Past exam of the physics A course of the University of Cambridge / 2018 / Paper 1 / Section D / D12 / d

For a candidate attenuation coefficient, the one-second expected count is ${\mu }_n=2018.5e^{-bn}$. The model predictions areUse first the conservative signal-dominated range $n=1,\dots ,4$, where the count approximation is reliable and background is a small correction. Using the estimated errors from part (c), the weighted residual statistic isIt gives $\boxed{Q_{\mathrm{obs}}(1)=4.45,Q_{\mathrm{obs}}(1.1)=12.74}$. Since both $b$ candidates are fixed and $N_0$ was calibrated independently, neither was fitted to these four counts: the approximate reference has $\nu =4$, not three. The upper-tail probabilities are approximately $0.349$ and $0.0126$, respectively. Thus $b=1$ is compatible with the scatter, whereas $b=1.1$ is disfavored at the usual $5\%$ level in this approximate test.

A Pearson chi-squared statistic, using model variances instead of estimated observed variances, gives the same preference:Including the fifth point also leaves the decision unchanged: the estimated-error statistics are $8.37$ and $12.74$ on five degrees of freedom, with approximate upper-tail probabilities $0.137$ and $0.0259$; the Pearson statistics are $6.34$ and $15.63$. HenceThe small uncertainty in $N_0$ contributes correlated prediction uncertainty, as in chi-squared comparison of a calibrated attenuation model, but is negligible compared with the one-second counting errors here. Choosing the test range after inspecting the observations limits a strict interpretation of its nominal significance; the numerical comparison and its stated range are the relevant evidence.

If all eight observations are inserted into the estimated-error formula, the totals are $14.99$ and $22.48$. Their nominal eight-degree-of-freedom calibration is unreliable because the final expected counts are much smaller than one for some predictions. In fact the full-range Pearson statistics are $43.34$ and $118.25$, exposing the late excess much more strongly. The full Poisson deviance values are $24.09$ and $47.03$, again favoring $b=1$ but signaling that a background-free exponential is inadequate over the complete range. These sparse-count comparisons should not be used to claim that either unmodified model explains all eight measurements.