Question

Give a two-way table and specify a particular cell for that table. In each case find the expected count for that cell and the contribution to the chi- square statistic for that cell. \((\) Group \(2,\) No \()\) $$ \begin{array}{l|rr} \hline & \text { Yes } & \text { No } \\ \hline \text { Group 1 } & 720 & 280 \\ \text { Group 2 } & 1180 & 320 \\ \hline \end{array} $$

Short answer

The expected count for the 'No' cell in 'Group 2' is 360 and its contribution to the Chi-square statistic is approximately 4.44.

Step-by-step solution

Figure out Row, Column and Total Totals

Start by calculating the totals for each row, column and the overall total. That is, the total for 'Group 1', 'Group 2', 'Yes', 'No', and the grand total: \\- For Group 1, add up the numbers of 'Yes' and 'No': 720 + 280 = 1000. \\- For Group 2 do the same: 1180 + 320 = 1500. \\- For 'Yes', add up the numbers from Group 1 and Group 2: 720 + 1180 = 1900. \\- Similarly for 'No': 280 + 320 = 600. \\- The grand total is the sum of all four cells: 1000 (Group 1) + 1500 (Group 2) = 2500.

Calculate the Expected Count

The expected count for a given cell in a two-way table is calculated by multiplying the row total for that cell by the column total, and then dividing by the grand total. \\- For the cell under 'No' in 'Group 2' (our point of interest), this would be: (1500 * 600) / 2500 = 360.

Determine the Chi-Square Statistic

The chi-square statistic for a specific cell is calculated as: \\- \( (observed-expected)^2 / expected \) \\The observed count is the actual count given in the cell, while the expected count is what we calculated in the previous step. So for 'Group 2’ and 'No': \\- \( (320-360)^2 / 360 \approx 4.44 \)