= Solution
The command fits the <simple linear regression>
$$
Y_i=\beta_0+\beta_1t_i+\varepsilon_i,\qquad \varepsilon_i\ \hbox{independent }N(0,\sigma^2),
$$
using <ordinary least squares>. The fitted conditional mean is a straight line in time. The <regression residuals> are mainly positive at the beginning and end and negative in the middle, so the mean function needs curvature.
A direct improvement is <polynomial regression> of degree two:
``
fit2 <- lm(Counts ~ Time + I(Time^2), data = geiger)
anova(fit1, fit2)
``
The nested-model <F-test> tests $H_0:\beta_2=0$ against a nonzero quadratic coefficient. If $N$ measurements were used, its statistic is
$$
\boxed{F=\frac{\operatorname{RSS}_1-\operatorname{RSS}_2}{\operatorname{RSS}_2/(N-3)}\sim F_{1,N-3}\quad\hbox{under }H_0.}
$$
A small reported $p$-value supports the quadratic term. The new <regression residuals> should then be checked for remaining systematic structure, changing variance and temporal dependence, because the <F-test> assumes independent errors of common variance. Radioactive decay also motivates an exponential mean, but the displayed curvature already justifies this simple nested comparison.
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