= Solution
Interpret the stipulated model as a <freely jointed chain> with $N$ independent, isotropically oriented segments of length $b$. With end-to-end vector $\mathbf R=\sum_{j=1}^N\mathbf b_j$, cross terms vanish on averaging:
$$
\boxed{\langle\mathbf R\rangle=0,\qquad
\langle R^2\rangle=Nb^2,\qquad R_{\rm rms}=b\sqrt N.}
$$
Using only the model's supplied segment size and count gives
$$
\boxed{R_{\rm rms}=0.32\sqrt{1000}\,\mathrm{nm}\simeq10.1\,\mathrm{nm}.}
$$
The mean vector is zero, not the mean magnitude. In the three-dimensional large-$N$ <Gaussian chain> limit,
$$
\langle R\rangle=b\sqrt{\frac{8N}{3\pi}}\simeq9.32\,\mathrm{nm}.
$$
Thus the common root-mean-square interpretation of an average chain size is $10.1\,\mathrm{nm}$, while the mean scalar distance is about $9.32\,\mathrm{nm}$. The contour length $Nb=320\,\mathrm{nm}$ is a different quantity. A count of $999$ links between $1000$ positions changes these estimates by less than $0.1\%$.
These are results of the explicitly idealized statistical chain, not a prediction about the conformation or <persistence length> of an actual biological molecule. TODO (user-deferred biology): material-specific biological interpretation of that model.
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