
%>% mutate(Ybar = map_dbl(Y_sample, mean))
%>% mutate(fU = dnorm(Ybar, mean = 4, sd = sqrt((8-0)^2/12) / sqrt(n)),
          FU = pnorm(Ybar, mean = 4, sd = sqrt((8-0)^2/12)/sqrt(n)),
          Fhat = cume_dist(Ybar),
          .by = n
)    
)
names(clt_Unif_sims)

[1] "n"        ".trial"   "Y_sample" "Ybar"     "fU"       "FU"       "Fhat"    

ggplot(data = clt_Unif_sims, aes(x = Ybar))+
  geom_histogram(aes(y = after_stat(density)),
                 binwidth = .2, fill = 'goldenrod')+
  geom_line(aes(y = fU), col = 'cornflowerblue')+
  facet_wrap(~n, labeller = label_both, scales = 'free_y')+
  labs(x = expression(bar(Y)))+
  theme_classic()

ggplot(data = clt_Unif_sims, aes(x = Ybar))+
  geom_step(aes( y = FU, col = 'Analytic Normal CDF'))+
  geom_step(aes(y = Fhat, col = 'Empirical CDF'))+
  facet_wrap(~n, labeller = label_both, scales = 'free')+
  labs(color = '',
       x = expression(bar(Y)), 
       y = 'CDF')+
  theme_classic()

You can add options to executable code like this

library(tidyverse)
library(purrrfect)

N <- 10000
clt_gamma_sims <-( parameters( ~n, c(5,10,20,40,80,160))
               
%>% add_trials(N) 
%>% mutate(Y_sample = map(n, ~ rgamma(.x,shape = 2, scale = 2)))
%>% mutate(Ybar = map_dbl(Y_sample, mean))
%>% mutate(fU = dnorm(Ybar, mean = 4, sd = sqrt(2*2^2) / sqrt(n)),
          FU = pnorm(Ybar, mean = 4, sd = sqrt(2*2^2/sqrt(n))),
          Fhat = cume_dist(Ybar),
          .by = n
)    
)


ggplot(data = clt_gamma_sims, aes(x = Ybar))+
  geom_histogram(aes(y = after_stat(density)),
                 binwidth = .2, fill = 'goldenrod')+
  geom_line(aes(y = fU), col = 'cornflowerblue')+
  facet_wrap(~n, labeller = label_both, scales = 'free_y')+
  labs(x = expression(bar(Y)))+
  theme_classic()

ggplot(data = clt_gamma_sims, aes(x = Ybar))+
  geom_step(aes( y = FU, col = 'Analytic Normal CDF'))+
  geom_step(aes(y = Fhat, col = 'Empirical CDF'))+
  facet_wrap(~n, labeller = label_both, scales = 'free')+
  labs(color = '',
       x = expression(bar(Y)), 
       y = 'CDF')+
  theme_classic()

library(tidyverse)
library(purrrfect)

N <- 10000
clt_exp_sims <-( parameters( ~n, c(5,10,20,40,80,160))
               
%>% add_trials(N) 
%>% mutate(Y_sample = map(n, ~ rexp(.x,rate = 1/4)))
%>% mutate(Ybar = map_dbl(Y_sample, mean))
%>% mutate(fU = dnorm(Ybar, mean = 4, sd = 4 / sqrt(n)),
          FU = pnorm(Ybar, mean = 4, sd = 4/sqrt(n)),
          Fhat = cume_dist(Ybar),
          .by = n
)    
)

ggplot(data = clt_exp_sims, aes(x = Ybar))+
  geom_histogram(aes(y = after_stat(density)),
                 binwidth = .2, fill = 'goldenrod')+
  geom_line(aes(y = fU), col = 'cornflowerblue')+
  facet_wrap(~n, labeller = label_both, scales = 'free_y')+
  labs(x = expression(bar(Y)))+
  theme_classic()

ggplot(data = clt_exp_sims, aes(x = Ybar))+
  geom_step(aes( y = FU, col = 'Analytic Normal CDF'))+
  geom_step(aes(y = Fhat, col = 'Empirical CDF'))+
  facet_wrap(~n, labeller = label_both, scales = 'free')+
  labs(color = '',
       x = expression(bar(Y)), 
       y = 'CDF')+
  theme_classic()

1. Uniform normality kicks in the fastest for all 3 of the distributions.
2. N = 80 is the n for which Y bar appears normal regardless of the population.