200 people are classified by whether they were vaccinated and whether they contracted a disease, as shown. Compute the chi-square statistic and comment on the strength of any association.

| | Contracted disease | No disease | Total |
|---|---|---|---|
| **Vaccinated** | 10 | 90

Question

200 people are classified by whether they were vaccinated and whether they contracted a disease, as shown. Compute the chi-square statistic and comment on the strength of any association.

| | Contracted disease | No disease | Total |
|---|---|---|---|
| **Vaccinated** | 10 | 90 | 100 |
| **Not vaccinated** | 40 | 60 | 100 |
| **Total** | 50 | 150 | 200 |

Accepted Answer

**Step 1 — Expected frequencies, $E_{ij} = R_i C_j / N$, $N = 200$.** Row totals both 100; columns 50 (disease), 150 (no disease).
$$E(	ext{Vacc, Disease}) = \dfrac{100 	imes 50}{200} = 25, \qquad E(	ext{Vacc, No disease}) = \dfrac{100 	imes 150}{200} = 75$$
$$E(	ext{Not vacc, Disease}) = \dfrac{100 	imes 50}{200} = 25, \qquad E(	ext{Not vacc, No disease}) = \dfrac{100 	imes 150}{200} = 75$$

**Step 2 — Tabulate $(O-E)^2/E$.**

| Cell | $O$ | $E$ | $O-E$ | $(O-E)^2$ | $(O-E)^2/E$ |
|---|---|---|---|---|---|
| Vacc, Disease | 10 | 25 | −15 | 225 | $225/25 = 9$ |
| Vacc, No disease | 90 | 75 | 15 | 225 | $225/75 = 3$ |
| Not vacc, Disease | 40 | 25 | 15 | 225 | $225/25 = 9$ |
| Not vacc, No disease | 60 | 75 | −15 | 225 | $225/75 = 3$ |
| **Total** | | | | | **24** |

**Step 3 — Sum.**
$$\chi^2 = 9 + 3 + 9 + 3 = 24$$ …

	Contracted disease	No disease	Total
Vaccinated	10	90	100
Not vaccinated	40	60	100
Total	50	150	200

Cell	$O$	$E$	$O-E$	$(O-E)^2$	$(O-E)^2/E$
Vacc, Disease	10	25	−15	225	$225/25 = 9$
Vacc, No disease	90	75	15	225	$225/75 = 3$
Not vacc, Disease	40	25	15	225	$225/25 = 9$
Not vacc, No disease	60	75	−15	225	$225/75 = 3$
Total					24

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