• Regardless of the original distribution, the distribution of sample means approaches a normal distribution as the sample size increases.
• Sample must be random and each observation independent
• \[ SE = \frac{\sigma}{\sqrt{n}} \] shows how as sample size increases, standard error decreases / fluctation from true mean

• As the sampling distribution of means becomes normalized, we can apply statistical methods that rely on normality

• confidence intervals and hypothesis tests are statistical methods that rely on normality

• With a sufficiantly large sample size, we can normalize skewed data sets

• \[z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\] is the Z-score formula that allows for normal statistical methods

In the next slides, we will visualize just how taking a distribution of sample means will normalize this distribution, starting with 200 sample means, and then 1000

set.seed(1)

sample_means <- replicate(1000, mean(sample(mpg$hwy, 30, replace = TRUE)))

ggplot(data.frame(sample_means), aes(x = sample_means)) +
  geom_histogram(binwidth = .5, color = "blue")