Past exam of the mathematics course of the University of Cambridge / 2008 / ib / Paper 4 / 19H / ii / Solution

Set $x_i=\sin (2{\theta }_i)$. The maximum likelihood estimation formulas for simple linear regression giveUsing the supplied rounded summaries gives $\hat{\beta }\approx 21082.7$ metres and $\hat{\alpha }\approx 312.0$ metres. Recomputing directly from the printed table gives approximately $21082.4$ and $312.0$ metres, respectively, so the small summary rounding differences do not affect the conclusion.

I would not accept the model merely because the Student t-test fails to reject a zero regression intercept. That hypothesis test tests one parameter restriction within the assumed model, not the adequacy of its error assumptions or its functional form. In particular, complementary elevations $\theta$ and $90^{\circ }-\theta$ have exactly the same $x_i$ and hence the same fitted mean, but their observed low-angle minus high-angle ranges arefor the four corresponding pairs. All four are positive, suggesting an elevation-dependent effect omitted by the mean model, such as air resistance, rather than random scatter determined only by $\sin (2\theta )$. This is a regression diagnostics concern, not a claim that these finite data logically rule out the normal-error model. The correct conclusion is nonrejection of the intercept restriction does not validate the regression model; further checks of the residual pattern and error assumptions are needed.