Confidence Intervals

The idea

A single sample mean is a point estimate — almost certainly a little off. A confidence interval instead gives a range of plausible values for the parameter, together with a stated level of confidence.

Every confidence interval has the same shape

$\text{estimate} \pm \text{margin of error}$ $\text{margin of error} = (\text{critical value}) \times (\text{standard error})$

Confidence interval for a mean

When $\sigma$ is known (or $n$ is large):

$\bar{x} \pm z^{*}\, \frac{\sigma}{\sqrt{n}}$

The critical value $z^{*}$ depends on the confidence level:

Confidence	$z^{*}$
90%	1.645
95%	1.96
99%	2.576

Worked example

A sample of $n = 100$ has $\bar{x} = 50$, with $\sigma = 8$. A 95% confidence interval:

$50 \pm 1.96 \cdot \frac{8}{\sqrt{100}} = 50 \pm 1.96(0.8) = 50 \pm 1.568$

The interval is $(48.43,\ 51.57)$. We're 95% confident the true mean lies in this range.

What "95% confident" actually means

• Correct: if we repeated the whole sampling process many times, about $95\%$ of the intervals we built would contain the true $\mu$.
• Wrong: "there's a $95\%$ probability that $\mu$ is in this interval." The parameter $\mu$ is fixed; the interval is the random thing.

The trade-offs

• Higher confidence → larger $z^{*}$ → wider interval (more sure, less precise).
• Larger $n$ → smaller standard error → narrower interval.
• To halve the margin of error, you need four times the sample size.

Key takeaways

• Confidence interval $=$ estimate $\pm$ (critical value)(standard error).
• For a mean: $\bar{x} \pm z^{*}\,\sigma/\sqrt{n}$, with $z^{*} = 1.96$ for 95%.
• "95% confident" describes the long-run method, not one specific interval.
• More confidence widens the interval; more data narrows it.