Interval estimation provides a range of values for an unknown population parameter.

A confidence interval (CI) pairs the estimate with a confidence level (95%).

Contrast with a point estimate, which is a single number.

If the population standard deviation \(\sigma\) is known, a \((1-\alpha)\times 100\%\) CI for the mean is \[ \bar{X} \;\pm\; z_{\alpha/2}\,\frac{\sigma}{\sqrt{n}} \, . \]

Where \(\bar{X}\) is the sample mean, \(n\) is the sample size, and \(z_{\alpha/2}\) is the standard normal critical value.

If \(\sigma\) is unknown, replace it with the sample SD \(s\) and use the \(t\)-distribution: \[ \bar{X} \;\pm\; t_{1-\alpha/2,\,df=n-1}\,\frac{s}{\sqrt{n}} \, . \]

This is the most common confidence interval for a mean in practice.

## $n
## [1] 100
## 
## $mean
## [1] 50.90406
## 
## $sd
## [1] 9.128159
## 
## $t_crit
## [1] 1.984217
## 
## $CI
## [1] 49.09283 52.71528

“There’s a 95% confidence that the true mean lies between 49.09 and 52.72.”

A larger \(n\) makes a narrower Confidence Interval.

More variability creates a wider Confidence Interval.

Higher confidence (for example, 99% vs 95%) generates a wider Confidence Interval.

Confidence intervals give a range of plausible values for parameters.

We use t-based confidence intervals for the mean when \(\sigma\) is unknown.

Larger samples and lower variability yield tighter intervals.

Higher confidence levels create wider intervals.