54737612
A \(2 \times 3\) table has row totals \(40\) and \(60\), column totals \(30\), \(50\), and \(20\), and grand total \(100\). The observed first row is \((18, 14, 8)\).
a) Find all expected cell counts under independence.
b) Compute observed minus expected for every cell.
c) Identify which column shows the largest absolute departure in the first row.
Hints
- Under independence, expected counts are determined entirely by the margins.
- Recover the unlisted observed row from the column totals.
- Residuals compare observed counts with independence expectations cell by cell.
Solution
1. Expected counts are row total times column total divided by the grand total. The first row expectations are \((12, 20, 8)\), and the second row expectations are \((18, 30, 12)\).
2. The observed second row is obtained from column totals: \((12, 36, 12)\).
3. Residuals are first row \((6, -6, 0)\) and second row \((-6, 6, 0)\).
4. The first two columns tie for the largest absolute first-row residual, each with magnitude \(6\).
Answer
a) Expected table: first row \((12, 20, 8)\), second row \((18, 30, 12)\).
b) Residuals: first row \((6, -6, 0)\), second row \((-6, 6, 0)\).
c) Columns 1 and 2 tie, with absolute residual \(6\).
