Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 206 / 1 / d

In the larger normal linear model, add laboratory intercept offsets ${\gamma }_B{I}_{Bi}+{\gamma }_C{I}_{Ci}$ to the mean in part (a). The nested-model F-test comparesThe smaller and larger models fit four and six mean coefficients respectively. Under $H_0$, independent Gaussian errors with common variance give independent quantities $({\mathrm{RSS}}_1-{\mathrm{RSS}}_2)/{\sigma }^2\sim {\chi }_2^2$ and ${\mathrm{RSS}}_2/{\sigma }^2\sim {\chi }_{24}^2$. Thus the test statistic isThe p-value is the upper-tail probability $\mathrm{Pr}(F_{2,24}\ge 1.9253)=0.1677$. At the conventional 5% significance level, do not reject the common-intercept restriction. Prefer the simpler model tumour1 on this comparison. This does not prove the intercepts coincide, nor address the curvature identified in part (b); the exact F-test calibration assumes the underlying error and mean assumptions. The one-decimal RSS values give $818.7$ when subtracted, whereas the table's difference uses unrounded values.