= Solution
From <Snell's law>, $n=\sin\theta_i/\sin\theta_r=1.5204$. Treating the quoted angle errors as independent one-standard-deviation uncertainties and using radians in <derivatives>,
$$
\sigma_n^2=n^2\left(\cot^2\theta_i\,\sigma_i^2+\cot^2\theta_r\,\sigma_r^2\right),\qquad
\boxed{n=1.52\pm0.14}.
$$
This <refractive-index uncertainty from Snell's law> is first-order <propagation of uncertainty>, with air's index approximated by one. Correlated angular <calibration> errors add $-2n^2\cot\theta_i\cot\theta_r\operatorname{Cov}(\theta_i,\theta_r)$. If the quoted errors are hard bounds instead, their conservative first-order sum gives about $0.19$, rather than the statistical quadrature value.
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