 Solution

For a general assumed speed ratio $r=v_A/v_B$,When $r=2$, each station estimates $t_0$ by $t_A-\Delta t$. Averaging givesThe individual estimates have sample standard deviation $2.74\mathrm{s}$ and range $11$ to $19\mathrm{s}$. If station clock errors are independent with zero mean, the standard error of their mean is $2.74/\sqrt{13}=0.76\mathrm{s}$. Thus one may quote $15.0\pm 0.8\mathrm{s}$ as a conditional one-standard-error mean, with about $\pm 3\mathrm{s}$ scatter for an individual station. The problem does not bound a common clock bias: adding the same offset to every clock shifts this inferred origin time by that offset without changing any time difference. Therefore no unconditional absolute-time accuracy follows from the supplied data.

The speed-ratio check is an origin-time and speed-ratio regression of $t_A$ on $\Delta t$. It giveswhere the uncertainty is a residual-based standard error of slope. The factor-two hypothesis predicts slope $1$, about $1.13$ standard errors from the fitted value: the ratio 2 is satisfactory at the accuracy allowed by these clocks, but is not precisely verified. Formally $r=1+1/\text{slope}\simeq 2.37$, with a local propagated standard error about $0.44$. This inference assumes clock errors have no systematic dependence on station position or time difference. With completely unrestricted station offsets, a regression cannot identify $r$ at all.

Let $K=(1/v_B-1/v_A)=\sqrt{M}$. The difference curve determines $K$, $x_0$, and $z_0$, regardless of the particular value assigned to $r$. Changing the ratio to $2.25$ while refitting the individual speeds givesHenceThe conditional standard error of this new mean is $0.72\mathrm{s}$. Its predicted slope $0.8$ is also compatible with the regression. Holding one speed fixed while changing $r$ would alter the depth, but that adds a constraint not present in this question: here both speeds are inferred from the same difference data and chosen ratio.