= Solution
\b[Greater technology distrust is associated with lower efficacy in the overall sample. The omnibus evidence against <independence> is strong, while the unsigned association magnitude is modest: $X^2\approx107.727$ on 6 <statistical degrees of freedom>, $p\approx6.10\times10^{-21}$ and <Cramér's V> about 0.148.] The <observational study> does not establish that changing trust would cause efficacy to change.
Rows identify the technology response and columns identify efficacy. Each body <percentage> is a within-row <conditional probability> estimate, not a column <percentage>. For example, $166/336=49.40\%$ of those who strongly disagree with the distrust statement have High efficacy, whereas $199/718=27.72\%$ of those who agree do. The difference is about 21.69 <percentage points>. High efficacy decreases across all four rows: 49.40%, 42.41%, 31.02%, 27.72%. Low efficacy is 20.54% in the strongly-disagree row, 18.00% in Disagree, 21.43% in Neutral and 34.26% in Agree. Thus the overall pattern is clear without claiming that every category changes monotonically: the low-efficacy proportion initially falls, and the Medium proportion peaks at 47.55% among neutral respondents.
For the <Pearson chi-squared test of independence>, the <null hypothesis> says that the two categorical variables are <independent>, equivalently that the efficacy distribution is the same in every technology-response row. With row totals $n_{i+}$ and column totals $n_{+j}$, the <expected frequency> under that hypothesis is $E_{ij}=n_{i+}n_{+j}/N$, and
$$
X^2=\sum_{i,j}\frac{(O_{ij}-E_{ij})^2}{E_{ij}},\qquad \mathrm{df}=(4-1)(3-1)=6.
$$
All twelve <expected frequencies> exceed 79, with minimum about 79.844, so sparse expected cells are not the concern here. Under <independent> sampling and the usual large-sample calibration, the very small <p-value> rejects <independence> at conventional <significance levels>. This is an omnibus finding: it does not provide a separate adjusted significance test for every pair of row <percentages>.<Pearson's Chi-squared Test for Count Data in the R documentation>[Details: product-marginal null hypothesis and asymptotic chi-squared calibration]{ref}
The <likelihood-ratio test of independence in a contingency table> uses $G^2=2\sum_{O_{ij}>0}O_{ij}\log(O_{ij}/E_{ij})$. The supplied $G^2\approx105.146$ with 6 <statistical degrees of freedom> gives $p\approx2.11\times10^{-20}$. Its displayed 0.000 is rounded, not an exact zero. It corroborates the omnibus finding. The <generalized phi coefficient> is $\sqrt{X^2/N}\approx0.209$, the <contingency coefficient> is about 0.205, and <Cramér's V> is $\sqrt{X^2/[N\min(3,2)]}\approx0.148$. These are unsigned measures of association strength rather than three additional significance tests; they do not describe <causal effects> or <percentages> of <variance> explained. Their magnitude and the small <p-value> answer different questions, especially with a large <sample size>.
The ordinal measures use the displayed order: increasing row index means increasing distrust, while increasing column index means decreasing efficacy. There are 882033 <concordant pairs> and 552543 <discordant pairs>. Hence <Goodman and Kruskal's gamma> is
$$
\gamma=\frac{882033-552543}{882033+552543}\approx0.230.
$$
Its positive sign therefore means that increasing distrust tends to accompany movement toward \i[lower] efficacy. Under an ascending efficacy order its sign would be negative, by <sign reversal under reversal of one ordinal ordering>. <Goodman and Kruskal's gamma> ignores tied pairs; it is not a claim that 23% of efficacy variation is explained. The displayed pair counts partition all $\binom{2466}{2}=3039345$ unordered pairs correctly, including ties. <Somers' D> for efficacy given technology is about 0.151, <Kendall's tau-b> about 0.158, and <Stuart's tau-c> about 0.163. Their different denominators handle ties or table dimensions differently, so their differing magnitudes do not constitute disagreement about the direction.<DescTools documentation for gamma, Somers' D and lambda>[GoodmanKruskalGamma and SomersDelta, Details: tied-pair exclusions and the directional denominator]{ref}<Ordinal association measures in the DescTools documentation>[KendallTauB and StuartTauC: tie and table-size adjustments]{ref}
<Goodman and Kruskal's lambda> concerns the proportional reduction in errors when predicting a most frequent category, not ordinal movement. Without technology information the modal efficacy category is Medium, producing $2466-972=1494$ errors. Choosing a separate modal efficacy category in each technology row yields $2466-(166+391+233+273)=1403$ errors. Thus $\lambda_{\mathrm{efficacy}\mid\mathrm{technology}}=91/1494\approx0.061$. In the reverse prediction direction the reduction is $80/1544\approx0.052$, matching the other printed lambda. The improvement in modal classification is small even though <independence> is decisively rejected. Both the conditional <percentages> and these distinct summaries support a real sample association; ethnicity and other possible <confounders> still require attention before any causal account.
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