= Solution
{c}
Compute fitted responses $\widehat y_i=\widehat\alpha+\widehat\beta x_i$ and <regression residuals> $e_i=y_i-\widehat y_i$. A conventional <residual plot> shows $e_i$ against $\widehat y_i$ or $x_i$; plotting against observation order additionally looks for runs. One checks for curvature, changing spread and other structure rather than random scatter around zero.
\b[An ordinary <residual plot> need not reveal this failure of <independence>.] The errors have constant <mean> zero, constant <variance> one and zero pairwise <covariances>, so the usual trend and changing-<variance> clues need not appear. A visually patternless <residual plot> cannot validate <independence>; fitted <regression residuals> themselves also share the fitted coefficients even when the true errors are <independent>.
There is detectable dependence in the magnitudes. The <Gaussian fourth moment> is $\mathbb E\eta_i^4=3$, giving for every adjacent pair
$$
\boxed{\operatorname{Cov}(\epsilon_i^2,\epsilon_{i+1}^2)=3-1=2.}
$$
For $i=1$, the product is $\eta_1^4\eta_2^2$; for $i\ge2$, it is $\eta_{i-1}^2\eta_i^4\eta_{i+1}^2$, with the same <expected value> 3. Large magnitudes tend to occur together. A lagged plot of squared <regression residuals> or a test directed at dependence in their squares is more informative. Thus recognition is possible with suitable diagnostics, but is not guaranteed by the standard <residual plot> requested here.
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