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

For a sample of size $n\ge 2$, defineFor the zero-mean Gaussian sample, the standardized vector $(X_1/\sigma ,\dots ,X_n/\sigma )$ has independent standard normal coordinates. Apply an orthogonal transformation whose first row is $(1,\dots ,1)/\sqrt{n}$. The resulting coordinates $Z_1,\dots ,Z_n$ remain independent standard normals: their joint density depends only on squared Euclidean length and the transformation has unit absolute Jacobian. Its first coordinate is $Z_1=\sqrt{n}\overline{X}/\sigma$, while orthogonal decomposition givesThusThe ratio of the first independent standard normal and the square root of the normalized chi-square variable has a Student probability distribution, giving

For the paired height observations, form one cross-minus-self difference per pot. The sampling units for this test are the fifteen differences, not thirty separate plant heights. Under the usual independent normal model for differences, a null mean of zero gives a $t_{14}$ statistic. The observed statistic isFor the two-sided alternative, reject at significance level $\delta$ if $|t_{\mathrm{obs}}|>t_{14,1-\delta /2}$. Its two-sided $p$-value is $2[1-F_{t_{14}}(|t_{\mathrm{obs}}|)]$. This uses the recorded values without performing the numerical calculation forbidden by the question. The normal difference model is what makes the small-sample Student calibration exact.

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.