= Solution
The additive model gives useful evidence for both predictors. To test for no adjusted sex effect, use $H_0:\beta_M=0$ against $H_1:\beta_M\ne0$. The statistic $T=\widehat\beta_M/\operatorname{se}(\widehat\beta_M)$ has a <Student t-distribution> with 97 <statistical degrees of freedom> under the <normal linear model> and the <null hypothesis>. Here $T=3.016$ and $p=0.00327$. Similarly, $H_0:\beta_h=0$ against a nonzero common slope gives $T=4.397\sim t_{97}$ under the <null hypothesis>, with $p=2.81\times10^{-5}$. Thus, at a fixed height males have a higher fitted mean weight, and within sex greater height is associated with greater weight.
The interaction model adds $\delta M(x-165)$. Its female height slope is $0.6696$ and its male height slope is $0.6696-0.0977=0.5719$. Test $H_0:\delta=0$ against $H_1:\delta\ne0$ using $T=-0.344\sim t_{96}$ under the <null hypothesis>, or equivalently $T^2\sim F_{1,96}$. The $p$-value is 0.73139. \b[The additive model is a reasonable parsimonious choice; the data give no evidence for different height slopes between sexes.] This does not establish exact equality of population slopes. The additive model also has the higher adjusted <coefficient of determination>, 0.405 versus 0.3996, and a slightly smaller <residual standard error>.
Ignoring sex gives a height slope of 0.8769, appreciably above the adjusted slope 0.6211. This is <omitted-variable bias> in the descriptive regression: a taller group with a larger intercept makes the pooled height association steeper. Algebraically, with $z=x-165$,
$$
\widehat b_{\rm pooled}=\widehat\beta_h+\widehat\beta_M\frac{\sum(z-\bar z)(M-\bar M)}{\sum(z-\bar z)^2}.
$$
The printed estimates imply a positive height-sex <covariance>. Pooled and adjusted slopes therefore describe different comparisons; the difference is not evidence that adding sex changed a physical effect of height. For predictions, retain both sex and height, check <regression diagnostics>, and assess performance on new students. The <residual standard error> near 5.7 remains substantial and the <coefficient of determination> near 0.42 does not promise precise individual prediction.
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