The cheat sheet organised by parameter

A grid, not a list

Most cheat sheets fail the same way: thirty formulas in the order the course met them, with no way to find the right one under time pressure. The paper's first mark is choosing the row — which parameter, which estimator, which distribution — so build the sheet as a grid indexed by parameter. One row per parameter; five columns: the estimator, its standard error, the distribution of the standardised estimator, the degrees of freedom, and the conditions under which the row is valid.

Worked example — the μ row, filled completely

Take the population mean, $\mu$.

• Estimator: $\bar{x}$, the sample mean.
• Standard error: $\sigma /\sqrt{n}$ when $\sigma$ is known; $s/\sqrt{n}$ when it is estimated from the sample.
• Distribution: $z$ when $\sigma$ is known; $t$ when $s$ replaces it.
• df: $n-1$ (for $t$ only; $z$ has none).
• Conditions: a simple random sample, and either a normal population or $n>30$ so that the Central Limit Theorem carries the sampling distribution.

Now the row can be used. A confidence interval is always

$$\text{estimate}\pm \text{critical value}\times \text{standard error}$$

so for $\mu$ with $\sigma$ unknown: $\bar{x}\pm t_{\alpha /2;n-1}\cdot s/\sqrt{n}$. A test statistic is always

$$\frac{\text{estimate}-\text{hypothesised value}}{\text{standard error}}$$

so for $\mu$: $t=(\bar{x}-{\mu }_0)/(s/\sqrt{n})$ on $n-1$ df. The same two grammars, read off the same row, produce Module 9's interval and Module 10's test.

The p row

For a population proportion $p$: estimator $\hat{p}=x/n$; standard error $\sqrt{p(1-p)/n}$ — with $p_0$ inside it for a test and $\hat{p}$ for an interval; distribution $z$ (never $t$: the SE of a proportion has no $s$ in it); no df; conditions $n\hat{p}\ge 5$ and $n(1-\hat{p})\ge 5$. The row says why a proportion is always a z problem: nothing in its standard error was estimated by a sample SD.

The rows you will fill later

The grid has six rows: $\mu$, $p$, ${\mu }_1-{\mu }_2$, ${\mu }_D$, $p_1-p_2$, ${\beta }_1$. The first two are filled here; the difference of two means comes in Module 11 (with the by-hand df rule $\min (n_1-1,n_2-1)$), paired differences reduce to the $\mu$ row applied to the differences, the two-proportion row is Module 12, and the slope row is Module 13 on $n-2$ df. At the end of Module 11 you will fill the ${\mu }_1-{\mu }_2$ row from memory — keep this page.

The four steps, as four lines

The second thing on the sheet is the marking skeleton itself, as four lines you can copy into every inference answer: hypotheses → statistic → table → conclusion. Type both grammars below until they are automatic.