 Solution

The null model's 99 residual degrees of freedom imply $n=100$. Each predictor has a one-degree-of-freedom linear contribution in the parametric table, plus the reported nonlinear contribution. Consequently the termwise effective degrees of freedom areInclude the intercept once, giving about $12.2$ total effective degrees of freedom. The more precise residual degrees of freedom in the output giveThere is no conflict: $8.8$ and $0.4$ are rounded, so they cannot recover the precise total. Individual unrounded termwise values are not provided.

The package's residual mean-square estimate of error variance ismatching the rounded $10328$ in the table. The nonlinear experience contribution is highly significant, while the smaller nonlinear education contribution has approximate $p=0.03923$.

This calculation uses the effective residual degrees of freedom reported by this fit. For a general non-idempotent linear smoother $S$, the noise contribution to expected RSS is ${\sigma }^2\mathrm{tr}[(I-S{)}^T(I-S)]$, and the expectation can additionally contain squared smoothing bias. Thus dividing by $n-\mathrm{tr}(S)$ is not a universal exact-unbiasedness theorem. The output's separation of linear and nonlinear term degrees of freedom is documented at cran.r-project.org/web/packages/gam/gam.pdf.