Confidence Intervals and Estimation

Point to Interval Estimation

A single sample number (point estimate, like $\bar{x}$) almost never hits the true population parameter exactly. A confidence interval gives a range of plausible values paired with a confidence level, honestly capturing our uncertainty.

The general formula is: $$\bar{x} \pm (\text{critical value}) \times \text{standard error}$$

This equals your point estimate plus or minus a margin of error. The margin multiplies a critical value (set by your confidence level, like $1.96$ for 95%) by the standard error $\frac{\sigma}{\sqrt{n}}$.

Decoding Confidence

Common Pitfall: Thinking a 95% interval means a "95% probability the true value is in this specific interval." The true parameter is fixed, and your calculated interval either contains it (1) or doesn't (0). Probability doesn't apply after the interval is built.

Instead, the confidence level describes the procedure: if you draw repeated samples and build an interval each time, about 95% of those intervals will contain the true parameter. Confidence belongs to the method, not the single interval.

$$\bar{x} \pm z \cdot \frac{\sigma}{\sqrt{n}}$$