One template for every interval

The formula sheet handed out on exam day carries two formulas for inference. Not twelve, and not one per procedure. Two:

\[\text{statistic} \pm (\text{critical value})(\text{standard error}), \qquad \frac{\text{statistic} - \text{parameter}}{\text{standard error}}.\]

The first is every confidence interval in the course. The second is every standardized test statistic. What follows them on the sheet is not more formulas — it is a table of standard errors, one row per sampling distribution, and choosing the row is the whole of what looks like memorising a dozen procedures.

Assemble one

Pick a procedure and watch which of the three slots changes.

Only one slot moves

Hold the purpose at confidence interval and step through the six families. The shape never changes. A statistic, a critical value, a standard error, and a plus-or-minus. What changes is the third slot, and every value it takes is a row of the sheet’s table: one proportion, two proportions, one mean, two means. There is no separate formula for a paired interval, because a paired design is a one-sample procedure on the differences and takes the one-mean row with \(n\) counting pairs.

That is why the sheet does not print twelve formulas. It prints two, and expects them to be assembled.

Where the two purposes disagree

Now hold the family fixed and switch between interval and test. For every procedure about means, nothing changes but the critical value, because the standard error is \(s/\sqrt{n}\) either way.

For proportions, the standard error changes too, and the sheet says why in a way that is easy to read past. It gives two columns rather than one: a standard deviation built from the parameter, and a standard error built from the statistic. A test has a null hypothesis, so it knows a value for the parameter and can use the first. An interval has no such value, so it must estimate with the second.

This is also the honest answer to a question students ask constantly: why do proportions get \(z\) and means get \(t\)? Because a proportion’s spread is determined by the proportion itself, so a hypothesised \(p_0\) supplies the standard deviation exactly. A mean’s spread depends on \(\sigma\), which no hypothesis ever supplies, so \(s\) must stand in — and the extra uncertainty in that substitution is precisely what the \(t\) distribution is for.

The two-proportion test carries the same idea one step further. Its null says the two proportions are equal, so a test may pool the two samples into a single combined estimate \(\hat{p}_c\) before computing the standard error. The interval, assuming nothing, keeps them apart. Same statistic on top, two different denominators, and the difference is entirely about what the null hypothesis was willing to tell you.

The exception, and what it tells you

Chi-square does not fit either template, and the sheet prints it separately for that reason. It is a sum over cells rather than a standardized distance, there is no parameter subtracted, and there is no chi-square interval anywhere in the course.

That is worth noticing rather than filing away. Every other procedure measures how far one number sits from another in standard errors, which is why every other procedure has both an interval and a test. Chi-square measures total disagreement across a whole table, and a total has no natural interval attached. The shape of the formula predicts what questions the procedure can answer.