Confidence Intervals for a Mean

A confidence interval is simply a range of plausible values for something you can't measure directly, like the average capacity of every battery a factory makes, built from one sample. It's the "give or take" you'd add to an honest estimate. The formula $\bar x \pm z^*\frac{\sigma}{\sqrt n}$ is three pieces: $\bar x$ (your best single guess, the sample mean), $\frac{\sigma}{\sqrt n}$ (the standard error, how much sample means typically wander) and $z^*$ (how many of those standard errors to go out, more for more confidence). Best guess, plus or minus a few wobbles. That's all it is.

Where it comes from. By the CLT, $\bar x$ falls within $1.96$ standard errors of $\mu$ in 95% of samples. Now think about that the other way around. If $\bar x$ is within $1.96$ standard errors of $\mu$, then $\mu$ is within $1.96$ standard errors of $\bar x$; distance works the same in both directions. So in 95% of samples, the range $\bar x \pm 1.96\,\text{SE}$ catches $\mu$.

What "95% confident" means. The method works 95% of the time. Any one interval either contains $\mu$ or doesn't; we just don't know which. There's a picture further down with thirty intervals from thirty samples, and you can count the ones that miss.

Width trade-offs: more confidence widens the interval; more data narrows it (like $\frac{1}{\sqrt n}$). It makes sense: to be surer of catching $\mu$ you need a wider net, and more data makes $\bar x$ steadier, so you can afford a smaller one. The sample-size formula just solves $E = z^*\frac{\sigma}{\sqrt n}$ for $n$.

This version assumes $\sigma$ is known, which is rare in practice; the next lesson handles the usual case.

A common mistake. It's tempting to say "there's a 95% chance $\mu$ is between 67 and 69." It isn't, strictly. $\mu$ is a fixed number, not a random one, so it's either in that interval or it isn't. The 95% describes the method: intervals built this way catch $\mu$ 95% of the time.

Lab results, engineering tolerances and economic forecasts are reported with intervals; a single number without its margin of error hides how uncertain it is.