Scenario
The Provincial Training Academy rolled out a voluntary “Digital Records” short course. After six months, M&E staff classify enrolled local-level staff by location and completion:
| |
Completed |
Did not complete |
Total |
| Urban |
48 |
32 |
80 |
| Rural |
22 |
48 |
70 |
| Total |
70 |
80 |
150 |
The Academy director claims urban and rural staff complete at the same rate. A board paper needs a clear statistical answer at $\alpha = 0.05$ and a short memo on whether outreach should be redesigned for rural offices.
(Alternate framing you may mention in the memo: if completion is later linked to a continuous performance score, regression could be a follow-up—but this dataset is categorical uptake.)
Your tasks
- Choose the appropriate procedure and say why (not a two-mean t-test).
- State hypotheses in words and symbols where helpful.
- Compute expected counts, $\chi^2$ (or outline steps), degrees of freedom, and decision.
- Report row percentages (completion rates) for practical interpretation.
- Write a short PA memo using the rubric below.
Memo rubric
Use these five beats (a few sentences each is enough):
| Beat |
What to include |
| Claim |
Restate the “equal completion” claim and the policy question (redesign rural outreach?). |
| Evidence |
Table totals, test name, $\chi^2$, df, α, and urban vs rural completion rates. |
| Decision |
Reject or fail to reject independence / equal association structure as framed. |
| Practical meaning |
Whether rural uptake looks weaker and what that suggests for targeting. |
| Caveats |
Voluntary enrollment bias, urban/rural definition, confounding (connectivity, workload), association ≠ causation. |
Hints (optional)
- Two categorical variables in a $2 \times 2$ table → chi-square test of independence (or equivalent two-proportion z-test).
- Goodness of fit would apply if you had one variable and fixed expected shares—not this cross-classification.
- Check expected cell counts $\ge 5$ before trusting the $\chi^2$ approximation.
Full solution (click to reveal)
### 1. Method
**Chi-square test of independence** for location × completion.
(Equivalently: **two-proportion z-test** for $p_{\text{urban}}$ vs $p_{\text{rural}}$.)
### 2. Hypotheses
$$H_0:\ \text{completion and location are independent}$$
$$H_1:\ \text{completion and location are associated}$$
Or: $H_0: p_U = p_R$ vs $H_1: p_U \neq p_R$.
### 3. Observed rates
- Urban completion: $48/80 = 0.60$ (60%)
- Rural completion: $22/70 \approx 0.314$ (31.4%)
### 4. Expected counts under independence
$$E_{ij} = \frac{(\text{row total})(\text{column total})}{n}$$
| | Completed | Did not complete |
| --------- | ------------------------ | ------------------------ |
| **Urban** | $80\times70/150 = 37.33$ | $80\times80/150 = 42.67$ |
| **Rural** | $70\times70/150 = 32.67$ | $70\times80/150 = 37.33$ |
All $E_{ij} > 5$.
### 5. Test statistic
$$\chi^2 = \sum \frac{(O-E)^2}{E}$$
$$
\begin{align*}
&= \frac{(48-37.33)^2}{37.33} + \frac{(32-42.67)^2}{42.67} \\
&\quad + \frac{(22-32.67)^2}{32.67} + \frac{(48-37.33)^2}{37.33} \\
&\approx 3.05 + 2.67 + 3.49 + 3.05 \approx 12.26
\end{align*}
$$
$df = (2-1)(2-1) = 1$. Critical value $\chi^2_{0.05,1} = 3.84$.
Since $12.26 > 3.84$, **reject $H_0$**.
### 6. Sample memo (model)
**To:** Academy Director / Board
**From:** M&E unit
**Re:** Urban–rural differences in Digital Records course completion
**Claim.** The program assumed urban and rural enrollees complete at similar rates. We tested whether completion is associated with location in the six-month cohort ($n = 150$).
**Evidence.** Urban completion was 60% (48/80) vs about 31% rural (22/70). A chi-square test of independence gave $\chi^2 \approx 12.3$ on 1 df ($α = 0.05$; critical value 3.84).
**Decision.** We reject independence: completion rates differ by location in this cohort.
**Practical meaning.** Rural uptake looks substantially weaker. Prioritize rural-friendly scheduling, connectivity support, or local mentoring before scaling the same delivery model nationwide.
**Caveats.** Enrollment is voluntary, so differences may reflect who signs up, not only course quality. Urban/rural labels can mask road access and office IT. Association does not prove that “being rural” causes dropout—follow up with interviews and, if performance scores exist later, a regression that controls for confounders.
