The paper and the four-step mark

Two papers, one skeleton

AFM 113 is examined by a written midterm (weeks 1–6: descriptives, correlation, probability, the discrete and normal distributions) and a cumulative final that adds the inference half (sampling distributions, confidence intervals, tests for one and two means, proportions, and regression inference). Both are sat by hand: a z-table, a t-table, a one-page cheat sheet and a non-programmable calculator. Where a question needs software, the R output is printed on the paper and you read it — you never write R under exam conditions.

Every inference question, whatever the parameter, is marked on the same four steps. Learn the skeleton now, and Modules 8–13 become variations on one theme.

Worked example — one finished answer, dissected

Grand River Packaging, a Waterloo distributor, has always taken an average of 45 days to collect an invoice. After introducing an early-payment discount, a random sample of 20 invoices shows a mean of 41.2 days with a sample standard deviation of 8.5 days. Has the average collection time fallen? Use $\alpha =0.05$.

Step 1 — parameter and hypotheses (the parameter mark). The parameter is $\mu$, the mean days-to-pay across all invoices under the discount; its estimator is $\bar{x}=41.2$. $H_0:\mu =45$ versus $H_a:\mu <45$. The direction comes from the question ("has it fallen?"), never from the data.

Step 2 — the statistic (the statistic mark). $\sigma$ is unknown, so the test statistic is

$$t=\frac{\bar{x}-{\mu }_0}{s/\sqrt{n}}=\frac{41.2-45}{8.5/\sqrt{20}}=\frac{-3.8}{1.9007}=-2.00$$

with $df=n-1=19$.

Step 3 — the table (the table mark). Row 19 of the t-table, column 0.05: $t_{0.05;19}=1.729$. The rejection region is $t<-1.729$, and $-2.00$ is inside it. Bracketing the p-value: row 19 has 1.729 at 0.05 and 2.093 at 0.025, so $0.025<p<0.05$.

Step 4 — the conclusion in context (the context mark). Reject $H_0$ at $\alpha =0.05$. There is evidence that the mean days-to-pay under the discount is below 45 days. The sentence names the parameter (mean days-to-pay), the direction (below 45), the level ($\alpha =0.05$) and the business setting. "Reject $H_0$" alone earns the decision, not the mark.

Where the marks sit

A 12-mark version of that question typically splits 3 / 4 / 2 / 3 across the steps. Notice that the arithmetic — the part most students rehearse — is one mark in four. The parameter mark is lost by writing $\bar{x}=45$ in $H_0$; the table mark by reading the 0.025 column for a one-sided 0.05 test; the context mark by stopping at "reject".

Budgeting time

The course's assumed paper is 150 minutes for 100 marks — 1.5 minutes per mark. A 12-mark question earns 18 minutes; a 3-mark "identify the parameter and statistic" earns four and a half. Budget from the marks before you start writing, and move on when the budget is spent.

Fast track

If you took Grade-12 Data Management and can already compute a mean, a standard deviation, $r$ and a binomial probability by hand, finish this module and jump to Module 7. Modules 2–6 are that material with an accounting accent. Nobody skips Module 1: the conventions fixed here (cumulative z-table, the ∞ row above $df=30$, bracketed p-values, $N(\mu ,{\sigma }^2)$) are used in every later lesson.