Past exam of the options II course of the University of Oxford / 2005 / O / 5 / ii

For the positive-direction alternative, use the same paired statistic $t_{\mathrm{obs}}=2.607/(4.713/\sqrt{15})$, but reject at level $\delta$ only when $t_{\mathrm{obs}}>t_{14,1-\delta }$. Its one-sided $p$-value isBecause the observed mean difference is positive and the null Student probability distribution is symmetric, the two-sided $p$-value is twice this upper-tail probability, consistent with the supplied $0.05$ and $0.025$.

The two-sided value is the null-model probability of a test statistic at least as far from zero in either direction as the observation. The one-sided value uses only the direction of larger cross-fertilized heights. They are probabilities under the assumed null sampling distribution, not posterior probabilities that the null hypothesis is true.

At the reported precision, the two-sided result is on the conventional 5 percent boundary; rounding can matter for a strict numerical threshold. The one-sided result is significant at5 percent and supports a positive mean difference if that directional alternative was chosen in advance. Choosing the direction after observing the sign would not justify using the smaller one-sided probability as a prespecified test. The paired design controls shared pot conditions, while the inference still relies on the independence and probability distribution assumptions for the differences.