An interval estimation is a topic in statistics that allows someone to use a range of potential values to estimate a population parameter rather than giving a single prediction.

This method was developed in the 1930s and continues to be a fundamental piece of statistics.

There are three different confidence intervals that we are able to use to estimate a parameter’s value - a CI for the mean with a known population standard deviation, a mean without a known population standard deviation, and a CI for the population proportion.

When the population standard deviation \(\sigma\) is known, a CI for the mean \(\mu\) is:

\[ \bar{x} \pm Z_{\alpha/2}\frac{\sigma}{\sqrt{n}} \]

where \(z_{\alpha/2}\) is the \((1-\alpha/2)\) quantile of the standard normal distribution.

(Notation: \(\bar x\) = sample mean, \(n\) = sample size.)

If \(\sigma\) is unknown and the sample comes from a Normal population, you can replace \(z\) with the \(t\)-quantile:

\[ \bar{x} \pm t_{\,\alpha/2}\frac{s}{\sqrt{n}} \]

where \(s\) is the sample standard deviation and \(t_{\,\alpha/2}\) is the \((1-\alpha/2)\) quantile of Student’s \(t\).

For a proportion \(p\), approximate \((1-\alpha)\) CI is:

\[ \hat p \pm Z_{\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}} \]

This is the code I used to create the 3D graph in the next slide

library(plotly)
alpha = 0.05
z = qnorm(1-alpha/2)

n_vals = seq(5,200, length.out = 50)
sig_vals = seq(0.5,5, length.out = 50)

# create grid
z_mat = outer(n_vals, sig_vals, function(n, s) 2*z*s/sqrt(n))

surf = plot_ly(x = ~n_vals, y = ~sig_vals, z = ~z_mat) %>% add_surface() %>%
  layout(scene = list(xaxis = list(title = "n"), yaxis = list(title = "sigma"), zaxis = list(title = "CI width")),
         title = "CI width surface: 95% CI width for mean (known sigma)")

surf

This interactive plot shows how CI width decreases with sample size and increases with standard deviation.

Confidence Interval | Definition, Interpretation, Example, & Facts | Britannica, www.britannica.com/science/confidence-interval. Accessed 17 Nov. 2025.

“Interval Estimation.” GeeksforGeeks, GeeksforGeeks, 2 Sept. 2024, www.geeksforgeeks.org/maths/interval-estimation/.

“Interval Estimation.” Wikipedia, Wikimedia Foundation, 11 Oct. 2025, en.wikipedia.org/wiki/Interval_estimation.