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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 $(\widehat{\Delta C},\widehat{\Delta E})$ isThe 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 $q={\Phi }^{-1}(0.975)\approx 1.96$, retain $r$ whenExpanding givesThe 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 approximatelyIt is nonpositive between its two real roots. The resulting approximate $95\%$ Fieller confidence set is therefore a bounded confidence interval: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 $(1/\Delta E,-\Delta C/\Delta E^2)$, givingThe resulting approximate Wald confidence interval is $40\pm 1.96\sqrt{133.333}=(17.37,62.63)$ 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.