---
title: "Stat. 316: Sampling"
author: "Prof. Eric A. Suess"
date: "3/20/2024"
format: 
  html:
    self-contained: true
---

# Central Limit Theorem (CLT)

The CLT states that when taking a random sample $X_1, X_2, ..., X_n$ from any population with population mean $\mu$ and population standard deviation $\sigma$, then sample mean $\bar{X}$ is Normally distributed with mean $\mu$ and standard deviation $\sigma/ \sqrt{n}$.

Or in repeated sampling the z-score is distributed standard normal.

$$ Z = \frac{\bar{X} - \mu}{\sigma/ \sqrt{n}} $$

Simulation:

Suppose we repeatedly take a samples of size $n = 12$ from a Normal population with mean $\mu = 1$ and standard deviation $\sigma = 3$.


```{r}
B <- 10000

n <- 12
mu <- 1
sigma <- 3

Z <- replicate(B, {
  x <- rnorm(n, mu, sigma)  # I need mu to simulate the data
  Xbar <- mean(x)           # Now assume I do not know mu
  (Xbar - mu) / (sigma/sqrt(n))
})

hist(Z)

plot(density(Z),
     main = "Standardized mean of 12 normal rvs", xlab = "Z"
)
curve(dnorm(x), add = TRUE, col = "red")

```

# T distribution, Sampling Distribution

Substitute the sample standard deviation for the population standard deviation.  


Simulation:

Suppose we repeatedly take a samples of size $n = 12$ from a Normal population with mean $\mu = 1$ and standard deviation $\sigma = 3$.

Or in repeated sampling the z-score is distributed standard normal.

$$ T = \frac{\bar{X} - \mu}{S/ \sqrt{n}} $$


```{r}
B <- 10000

n <- 12
mu <- 1
sigma <- 3

T <- replicate(B, {
  x <- rnorm(n, mu, sigma)
  Xbar <- mean(x)
  Xsd <- sd(x)
  SE <- Xsd / sqrt(n)
  (Xbar - mu) / SE
})

hist(T)

plot(density(T),
     main = "Standardized mean of 12 normal rvs", xlab = "T")
curve(dt(x, df = n-1), add = TRUE, col = "red")
```