200 buyers are classified by region (Urban / Rural) and by which of three brands X, Y, Z they bought, as shown. Compute the chi-square statistic.

| | X | Y | Z | Total |
|---|---|---|---|---|
| **Urban** | 20 | 30 | 50 | 100 |
| **Rural** | 30 | 20 | 50 | 100 |
| **Total** | 50

Question

200 buyers are classified by region (Urban / Rural) and by which of three brands X, Y, Z they bought, as shown. Compute the chi-square statistic.

| | X | Y | Z | Total |
|---|---|---|---|---|
| **Urban** | 20 | 30 | 50 | 100 |
| **Rural** | 30 | 20 | 50 | 100 |
| **Total** | 50 | 50 | 100 | 200 |

Accepted Answer

**Step 1 — Compute expected frequencies, $E_{ij} = R_i C_j / N$, with $N = 200$.** Both row totals are 100; column totals are 50 (X), 50 (Y), 100 (Z).
$$E(	ext{Urban, X}) = \dfrac{100 	imes 50}{200} = 25, \quad E(	ext{Urban, Y}) = \dfrac{100 	imes 50}{200} = 25, \quad E(	ext{Urban, Z}) = \dfrac{100 	imes 100}{200} = 50$$
By symmetry (Rural also has row total 100), $E(	ext{Rural, X}) = 25$, $E(	ext{Rural, Y}) = 25$, $E(	ext{Rural, Z}) = 50$.

**Step 2 — Tabulate $(O-E)^2 / E$ for every cell.**

| Cell | $O$ | $E$ | $O-E$ | $(O-E)^2$ | $(O-E)^2/E$ |
|---|---|---|---|---|---|
| Urban, X | 20 | 25 | −5 | 25 | $25/25 = 1$ |
| Urban, Y | 30 | 25 | 5 | 25 | $25/25 = 1$ |
| Urban, Z | 50 | 50 | 0 | 0 | $0/50 = 0$ |
| Rural, X | 30 | 25 | 5 | 25 | $25/25 = 1$ |
| Rural, Y | 20 | 25 | −5 | 25 | $25/25 = 1$ |
| Rural, Z | 50 | 50 | 0 | 0 | $0/50 = 0$ |
| **Total** | | | | | **4** |

**Step 3 — Sum the last column.**
$$\chi^2 = 1 + 1 + 0 + 1 + 1 + 0 = 4$$ …

Cell	$O$	$E$	$O-E$	$(O-E)^2$	$(O-E)^2/E$
Urban, X	20	25	−5	25	$25/25 = 1$
Urban, Y	30	25	5	25	$25/25 = 1$
Urban, Z	50	50	0	0	$0/50 = 0$
Rural, X	30	25	5	25	$25/25 = 1$
Rural, Y	20	25	−5	25	$25/25 = 1$
Rural, Z	50	50	0	0	$0/50 = 0$
Total					4

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