When you want to know if a predictor truly helps explain the outcome beyond what the other predictors already explain, you compare the full model (including that predictor) to a reduced model (excluding it) using a partial F-test. The F-statistic used for this purpose is the ratio of the mean square attributed to that predictor to the mean square error from the full model. The numerator, the mean square due to the predictor, reflects the incremental increase in explained variation when you add that predictor to the model, adjusted for the other predictors. Since a single predictor contributes one degree of freedom, this MSR is SSR for that predictor divided by 1. The denominator is the residual mean square from the full model (the MSE), which estimates the typical squared size of the errors after fitting all the predictors. If the predictor has no real effect, the extra explained variance is small and the F statistic is near 1. If the predictor does have a real effect, the predictor’s MSR is large relative to the MSE, and the F statistic is large, signaling significance. So the statement is true: the F-statistic for testing a predictor in regression is the ratio of its mean square (the incremental, adjusted contribution) to the mean square error from the full model.

In a smoothing spline, lambda controls how strongly we favor smoothness versus fidelity to the data. A larger lambda puts more weight on the roughness penalty, so the fitted curve becomes smoother and less flexible. That reduced flexibility means the model may not capture all the true features of the underlying function, which increases the systematic error—or bias—of the fit. At the same time, variance tends to decrease because the fit is less sensitive to random noise. So as lambda increases, bias increases. In the extreme, a very large lambda forces a very smooth (often nearly straight) curve, which typically has higher bias if the true function is more complex. Therefore the statement is true.

When coding a categorical predictor with w levels, you use w-1 binary indicators to keep the model estimable. If you included all w dummies plus an intercept, the columns would be perfectly linearly related: for every observation the dummies sum to one, so one column would be a linear combination of the others and the intercept. This creates perfect multicollinearity, making the coefficients non-identifiable. By leaving out one category (the reference), the intercept plus the w-1 dummies provide a full-rank design matrix. The intercept captures the baseline category, and each included dummy measures the difference from that baseline. This is why the w-1 coding is used. The other ideas—increasing complexity, improving sparsity, or changing the response scale—don’t address the identifiability issue.

The mean squared error is driven by the residuals between observed and predicted values. If the model equation is misspecified, predictions don’t capture the true relationship, leaving large and systematic residuals that inflate MSE. When errors are heteroscedastic, their variance changes with the level of the data, so the usual assumption of constant variance is violated and the standard interpretation of MSE as a uniform measure of predictive error becomes unreliable. Outliers push squared residuals up dramatically, disproportionately distorting the MSE and giving a misleading view of typical predictive performance. Multicollinearity, while it makes coefficient estimates unstable and inflates their standard errors, does not directly distort the overall mean squared error of the model’s predictions. Thus, multicollinearity is not a direct cause of unreliable MSE estimates.

In lasso regression, the penalty term uses the L1 norm, which is the sum of the absolute values of the coefficients. This choice is key because the L1 penalty tends to produce sparse solutions, pushing some coefficients to exactly zero, effectively selecting a subset of features. The geometry of the L1 ball has corners aligned with the axes, making it more likely for the optimization to hit a point where some coefficients are zero when combined with the least-squares objective. By contrast, an L2 penalty (ridge) shrinks coefficients but rarely makes them exactly zero, since it uses the sum of squares. L0 focuses on the count of nonzero coefficients and is non-convex and hard to optimize, while L∞ would constrain the largest coefficient in absolute value, not create sparsity in the same way. Hence, the penalty term in lasso is the L1 norm.