 Solution

Read the body percentages within the populism rows. There are 50 Yes responses out of 447 Not-populist respondents and 218 out of 1182 Populist respondents. The corresponding No percentages are 76.73% and 75.38%, while Not know falls from 12.08% to 6.18%. The overall distribution therefore changes in more than just the Yes category. The row-total percentages, 27.44% and 72.56%, describe group composition in the complete-case table, not the conditional blocking rates. Their slight difference from Question 2's 28.25% and 71.75% is consistent with different missing-value patterns.

The Pearson chi-squared test of independence compares the two full response distributions. Under the null hypothesis of independence, each cell has expected frequency $E_{ij}=n_{i+}{n}_{+j}/N$, andThe minimum expected count is about 34.849, comfortably above the usual sparse-cell concern. With independent observations and a suitable sampling model, the tiny p-value rejects that omnibus null at conventional significance levels. The likelihood-ratio test of independence in a contingency table gives $G^2\approx 24.6044$, also on 2 statistical degrees of freedom, with calculated $p\approx 4.54\times 10^{-6}$. Its printed 0.000 is a rounded value, not zero.^[ref]

The generalized phi coefficient and Cramér's V both equal $\sqrt{X^2/N}\approx 0.124$ because the smaller nontrivial table dimension is one. The contingency coefficient is $\sqrt{X^2/(X^2+N)}\approx 0.123$. These are unsigned magnitude measures, not further significance tests or percentages of explained variation. A modest magnitude can be highly significant with a large sample size.

The omnibus test does not, by itself, identify which response contrast differs. To target reported Yes specifically, an explicitly derived binary table combines No and Not know as non-Yes reports, producing rows $(50,397)$ and $(218,964)$. An uncorrected two-by-two Pearson test gives $X^2\approx 12.4292$ on one statistical degree of freedom, $p\approx 0.000423$. A large-sample confidence interval for the difference of reported-Yes proportions is approximately $[0.0359,0.1092]$, or 3.59 to 10.92 percentage points. These are additional count-derived calculations under independent binomial sampling, not values printed in the exam. They address the selected Yes contrast and do not pretend that Not know and No are substantively identical in every respect. As aggregation can change a contingency-table independence test explains, the binary and three-category tests have different null hypotheses.

The printed ordinal coefficients, Goodman and Kruskal's gamma 0.295, Somers' D 0.116, Kendall's tau-b 0.117 and Stuart's tau-c 0.093, are arithmetically consistent with the displayed ordering. For example, 134660 concordant and 73239 discordant pairs give $\gamma =(134660-73239)/(134660+73239)\approx 0.295$. The full tie counts also partition all $\left(\genfrac{}{}{0ex}{}{1629}{2}\right)=1326006$ unordered pairs correctly. However, the three response labels do not establish an ordinal scale. Their arithmetic sign therefore cannot replace the direct Yes comparison or be called the proportion of variation explained. Changing the arbitrary order of Not know and No could change the ordinal summary without changing the nominal association.^[ref]

Both directional values of Goodman and Kruskal's lambda are zero. No is the modal blocking answer in both populism groups, and Populist is the modal classification within each blocking column. Knowing the other variable therefore changes no modal prediction enough to reduce the number of classification errors. This does not contradict the unequal conditional distributions or the chi-squared result. Before generalizing, missing outcomes, missing classifications, the sampling design and possible confounding must be considered; a p-value does not supply those assumptions.