Confidence Intervals

Suppose you want the average height of every adult in a city. You cannot measure everyone, so you take a sample and compute its average — say 68 inches. That sample average is a point estimate: a single best guess for the true population mean.

But a different sample would have given a slightly different number. The point estimate is almost certainly a little bit off, and reporting it alone hides that wobble. The honest move is to report a range that reflects how much the estimate could plausibly bounce around:

How wide should the cushion be? That depends on how much sample averages naturally vary — which is exactly the standard error, \( \text{SE} = \sigma / \sqrt{n} \), from sampling distributions. Because the sampling distribution of the mean is approximately normal, we know that most sample means land within a couple of standard errors of the truth. We turn that into an interval by stepping out a chosen number of standard errors on each side. That multiplier is the critical value \( z^{*} \):

The critical value \( z^{*} \) is fixed by the confidence level you choose — how often you want the recipe to succeed. Larger confidence means reaching further out, so \( z^{*} \) grows:

• 90% confidence uses \( z^{*} \approx 1.645 \)
• 95% confidence uses \( z^{*} \approx 1.96 \)
• 99% confidence uses \( z^{*} \approx 2.576 \)

These come from the standard normal curve: for 95%, \( z^{*} = 1.96 \) is the point that leaves 2.5% in each tail, so 95% of the area sits in the middle.

Here is the idea people most often get wrong. A 95% interval does not mean 'there is a 95% probability the true mean is inside this interval'. Once you have collected your data and computed the interval, the true mean is either in it or it is not — there is no more chance involved.

The 95% describes the procedure, not the single interval. If you repeated the whole process — take a fresh sample, build the interval — over and over, about 95% of those intervals would capture the true mean. Any one interval you actually hold is one draw from that long-run game. The simulation below lets you play the game hundreds of times and watch the capture rate settle in.

The half-width of the interval is the margin of error, \( z^{*}\,\sigma/\sqrt{n} \). Two knobs move it:

• Confidence level — more confidence means a bigger \( z^{*} \), so the interval gets wider. Demanding to be right more often forces you to cast a bigger net.
• Sample size n — because of the \( \sqrt{n} \), more data shrinks the standard error, so the interval gets narrower. But it is a square-root law: cutting the width in half takes four times the data.

To get a tighter interval without giving up confidence, the honest lever is collecting more data.