= Solution
These are row percentages, so the comparisons estimate the conditional <probability> of recorded <vaccination> within each NHS trust category. The percentages increase across the ordered trust categories as
$$
41.94\%,\quad55.88\%,\quad62.32\%,\quad88.44\%,\quad91.25\%.
$$
The extreme-category difference is approximately \b[49.31 <percentage points>], rather than a 49.31% relative increase. The <ordinal categorical variable> records an order; the categories do not establish equal numerical trust increments.
Under the <null hypothesis> of <independence>, the expected count in a cell is its row total times its column total divided by 748. The <Pearson chi-squared statistic> compares observed and expected counts through
$$
X^2=\sum_{i,j}\frac{(O_{ij}-E_{ij})^2}{E_{ij}}=100.84896.
$$
There are $(5-1)(2-1)=4$ <statistical degrees of freedom>, giving $p\approx6.49\times10^{-21}$. The minimum expected count is approximately 5.015, so the usual expected-count concern is much less acute here than in the income strata. \b[The table gives strong evidence of a statistical association.] The <likelihood-ratio test of independence in a contingency table> agrees: its statistic is approximately 81.361 on four <statistical degrees of freedom>, with a very small <p-value>. The displayed `0.000` is rounding, not a zero probability.
For magnitude, <Cramér's V> is
$$
V=\sqrt{100.84896/748}\approx0.3672.
$$
It describes the size of the departure from <independence> without using the category order or giving a direction. Calling it “moderate” is only a rough contextual description; the actual percentage differences are easier to interpret.
The ordered coefficients describe a positive direction. <Goodman and Kruskal's gamma> uses only <concordant pairs> and <discordant pairs>:
$$
\gamma=\frac{44257-12957}{44257+12957}\approx0.5471.
$$
It excludes <tied pairs>, which are frequent when the response has only two categories. It is therefore not a claim that 54.7% of respondents were vaccinated because of their trust. The directional <Somers' D> for vaccination as response includes pairs tied on that response in its denominator:
$$
D_{Y\mid X}=\frac{44257-12957}{44257+12957+124681}\approx0.1721.
$$
The smaller value reflects the retained ties; it does not contradict the positive <statistical association>.
The small column-dependent <Goodman and Kruskal's lambda> has a useful prediction interpretation. Without trust information, always predicting the more frequent vaccinated category gives 121 mistakes. Predicting the majority category separately in each trust row gives $13+15+26+34+28=116$ mistakes. Hence
$$
\lambda_{Y\mid X}=\frac{121-116}{121}=\frac5{121}\approx0.0413.
$$
Trust reduces this particular modal-prediction error by only about 4.13%, despite the strong evidence against <independence>. Most rows retain the same majority category. \b[Evidence of association, its ordinal direction and improvement in a chosen prediction rule are different questions.] None establishes that changing trust would change an individual's <vaccination status>.
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