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 3. Yes) cell $$ \begin{array}{l|rr|r} \hline & \text { Yes } & \text { No } & \text { Total } \\ \hline \text { Group 1 } & 56 & 44 & 100 \\ \text { Group 2 } & 132 & 68 & 200 \\ \text { Group 3 } & 72 & 28 & 100 \\ \hline \text { Total } & 260 & 140 & 400 \\ \hline \end{array} $$

Short answer

The expected count for the cell 'Group 3, Yes' is 65 and the contribution of this cell to the Chi-square statistic is 0.74.

Step-by-step solution

Calculate the Expected Count

The expected count for a cell in a contingency table is calculated by taking the product of the sum of rows and sum of columns for that cell, divided by the total sum. Hence, for Group 3, Yes cell, the Expected count \(E_{ij}\) is calculated as: \[E_{ij} = \left(\frac{{\text{Sum of row 3} * \text{Sum of 'Yes' column}}}{\text{Total}}\right)\] \[E_{ij} = \left(\frac{{100 * 260}}{400}\right) = 65\]

Calculate the Contribution to the Chi-square Statistic

The contribution to the Chi-square statistic for a particular cell is given by: \[X^2_{ij} = \frac{{(O_{ij} - E_{ij})^2}}{E_{ij}}\] where \(O_{ij}\) is the observed count (actual value from the table). Thus, for the Group 3, Yes cell, the Chi-square contribution \(X^2_{ij}\) is: \[X^2_{ij} = \frac{{(72 - 65)^2}}{65} = 0.74\]

Interpret the Results

The expected count for the cell corresponding to Group 3, Yes is 65. This is the count that we would expect if there was no association between the group and the response. The Chi-square contribution from this cell is 0.74. This value contributes to the total Chi-square statistic, which helps evaluate if there is a significant association between the variables in the table. If the Chi-square statistic is large, it means the observed counts deviate significantly from the expected counts, implying a potential association.