= Solution
Assume independent approximately Gaussian log-rate errors with known common <standard deviation> $s=0.15$. The <weighted least squares> objective is
$$
\chi^2(m,c)=\frac1{s^2}\sum_{i=1}^{6}(y_i-mx_i-c)^2.
$$
Since all weights agree, the <normal equations for linear least squares> reduce to
$$
\widehat m=\frac{S_{xy}}{S_{xx}},\qquad
\widehat c=\bar y-\widehat m\bar x,
\quad S_{xx}=\sum_i(x_i-\bar x)^2,\quad
S_{xy}=\sum_i(x_i-\bar x)(y_i-\bar y).
$$
For these data, $\bar x=1.5\,\mathrm{ps}$, $\bar y=4.1596533$, $S_{xx}=4.375\,\mathrm{ps}^2$ and $S_{xy}=-10.3782409\,\mathrm{ps}$. Hence $\widehat m=-2.3721693\,\mathrm{ps}^{-1}$ and $\widehat c=7.7179073$.
The <covariance matrix> of the fitted slope and intercept follows from the inverse weighted <design matrix> product, or directly by writing both estimates as linear combinations of the independent $y_i$:
$$
\operatorname{Var}(\widehat m)=\frac{s^2}{S_{xx}},\qquad
\operatorname{Var}(\widehat c)=s^2\left(\frac16+\frac{\bar x^2}{S_{xx}}\right),\qquad
\operatorname{Cov}(\widehat m,\widehat c)=-\frac{s^2\bar x}{S_{xx}}.
$$
This gives
$$
\boxed{\widehat m=(-2.372\pm0.072)\,\mathrm{ps}^{-1},\qquad
\widehat c=7.718\pm0.124.}
$$
The parameter conversion and first-order <propagation of uncertainty> are
$$
\widehat\tau=-\frac1{\widehat m},\quad
\sigma_\tau=\frac{\sigma_m}{\widehat m^2},\qquad
\widehat R_0=R_*e^{\widehat c},\quad
\sigma_{R_0}=\widehat R_0\sigma_c.
$$
Consequently,
$$
\boxed{\widehat\tau=(0.4216\pm0.0127)\,\mathrm{ps},\qquad
\widehat R_0=(2.25\pm0.28)\times10^3\,\mathrm{counts\,day^{-1}}.}
$$
These are one-<standard deviation> errors. The estimates are correlated, since $\operatorname{Cov}(\widehat m,\widehat c)=-0.0077143\,\mathrm{ps}^{-1}$. The minimum is $\chi^2_{\min}=4.6537$ for four residual degrees of freedom. Because $s$ is supplied as a known measurement uncertainty, it should not be rescaled by the observed reduced $\chi^2$. This is a <log-linear exponential-decay fit>.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2021/paper-1-lifetime-fit.png]
{title=Log-rate fit and the effect of imposing the alternative lifetime}
{description=Original calculation from the exam measurements. The solid line fits both lifetime and normalization; the dashed line fixes $\tau=0.478\,\mathrm{ps}$ and refits its normalization. The lower panel shows standardized log-rate residuals for the two fits. No error bar is assigned to proper time.}
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