PS 7.5

For each combination of n∈{4,8,15}, μ∈{-2,0,2}, and σ∈{1,2,3}, simulate 10,000 realizations of (Y ̅-μ)/(S/√n) . Verify that this has a t distribution with n-1 degrees of freedom by plotting the simulated densities superimposed by the analytic densities. Submit a link to your rendered solutions published on rpubs.

library(tidyverse)

── 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   3.5.2     ✔ tibble    3.3.0
✔ lubridate 1.9.4     ✔ tidyr     1.3.1
✔ purrr     1.1.0     
── 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

t_sample <- parameters(   ~n, ~mu, ~sigma,
                          c(4, 8, 15), c(-2, 0, 2), c(1, 2, 3)) %>%
  add_trials(N) %>%
  mutate( y_sample = pmap(list(n, mu, sigma), \(nn, mm, ss) rnorm(nn, mm, ss))
  ) %>%
  mutate(
    Ybar = map_dbl(y_sample, mean), 
    S    = map_dbl(y_sample, sd), 
    Tval = (Ybar - mu) / (S / sqrt(n))) %>%
  mutate(
    f_t = pmap_dbl(list(Tval, n), \(x, nn) dt(x, df = nn - 1))
  )

ggplot(data = t_sample, aes(x = Tval)) +
  geom_histogram(aes(y = after_stat(density)),fill = "goldenrod",binwidth= .1 ,alpha = 0.6
  ) +
  geom_line(aes(y = f_t), col = "cornflowerblue") +
  facet_grid( n ~ sigma ~ mu , labeller = label_both, scales = "free_y"
  ) +
  coord_cartesian(xlim = c(-10, 10)) +
  theme_classic(base_size = 16) +labs(
    x = expression((bar(Y) - mu)/(S/sqrt(n))), y = "Density",
    title = "Simulated and Analytic Densities of the t Statistic"
  )