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

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.

Set $s=S_D/\sqrt{17}$. Symmetry of $t_{16}$ gives $P(T>-1.746)=0.95$, soThe requested upper confidence bound is thereforeThere is a direction inconsistency in connecting this particular interval to the stated treatment alternative. Since $D$ is before minus after, a blood-pressure reduction means $\mu >0$. An upper interval $(-\infty ,b)$ excludes the null value zero only when $b<0$, which instead supports $\mu <0$.

For the requested reduction alternative, use the lower 95% intervalIt excludes zero in the appropriate direction exactly whenThis is the correct level-5% paired one-sided $t$ test of no mean change against a reduction. The requested upper bound is valid as a confidence statement, but its ordinary exclusion-of-zero test has the opposite direction.

The sign follows the duality between confidence intervals and hypothesis tests for the paired t-test.