Making Smart Guesses

In statistics, we rarely know everything about a population. Instead, we use samples. Confidence intervals help us estimate population parameters (like the mean) and show how sure we are about our estimate.

What Is a Confidence Interval?

A confidence interval is a range of values, calculated from the sample, that’s likely to include the true population value. For example, a 95% confidence interval means we're 95% sure the real value is in our interval.

Calculating Confidence Intervals

The formula for the confidence interval for a mean is:

\[ \text{CI} = \bar{x} \pm z^* \left( \frac{s}{\sqrt{n}} \right) \]

where:

• \( \bar{x} \) = sample mean
• \( z^* \) = z-score for the chosen confidence level
• \( s \) = sample standard deviation
• \( n \) = sample size

Why It Matters

Confidence intervals let scientists, marketers, and pollsters say, “We’re pretty sure the truth is in this range!”