Most of the inference formulas in AP Statistics can be organized around two structures. A confidence interval has the form

\[\text{statistic} \pm (\text{critical value})(\text{standard error})\]

and a standardized test statistic has the form

\[\frac{\text{statistic}-\text{null parameter value}}{\text{standard error}}\]

The procedure determines which statistic, critical value, and standard error belong in those slots.

Assemble one

Pick a procedure and switch between interval and test.

The large line is the assembled result and the three boxes underneath are the slots it was built from. Step through the six families with the purpose held fixed and the template never moves: only the third box changes, and it changes to whichever row of the sheet's standard-error table matches the sampling distribution. Then hold the family fixed and switch purpose. For means nothing changes but the middle box. For proportions the third box changes as well, which is the one place the two purposes genuinely disagree.

The overall structure stays nearly fixed, and what changes most often is the standard error. For example, a one-proportion confidence interval uses \(\hat p \pm z^* \sqrt{\tfrac{\hat p(1-\hat p)}{n}}\), and a one-sample \(t\)-interval for a mean uses \(\bar x \pm t^* \tfrac{s}{\sqrt n}\). A two-sample interval changes the statistic and standard error but not the general form.

Paired data fit the same template

A matched-pairs problem is converted into one sample of differences. If \(d_i\) is the difference for pair \(i\), then inference concerns \(\mu_d\). The interval is therefore a one-sample \(t\)-interval applied to the differences, \(\bar d \pm t^* \tfrac{s_d}{\sqrt n}\). There is no need for an entirely separate formula family. The design changes the variable being analyzed.

Why means use \(t\)

For a population mean, the standard deviation \(\sigma\) is usually unknown, and the standard error is estimated using \(\tfrac{s}{\sqrt n}\). That substitution adds uncertainty, and the \(t\)-distribution accounts for it. As the degrees of freedom increase, \(t\) approaches the standard normal distribution.

Why proportion intervals and tests use different standard errors

A one-proportion confidence interval does not assume a population value for \(p\), so its standard error is estimated with the sample proportion, \(\sqrt{\tfrac{\hat p(1-\hat p)}{n}}\). A hypothesis test specifies a null value \(p_0\), and under the null, the sampling distribution is built using that hypothesized parameter, \(\sqrt{\tfrac{p_0(1-p_0)}{n}}\).

The difference comes from the question. The interval estimates an unknown parameter, and the test temporarily assumes a particular value.

Pooling in a two-proportion test

For a two-proportion interval, the two sample proportions remain separate in the standard error. A two-proportion test under \(H_0:p_1=p_2\) assumes the groups share one population proportion under the null, and that allows a pooled estimate \(\hat p_c = \tfrac{x_1+x_2}{n_1+n_2}\). The test standard error uses this combined value, and the interval does not pool because it is estimating the difference without assuming equality.

Chi-square is different

Chi-square procedures use

\[\chi^2 = \sum \frac{(O-E)^2}{E}\]

This is not a single standardized difference between a statistic and a parameter, and it accumulates discrepancies across several cells of a table. That is why chi-square sits outside the two general templates and appears only as a test in this course.

The reference sheet is easier to use once the formulas are seen this way, and most of the work is identifying the design and choosing the matching standard error.