 Solution

Each row of the contingency table is a conditional distribution of the safety category among respondents with that experience status. Thus the relevant comparison uses row percentages, rather than interpreting the larger number of respondents with some bad experiences as itself an effect on safety.

The sample proportions reporting feeling very safe are $39.59\%$ without bad experiences and $32.78\%$ with some bad experiences: the latter is lower by about $6.81$ percentage points. Feeling fairly safe rises from $28.49\%$ to $34.63\%$, a difference of about $6.14$ percentage points. Feeling unsafe changes only from $31.92\%$ to $32.59\%$, about $0.67$ percentage points. The main descriptive shift is therefore from very safe toward fairly safe, rather than a large increase in the unsafe category.

Pearson's chi-squared test tests the null hypothesis that experience status and safety category are independent. If that were true, the expected cell count would be its row total times column total divided by $n$. The Pearson chi-squared statistic sums squared observed-minus-expected count differences divided by their expected counts. HereThe expected counts are comfortably large, with minimum approximately $234.7$, so the ordinary large-sample chi-squared distribution approximation is appropriate under the sampling assumptions. There is evidence against independence even at the $1\%$ significance level. The reported likelihood-ratio result gives the same conclusion.

Magnitude is a separate question. Cramér's V giveswhich describes a weak statistical association. It does not mean that $7.7\%$ of safety is explained. Nor does the p-value measure the size of that association.

For ordinal direction, order the experience categories as none, some, and the safety categories from very safe toward unsafe. Goodman and Kruskal's gamma compares concordant pairs with discordant pairs after excluding tied pairs:Its small positive sign then corresponds to a weak tendency toward less safety among respondents with bad experiences. The sign would reverse if exactly one category ordering were reversed. Because this contingency table has many ties, the response-direction Somers' D, which retains response ties among pairs with different predictor values, is smaller:These coefficients describe ordered association; the omnibus Pearson's chi-squared test tests any departure from independence and does not supply a separate p-value for gamma.

The column-dependent Goodman and Kruskal's lambda makes the weak predictive improvement tangible. Predicting the overall most frequent safety category makes $1165$ errors in this table. Using a separate most frequent category within each experience row makes $1145$, giving $\lambda =20/1165\simeq 0.0172$, only about a $1.7\%$ reduction in those classification errors.

The sample supports a statistically detectable but weak association, chiefly a reduced sample proportion feeling very safe. This observational study does not establish that bad experiences cause the change: reverse direction, reporting differences or confounding could also contribute. The numerical p-value also presumes the sampling and complete-case assumptions appropriate to the observations used.