Point estimation involves using sample data to calculate a single value (known as a statistic) that serves as the best guess for an unknown population parameter.

\[ \begin{align*} \text{Sample Mean:}\quad & \bar{X} = \frac{1}{n} \sum_{i=1}^n X_i \\ \text{Sample Variance:}\quad & S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \bar{X})^2 \end{align*} \] These are unbiased estimators of the population mean and variance.

• Unbiasedness: \(E[\hat{\theta}] = \theta\)
• Consistency: \(\hat{\theta}_n \xrightarrow{p} \theta\)
• Efficiency: Among all unbiased estimators, the one with the smallest variance.

set.seed(123)
data <- replicate(1000, mean(rnorm(30, mean = 10, sd = 2)))
ggplot(data.frame(means = data), aes(x = means)) +
  geom_histogram(color = "black", fill = "skyblue", bins = 30) +
  labs(title = "Sampling Distribution of the Sample Mean",
       x = "Sample Mean", y = "Frequency")

• Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers (6th ed.). Wiley.
• Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
• R Documentation: ggplot2, plotly, and base R stats package.
• Simulation and plot examples were custom-generated for this presentation using R.