Confidence Intervals One- and Two-Sample Cheat Sheet

Key Facts

• The general confidence interval form is $\text{estimate} \pm \text{critical value} \cdot \text{standard error}$.
• For one population proportion, use $\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ when the success-failure condition is met.
• For one population mean with unknown $\sigma$, use $\bar{x} \pm t^*\frac{s}{\sqrt{n}}$ with $df = n - 1$.
• For two population proportions, use $(\hat{p}_1 - \hat{p}_2) \pm z^*\sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$.
• For two independent population means, use $(\bar{x}_1 - \bar{x}_2) \pm t^*\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$.
• A higher confidence level uses a larger critical value, which makes the confidence interval wider.
• A larger sample size decreases the standard error because standard error usually contains a denominator of $\sqrt{n}$.
• A $95\%$ confidence interval means the method would capture the true parameter in about $95\%$ of many repeated random samples.

Vocabulary

Confidence interval

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

Confidence level

The long-run percentage of intervals from the same method that would contain the true population parameter.

Margin of error

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

Standard error

An estimate of the standard deviation of a sampling distribution, such as $\frac{s}{\sqrt{n}}$ for a sample mean.

Critical value

A multiplier such as $z^*$ or $t^*$ chosen from a distribution based on the confidence level.

Degrees of freedom

A value used with the $t$ distribution that depends on sample size, often $df = n - 1$ for one sample mean.