• A point estimate is a single value that represents a “best guess” for an unknown population parameter.
  
• Examples: estimating population mean \(\mu\) with sample mean \(\bar{x}\), or population proportion \(p\) with sample proportion \(\hat{p}\).
  
• We want to use a sample parameter from a sample that is well representative of the population, meaning that it is unbiased and consistent.

The sample mean is given by:

\[ \bar{X} = \frac{1}{n}\sum_{i=1}^n X_i \]

The sample variance is:

\[ S^2 = \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar{X})^2 \]

set.seed(123)
sample_data <- rnorm(50, mean = 10, sd = 3)

mean_est <- mean(sample_data)
var_est  <- var(sample_data)

mean_est

## [1] 10.10321

var_est

## [1] 7.715117

Suppose we take many random samples of size \(n\) from a population with mean \(\mu\) and variance \(\sigma^2\).

The sampling distribution of the sample mean is:

\[ \bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \]

By the Central Limit Theorem (CLT):
- As \(n\) grows large, \(\bar{X}\) becomes approximately normal,
even if the original population is not normal.
- The standard error shrinks as \(n\) increases:

\[ SE = \frac{\sigma}{\sqrt{n}} \]

means <- replicate(1000, mean(rnorm(30, mean = 5, sd = 2)))
hist(means, main = "Sampling Distribution of the Mean", xlab = "Sample Mean", 
     col = "skyblue", border = "black")