Confidence Interval Calculator

Understanding Confidence Intervals

A **Confidence Interval** provides an estimated range of values which is likely to contain an unknown population parameter (in this case, the population mean $\mu$). The width of the interval tells us about the precision of our estimate.

Z-Interval vs. T-Interval

The choice of interval type depends on whether the population standard deviation ($\sigma$) is known:

• **Z-Interval (known σ):** Used when you know the standard deviation of the entire population. It relies on the standard normal ($Z$) distribution.
• **T-Interval (unknown σ):** Used when the population standard deviation is unknown, and you estimate it using the sample standard deviation ($s$). It relies on Student's t-distribution, which adjusts for the extra uncertainty of estimating the standard deviation.

The Formulas

The general formula for a confidence interval is: $$\text{Confidence Interval} = \bar{x} \pm \text{Margin of Error (ME)}$$

**Standard Error of Mean (SE):** $$\text{SE} = \frac{\text{Standard Deviation}}{\sqrt{n}}$$

**Margin of Error (ME):** $$\text{ME} = \text{Critical Value} \times \text{SE}$$ Where the critical value is:
- $z^*$ from standard normal distribution for Z-interval.
- $t^*$ from Student's t-distribution with $df = n - 1$ degrees of freedom for T-interval.