The cheat-sheet formula inventory

◈ 8 cards

One page, organised by parameter: for each row the estimator, its SE, the distribution with its df, and the conditions — the column the marks are actually for.

The sheet, finished

L1.2 promised a grid: one row per parameter, filled as the course went. It is full now. The paper lets you bring it, so the marks are not for remembering the formulas — they are for choosing the row, using the right SE, stating the df, and checking the conditions. A sheet that carries the formulas without the conditions leaves the fourth of those on the table.

One mean

$\overset{x}{ˉ} \pm z_{α /2} σ / n$ when $σ$ is known; $\overset{x}{ˉ} \pm t_{α /2; n - 1} s / n$ otherwise. Test: $(\overset{x}{ˉ} - μ_{0}) / (σ / n)$ or $/ (s / n)$ . Minimum $n = (z σ / m)^{2}$ , rounded up. Conditions: a random sample; normal population or $n > 30$ (t: no strong skew or outliers for $n < 40$ ).

One proportion

$\overset{p}{^} \pm z_{α /2} \overset{p}{^} (1 - \overset{p}{^}) / n$ ; minimum $n = z^{2} p^{*} (1 - p^{*}) / m^{2}$ with $p^{*} = 0.5$ if no prior; test $(\overset{p}{^} - p_{0}) / p_{0} (1 - p_{0}) / n$ . Conditions: $n \overset{p}{^} \geq 5$ and $n (1 - \overset{p}{^}) \geq 5$ (with $p_{0}$ in a test).

Two means, two proportions

Welch: $(\overset{x}{ˉ}_{1} - \overset{x}{ˉ}_{2}) \pm t s_{1}^{2} / n_{1} + s_{2}^{2} / n_{2}$ , df $= min (n_{1} - 1, n_{2} - 1)$ . Pooled, when equal variances are stated: $s_{p}^{2} = [(n_{1} - 1) s_{1}^{2} + (n_{2} - 1) s_{2}^{2}] / (n_{1} + n_{2} - 2)$ , SE $= s_{p} 1/ n_{1} + 1/ n_{2}$ , df $= n_{1} + n_{2} - 2$ . Paired: $\overset{ˉ}{d} \pm t_{n_{d} - 1} s_{d} / n_{d}$ . Two proportions: test with pooled $\overset{p}{^}_{c} = (x_{1} + x_{2}) / (n_{1} + n_{2})$ and SE $= \overset{p}{^}_{c} (1 - \overset{p}{^}_{c}) (1/ n_{1} + 1/ n_{2})$ ; interval with each $\overset{p}{^}$ in its own term. Conditions: independent random samples (or a paired design for $μ_{D}$ ); each sample normal-ish or large; five successes and failures per sample for proportions.

Regression

$s = S S E / (n - 2)$ ; $S E (\hat{β}_{1}) = s / S S_{xx}$ ; $t = \hat{β}_{1} / S E$ on $n - 2$ df; $\hat{β}_{1} \pm t_{α /2; n - 2} S E$ ; $R^{2} = S S R / S S T = 1 - S S E / S S T$ . Conditions: linearity, independence, constant variance, normal errors — checked on the residual plots.

How to use it in the room

Read the question; write the parameter line (L14.1); find the row; copy the SE with the question’s numbers in it; write the df beside the $t$ ; check the conditions in one clause (" $n \overset{p}{^} = 60 \geq 5$ "); then the four steps. The sheet is a map, not a calculator — every mark still comes from the working you write, and a formula copied without its df or its condition is a formula the marker cannot credit.

The four snippets below are the sheet in plain text, the way it fits on the page. Type them; then write the two-means block from memory — it is the one with the most branches, and the one most often chosen wrong.