= Solution
The three-component <finite Gaussian mixture with a common variance> has the largest displayed maximized <log-likelihood>, but it also has more fitted parameters. For $k$ components there are $k-1$ free weights, $k$ means and one common variance, making \b[$d=2k$] residual-distribution parameters. Applying the nominal <Akaike information criterion> and <Bayesian information criterion> to the three residual fits gives
$$
\begin{array}{c|r|r|r|r}
k&\ell&d&\mathrm{AIC}=-2\ell+2d&\mathrm{BIC}=-2\ell+d\log500\\\hline
1&-1205.2500&2&2414.500&2422.929\\
2&-1193.8265&4&2395.653&2412.511\\
3&-1192.2715&6&2396.543&2421.831
\end{array}
$$
Both criteria favour \b[the two-component model]. It improves the likelihood substantially over one normal component; the third component gains only $1.5550$ in log-likelihood at a cost of two further parameters. The <Akaike information criterion> difference between two and three components is small, about $0.89$, so that criterion alone gives only a slight preference; the <Bayesian information criterion> preference for two is stronger. Their nearly indistinguishable overlaid curves also support retaining the simpler two-component fit.
These calculations are model-selection aids, not an exact test. Ordinary chi-squared calibration of a <likelihood-ratio test> for the number of mixture components is invalid in general: under a smaller-component null, some weights lie on the boundary and extra-component parameters are unidentified. This is <nonregular mixture model selection>. A parametric bootstrap or predictive <cross-validation> is preferable for a formal comparison, and should account for the mean-adjustment stage. The table uses the provided residual likelihoods and their nominal parameter counts; first-stage regression uncertainty and possible local EM maxima remain qualifications. \b[On the supplied evidence, select two components without claiming that two distinct biological groups have been established.]
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