= Solution
Assume the two trial arms are independent samples, their cost and survival means have a valid joint <normal approximation>, and the supplied <standard errors> and <correlation coefficient> estimate the joint uncertainty in their mean differences. This could follow from a <central limit theorem> for patient-level cost/survival pairs, with finite second moments and suitably handled survival follow-up. The supplied <covariance matrix> for $(\widehat{\Delta C},\widehat{\Delta E})$ is
$$
\widehat\Sigma=\begin{pmatrix}300^2&0.5(300)(15)\\0.5(300)(15)&15^2\end{pmatrix}
=\begin{pmatrix}90000&2250\\2250&225\end{pmatrix}.
$$
The off-diagonal entry has units pounds times days, and must not be discarded. Independence of trial arms does not mean cost and survival increments are independent: patient-level cost and survival are paired within each arm.
For a proposed ratio $r$, the null assertion is $\Delta C-r\Delta E=0$. Its estimated <variance> is $90000-4500r+225r^2$. <Fieller's theorem> inverts the associated two-sided normal tests. With $q=\Phi^{-1}(0.975)\approx1.96$, retain $r$ when
$$
(1800-45r)^2\leq q^2(90000-4500r+225r^2).
$$
Expanding gives
$$
(2025-225q^2)r^2+(-162000+4500q^2)r+(3240000-90000q^2)\leq0.
$$
The leading coefficient is positive, since $45>q(15)$; the effect increment is separated from zero at this level. With $q^2=3.841459$, the polynomial is approximately
$$
1160.672r^2-144713.435r+2894268.706.
$$
It is nonpositive between its two real roots. The resulting approximate $95\%$ <Fieller confidence set> is therefore a bounded <confidence interval>:
$$
\boxed{\operatorname{ICER}\in[25.02,99.66]\text{ pounds per additional survival day}.}
$$
On the $365$-day life-year scale the endpoints are approximately £9,133 and £36,376. Exact coverage would need an exact pivot with its corresponding critical value; using estimated covariance and a <central limit theorem> makes this an asymptotic interval. In other data, a poorly determined denominator can give unbounded or disconnected <Fieller confidence sets>, which should not be replaced by an artificially bounded interval.
An alternative is a paired <bootstrap>: independently resample patients within each randomized arm, keeping each patient's cost and survival together; recompute both mean differences and their ratio in each replicate. The $2.5\%$ and $97.5\%$ empirical ratio quantiles give a <percentile bootstrap confidence interval>. The raw paired data are needed to calculate it, so the numerical endpoints cannot be obtained from this summary table alone. Near-zero bootstrap effect increments or sign changes warn that ordinary ratio-percentile intervals may be unreliable; one should retain the joint cost/effect uncertainty or invert a suitable bootstrap test instead.
A simpler first-order alternative is the <delta method>. For $r=\Delta C/\Delta E$, the gradient is $(1/\Delta E,-\Delta C/\Delta E^2)$, giving
$$
\widehat{\operatorname{Var}}(\widehat r)=\frac{90000+40^2(225)-2(40)(2250)}{45^2}=133.333.
$$
The resulting approximate <Wald confidence interval> is $40\pm1.96\sqrt{133.333}=(17.37,62.63)$ pounds per day. Its symmetric first-order form misses the denominator's nonlinear effect, explaining its marked difference from the <Fieller confidence set>; it is not the preferred ratio interval here.
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