Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 5J (Statistical Modelling)

The individual t-tests only test each regression coefficient after adjusting for the other covariate. Here neither covariate is individually significant at conventional levels, but from the abbreviated coefficient table alone one cannot conclude that neither explains the response: strong multicollinearity can make both standard errors large even when the covariates are jointly useful.

The full F-test has $p<2.2\times 10^{-16}$ and tests the null hypothesis that both slope coefficients vanish. It decisively rejects that hypothesis, while $R^2=0.8126$ says the fitted model explains about $81\%$ of the sample variation. Together with the insignificant individual t-tests, this is characteristic of multicollinearity: the two covariates explain the response jointly, but the output does not identify either one as having a separately estimable effect.

Here the full F-test has $p=0.9834$ and $R^2\approx 0$. There is no evidence against the joint null hypothesis that both slopes vanish, and hence no evidence in this model that either covariate explains the response.

The intercept-only model has $49$ residual degrees of freedom, so the sample size is $50$. The one-covariate model and full model therefore have respectively $48$ and $47$ residual degrees of freedom. The completed sequential analysis of variance table hasThus the missing sums of squares are $133.617$ and $0.014$, and the corresponding F-statistics are approximately $203.8$ and $0.0214$. The first p-value is below $2\times 10^{-16}$; the second is approximately $0.8845$. The tiny second increment is consistent with the reported t-test because, for one added coefficient, the partial F-statistic equals the squared t-statistic up to displayed rounding.