= Solution
For a general assumed speed ratio $r=v_A/v_B$,
$$
t_A=t_0+\frac{\Delta t}{r-1}.
$$
When $r=2$, each station estimates $t_0$ by $t_A-\Delta t$. Averaging gives
$$
\boxed{\widehat t_0=15.0\,\mathrm{s}.}
$$
The 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 \b[$15.0\pm0.8\,\mathrm{s}$ as a conditional one-standard-error mean], with about $\pm3\,\mathrm{s}$ 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 gives
$$
t_A\simeq20.00\,\mathrm{s}+(0.732\pm0.237)\Delta t,
$$
where 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: \b[the ratio 2 is satisfactory at the accuracy allowed by these clocks, but is not precisely verified]. Formally $r=1+1/\text{slope}\simeq2.37$, 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 gives
$$
v_A=\frac{r-1}{K}\simeq10.01\,\mathrm{km\,s^{-1}},\qquad
v_B=\frac{r-1}{rK}\simeq4.45\,\mathrm{km\,s^{-1}}.
$$
Hence
$$
\boxed{\Delta x_0=0,\quad\Delta z_0=0,\quad
\widehat t_0(2.25)=\overline{t_A-0.8\Delta t}=18.738\,\mathrm{s},\quad
\Delta\widehat t_0=+3.738\,\mathrm{s}.}
$$
The 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.
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