 Solution

The displayed percentages use row denominators. The relevant comparison is therefore the conditional probability of each response within each recorded gender group, rather than the gender composition within a response category.

For the male group, $364/1042\approx 34.93\%$ strongly agree and $286/1042\approx 27.45\%$ agree. For the female group, the corresponding percentages are $225/1093\approx 20.59\%$ and $243/1093\approx 22.23\%$. Combining the two agreement categories givesThe female group has more neutral responses, $24.25\%$ versus $17.47\%$, and a higher combined disagreement percentage, $32.94\%$ versus $20.15\%$. Thus the observed distributions differ in more than a single cell.

The Pearson chi-squared test assesses the null hypothesis of independence in the complete $2\times 5$ contingency table. Under that null, the expected frequency in a cell is its row total times its column total divided by 2,135. Recalculation givesThe reported p-value, approximately $9.97\times 10^{-19}$, gives very strong evidence against independence under the test's sampling assumptions. The minimum expected frequency is about $106.88$, so sparse expected cells are not a problem here. The test concerns the full five-category distribution, not just the combined agreement comparison.

Cramér's V summarizes the magnitude without imposing an ordering on the gender labels:The small p-value does not imply a nearly deterministic relationship. For example, the column-dependent Goodman and Kruskal's lambda is only about $0.026$: choosing the modal response separately within the two groups reduces prediction errors from 1,546 to 1,506, a relative reduction of $40/1546\approx 2.59\%$.

The positive Goodman and Kruskal's gamma must be interpreted using the table's chosen ordering. The agreement categories run toward less willingness, and gender has no intrinsic ordinal direction. A sign attached to an arbitrary group coding cannot replace the substantive comparison of conditional probabilities.

The NaN displayed for the likelihood-ratio calculation is not a valid p-value and is not evidence of no association. The Pearson chi-squared statistic can be independently recomputed from the positive cell counts. None of these calculations identifies a causal effect; the table has not controlled for income or other possible confounders.