• Define point estimation
• Compute and interpret the sample mean as a point estimator
• Visualize sample data with ggplot2
• Simulate sampling variability and show a 3D plotly plot
• Show mathematical properties (unbiasedness, variance)

A point estimate is a single-number summary used as an estimate of a population parameter.

For example, the sample mean is a point estimator for the population mean:

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

Marine biologists take a random sample of \(n=30\) trout from a lake and measure their weights (kg).
Question: what is the best single-number estimate of the population mean weight?

## # A tibble: 1 × 3
##       n sample_mean sample_sd
##   <int>       <dbl>     <dbl>
## 1    30        2.48     0.392

Below is the R code used to generate the sample and compute the point estimate.

set.seed(123)
weights <- rnorm(30, mean = 2.5, sd = 0.4)  # weights in kg
sample_mean <- mean(weights)
sample_sd <- sd(weights)
n <- length(weights)

# view results
sample_mean
sample_sd

##ggplot: histogram of sample weights

##ggplot: density with sample vs (assumed) true mean

Unbiasedness \[ E[\bar{X}] = \mu \]

Variance Assuming independent observations with variance \(\sigma^2\): \[ \mathrm{Var}(\bar{X}) = \frac{\sigma^2}{n} \]

Conclusion: as \(n\) increases, \(\mathrm{Var}(\bar{X})\) decreases — the estimator becomes more precise.

• The sample mean is unbiased and efficient for normal data.
• In practice, \(\sigma\) is unknown → use sample SD to form confidence intervals.
• Next: Interval Estimation and Hypothesis Testing for \(\mu\).

1. Knit the file to HTML in RStudio.
   
2. In RStudio: File → Publish to RPubs.
   
3. Copy the link and include it in the PDF for Canvas.

• Textbooks: any intro to mathematical statistics
  
• R packages: ggplot2, plotly, dplyr, tibble