Distinguish an income association from a causal claim

Plot aggregate horizontally and monthly income vertically, preserving each classmate’s pair. The independently reconstructed scatter rises almost linearly. From $\bar x=56.375$, $\bar y=3593.75$, $S_{xx}=741.875$, $S_{yy}=2047187.5$ and $S_{xy}=38868.75$, the coefficient is $0.9973702172\ldots$, agreeing with the printed $0.997$. The blue line is the fit calculated for part (iii), not additional source data.

The association among these eight former classmates does not isolate a causal effect of examination performance. Other factors such as later education, experience or occupation may affect income. These are possible explanations, not measured variables in the supplied table. The source asks for a comment but prints no explanation; this response is authored.

The least-squares gradient and intercept are

$b=\frac{38868.75}{741.875}=52.3925863521\ldots,$

$\begin{aligned}&a\\ & =3593.75-b(56.375)\\ & =640.117944398\ldots.\end{aligned}$

Thus $y=52.3925863521x+640.117944398$, reported as the printed $y=52.4x+640$. Use aggregate as explanatory and income as response for this requested direction. The intercept is part of the fitted model; it is not evidence about someone outside the observed aggregate range 42–72.