= Solution
One presentation is the <cost-effectiveness plane>, with $\Delta E$ horizontally and $\Delta C$ vertically. Plot the point estimate $(45,1800)$ with a joint <confidence region> or paired <bootstrap> cloud. Its position shows the tradeoff quadrant, while the cloud shows uncertainty, dependence and possible dominance. A willingness-to-pay threshold $\lambda$ is the line $\Delta C=\lambda\Delta E$ through the origin. Points below that line have positive <incremental net monetary benefit>, including points in quadrants where a ratio rule would fail. The point-estimated switch occurs at $\lambda=40$ pounds per day.
A second presentation is a <cost-effectiveness acceptability curve>, giving an uncertainty-based measure of cost-effectiveness for a range of thresholds. For paired <bootstrap> increments $(\Delta E^{*(b)},\Delta C^{*(b)})$, plot
$$
\widehat p(\lambda)=\frac1B\sum_{b=1}^B
\mathbf1\{\lambda\Delta E^{*(b)}-\Delta C^{*(b)}>0\}.
$$
This counts positive net benefits, not merely ratios below the threshold, so it handles all quadrants. Using <Bayesian inference> the curve is the <posterior probability> that <incremental net monetary benefit> is positive; a frequentist bootstrap curve is a resampling summary, not literally a probability that a fixed parameter is positive.
Here the estimated <incremental net monetary benefit> and its <standard error> are
$$
\widehat{\operatorname{INMB}}(\lambda)=45\lambda-1800,\qquad
s_{\mathrm{INMB}}(\lambda)=\sqrt{225\lambda^2-4500\lambda+90000}.
$$
Thus a <normal approximation> uncertainty curve is
$$
\widehat p(\lambda)\approx
\Phi\!\left(\frac{45\lambda-1800}{\sqrt{225\lambda^2-4500\lambda+90000}}\right).
$$
It passes through one half at £40 per day. Alternatively display the <incremental net monetary benefit> curve itself with approximate pointwise $95\%$ bands $45\lambda-1800\pm1.96s_{\mathrm{INMB}}(\lambda)$. \b[The plane preserves the joint uncertainty; the acceptability curve shows how the threshold changes the cost-effectiveness assessment.] Neither presentation chooses society's willingness-to-pay threshold.
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