Confidence Interval Master Reference Cheat Sheet

Key Facts

• The general confidence interval form is $\text{estimate} \pm \text{critical value} \cdot \text{standard error}$.
• For one population mean with unknown $\sigma$, the interval is $\bar{x} \pm t^*\frac{s}{\sqrt{n}}$ with $df = n - 1$.
• For one population proportion, the interval is $\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$.
• For two independent means, the interval is $(\bar{x}_1 - \bar{x}_2) \pm t^*\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$.
• For two independent proportions, the interval is $(\hat{p}_1 - \hat{p}_2) \pm z^*\sqrt{\frac{\hat{p}_1(1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2(1 - \hat{p}_2)}{n_2}}$.
• Common critical values for proportions are $z^* = 1.645$ for $90\%$, $z^* = 1.96$ for $95\%$, and $z^* = 2.576$ for $99\%$ confidence.
• Increasing the confidence level increases the critical value, so the confidence interval becomes wider.
• A correct interpretation says that the method captures the true parameter in about the stated percent of many repeated samples, not that one fixed interval has that probability.

Vocabulary

Confidence interval

A range of plausible values for an unknown population parameter based on sample data and a confidence level.

Point estimate

A single sample statistic, such as $\bar{x}$ or $\hat{p}$, used to estimate a population parameter.

Margin of error

The amount added to and subtracted from the point estimate, calculated as $\text{critical value} \cdot \text{standard error}$.

Critical value

A multiplier such as $z^*$ or $t^*$ that matches the confidence level and sampling distribution.

Standard error

The estimated standard deviation of a statistic, such as $\frac{s}{\sqrt{n}}$ for a sample mean.

Degrees of freedom

A value, often $df = n - 1$ for a one-sample $t$ interval, used to choose the correct $t^*$ value.