Which parameter, which estimator

◈ 5 cards

μ, p, μ₁ − μ₂, μ_D, p₁ − p₂ or β₁ — the first mark of every inference question is naming the parameter and its estimator; the selection tree then fixes σ known or not, the design, and the table.

The first mark

Every inference question on the final opens the same way, whether or not it says so: what is this question about? The answer is a parameter — one of six in this course — and the estimator that stands in for it. Get that line right and the rest is a formula row; get it wrong and every number that follows is the right arithmetic for the wrong question.

Wording	Parameter	Estimator
the mean of one population	$μ$	$\overset{x}{ˉ}$
the percent / rate / fraction with an attribute	$p$	$\overset{p}{^}$
the difference in means of two separate groups	$μ_{1} - μ_{2}$	$\overset{x}{ˉ}_{1} - \overset{x}{ˉ}_{2}$
the mean change on the same units, before and after	$μ_{D}$	$\overset{ˉ}{d}$
the difference in rates of two groups	$p_{1} - p_{2}$	$\overset{p}{^}_{1} - \overset{p}{^}_{2}$
how a response changes per unit of a predictor	$β_{1}$	$\hat{β}_{1}$

The tree

After the parameter, three questions in order, each answered from the question’s wording:

Is σ known? Only for a mean. A stated population SD → $z$ ; a sample $s$ → $t$ . A proportion is always $z$ (its variance is $p (1 - p)$ , nothing to estimate).
What is the design? One sample; two independent samples; the same units measured twice (paired); or an $x$ – $y$ pair per unit (regression).
Which table and df? $t_{n - 1}$ for one mean; Welch $t_{m i n (n_{1} - 1, n_{2} - 1)}$ unless the question says assume equal variances (then pooled $t_{n_{1} + n_{2} - 2}$ ); $t_{n_{d} - 1}$ paired; $z$ for any proportion, pooled $\overset{p}{^}_{c}$ in a two-proportion test; $t_{n - 2}$ for a slope. Any $t$ with df above 30 reads the ∞ row.

Eight scenarios through the tree

Is the mean processing time of Kitchener invoices above 5 days? σ unknown, n = 20. → $μ$ , $\overset{x}{ˉ}$ , $t_{19}$ .
Is the mean above 5 days, with σ = 1.2 days known from years of records? → $μ$ , $\overset{x}{ˉ}$ , $z$ .
Does the percent of invoices paid late exceed 15 %? → $p$ , $\overset{p}{^}$ , $z$ .
Do mean basket sizes differ between two chains, n = 15 and 18, nothing said about variances? → $μ_{1} - μ_{2}$ , $\overset{x}{ˉ}_{1} - \overset{x}{ˉ}_{2}$ , Welch $t_{14}$ .
Same, but "assume the two populations have equal variances". → $μ_{1} - μ_{2}$ , pooled $t_{31}$ → ∞ row.
Did eight stores sell more after the promotion than before? → $μ_{D}$ , $\overset{ˉ}{d}$ , $t_{7}$ .
Is the default rate different in two regions? → $p_{1} - p_{2}$ , $\overset{p}{^}_{1} - \overset{p}{^}_{2}$ , pooled $z$ .
Is income linearly related to credit score for 8 clients? → $β_{1}$ , $\hat{β}_{1}$ , $t_{6}$ .

The traps sit between rows. Scenario 6 is paired because the same stores appear twice — treating it as scenario 4 throws away the pairing and usually the significance. Scenario 3 is a proportion because each invoice is late or not; the mean number of days late would be scenario 1. Scenario 5 becomes pooled only because the question said so; nothing in the data decides that.

The sentence the marker wants

One line, before any arithmetic: "The parameter is $μ_{D}$ , the mean change in sales per store; the estimator is $\overset{ˉ}{d}$ ; the test statistic is $t$ on 7 df." Written that way it is the first mark, the choice of table, and the wording of the conclusion, all at once.

The selection tree. Parameter first, then σ known or not, then the design; the leaf is the table and its df. Any t with df above 30 reads the ∞ row.