= Solution
For each income band, compare the response distribution within the two gender groups. The table below combines agree and strongly agree only to make one comparable contrast visible. The <Pearson chi-squared statistic> and its <p-value> still use all five original response categories, with four <statistical degrees of freedom> in every band. Income labels are thousands of USD per year; finite upper bounds are excluded.
|| Income band
|| $n$
|| Male agreement
|| Female agreement
|| Difference in percentage points
|| $X^2_4$
|| $p$
|| <Cramér's V>
| Under 10k
| 117
| 38.64%
| 34.25%
| 4.39
| 3.433
| 0.4881
| 0.171
| 10k–20k
| 212
| 35.90%
| 35.07%
| 0.82
| 5.457
| 0.2435
| 0.160
| 20k–30k
| 245
| 52.17%
| 39.22%
| 12.96
| 5.893
| 0.2073
| 0.155
| 30k–40k
| 278
| 54.90%
| 37.50%
| 17.40
| 10.117
| 0.03851
| 0.191
| 40k–50k
| 178
| 54.74%
| 38.55%
| 16.18
| 6.687
| 0.1534
| 0.194
| 50k and above
| 1105
| 71.16%
| 50.21%
| 20.95
| 63.765
| $4.68\times10^{-13}$
| 0.240
In the lowest-income band, combined agreement differs by only about $4.39$ <percentage points>. The full distributions still differ descriptively: strong disagreement is $13.64\%$ in the male group and $27.40\%$ in the female group. The omnibus <Pearson chi-squared test> does not reject <independence> at 5%, but that decision does not prove equal distributions.
In the 10k–20k band, combined agreement is almost the same. Nevertheless, strong agreement is higher in the female group, $18.66\%$ versus $11.54\%$, while the less intense agreement category is higher in the male group, $24.36\%$ versus $16.42\%$. Collapsing the categories hides that distinction. The full-distribution <p-value> is approximately $0.2435$.
In the 20k–30k band, combined agreement is higher in the male group by about $12.96$ <percentage points>. Strong agreement alone is $27.17\%$ versus $18.30\%$. The observed difference is therefore larger than in the two lowest bands, despite its full-table <p-value> remaining above 5%.
In the 30k–40k band, combined agreement differs by about $17.40$ <percentage points>, with fewer male neutral responses, $15.69\%$ versus $26.14\%$. The full-table <p-value> of $0.03851$ is nominally significant at 5%. In the 40k–50k band, the combined agreement gap is a similar $16.18$ <percentage points>, but the full-table <p-value> is $0.1534$. These labels are not a direct comparison between the two effects: <significant and nonsignificant results need not differ significantly>.
In the highest-income band, strong agreement is $41.84\%$ in the male group and $22.15\%$ in the female group. Combined agreement differs by about $20.95$ <percentage points>. This band has both the largest <sample size> and the largest <Cramér's V> of the six tables, with a very small <p-value>. Its particularly strong evidence reflects both precision and an observed difference in the distribution.
All six minimum <expected frequencies> exceed five: approximately $7.15$, $9.57$, $12.02$, $15.04$, $7.93$ and $36.03$. Thus sparse expected counts do not provide a reason to reject the usual <chi-squared distribution> approximation here. If the six <hypothesis tests> form one exploratory family, a <Bonferroni correction> would use $0.05/6\approx0.00833$ for each test; only the highest-income result crosses that threshold. This qualifies the unadjusted 30k–40k result without turning its observed contrast into zero.
In the <elaboration model>, the larger agreement gaps in the middle and upper bands provide a descriptive pattern of specification: the conditional contrast varies with income. This is not a <Simpson's paradox> reversal, because the combined agreement difference has the same sign in every band. Nor is the larger high-income <effect size> established simply by comparing its <p-value> with the other <p-values>. A formal claim of <effect modification> requires a stated scale and an appropriate comparison of the effects.
The pooled comparison also mixes different income compositions. The highest-income band contains $631/1042\approx60.56\%$ of the male group but $474/1093\approx43.37\%$ of the female group. Average each group's six agreement proportions over the same pooled income weights using <standardization with common weights>. This gives
$$
\boxed{\widehat p_M^{(w)}\approx60.21\%,\qquad
\widehat p_F^{(w)}\approx43.94\%,\qquad
\widehat p_M^{(w)}-\widehat p_F^{(w)}\approx16.27\text{ percentage points}.}
$$
The crude difference was $19.56$ <percentage points>, so common weighting attenuates it by about $3.30$ <percentage points> but leaves a substantial descriptive contrast. With average-group rather than pooled-person weights, the <symmetric decomposition of a stratified mean difference> instead gives an exact split into approximately $16.29$ <percentage points> from conditional contrasts and $3.27$ <percentage points> from composition. These are different explicitly defined descriptive summaries, not estimates of a causally mediated share.
\Image[/past-exam-of-the-digital-social-research-statistics-core-course-of-the-university-of-oxford/2021-willingness-by-income.png]
{title=<Vaccination intention> by income and a comparison using <standardization with common weights>}
{description=Original calculations from the printed aggregate counts. The outcome combines agree and strongly agree; the two gender labels are those recorded in the paper. Error bars are individual approximate 95% <Wilson intervals>, not simultaneous intervals or a formal test of between-band <effect modification>. The right panel uses the same pooled income frequencies for both groups. No individual observations were reconstructed.}
An antecedent income variable could be considered in a <confounding> explanation, whereas an income variable on an assumed causal pathway would enter an interpretation through <causal mediation>. The conditional tables do not distinguish these causal structures. They also do not identify whether any adjustment introduces <collider bias>. \b[The supported conclusion is income-dependent descriptive association plus a measurable composition difference, with causal interpretation left conditional on additional assumptions.]
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