Past exam of the options I course of the University of Oxford / 2010 / O / O1 / a

Each before/after pair concerns the same individual, so subject-specific baseline blood pressure can affect both measurements. Analyzing the within-person difference retains this pairing and removes a common baseline contribution. Under the stated independent normal-difference model,The normal-sample variance identity givesMoreover $\overline{D}$ and $S_D^2$ are independent, because the sample-mean projection and centered residual projection of a Gaussian vector are orthogonal and hence independent. Thus the standardized normal variable $N=\sqrt{n}(\overline{D}-\mu )/\sigma$ is independent of $U=(n-1)S_D^2/{\sigma }^2\sim {\chi }_{n-1}^2$. ConsequentlyThis distribution is free of both unknown parameters.