Solutions

MATH-C10-C-001

The total is 200. Train uses 140 and validation uses 30:

So 30 examples remain for the test split.

MATH-C10-C-002

The mean is:

MATH-C10-C-003

The width is:

MATH-C10-C-004

In 4-fold cross-validation, each fold is held out once.

So there are 4 validation runs.

MATH-C10-C-005

Multiply prior weight by likelihood weight:

This is unnormalized because it may still need to be divided by the total weight across all possibilities.

MATH-C10-C-006

No. The test split is meant to estimate final performance after model choices have already been made.

If we repeatedly inspect the test score while choosing the model, the test split starts influencing the model-selection process. It becomes another validation set, and the final reported score can become too optimistic.

MATH-C10-C-007

The better first diagnosis is bias.

High training error and high validation error suggest the model is too limited to fit the pattern well. A high-variance model usually fits training data well but fails to transfer cleanly to validation data.

MATH-C10-C-008

No. A higher likelihood says the observed data is more plausible under that choice than under another choice. It does not prove that the model is the true explanation of the world.

MATH-C10-C-009

Yes. Bootstrap resampling draws with replacement, so the same example can appear more than once in one resampled dataset.

MATH-C10-C-010

No. A hypothesis test can provide evidence that a difference is unlikely under a baseline assumption. It does not decide whether the effect is large, useful, worth the cost, or safe enough for deployment.

MATH-C10-C-011

Enter 1.

A large dataset can still be narrow. If it misses the population or use case we care about, size alone does not make it strong evidence for that target.

MATH-C10-C-012

Enter 1.

An estimator is a rule. A reasonable rule can behave well across many samples and still produce a misleading estimate on one unlucky or unrepresentative sample.

MATH-C10-C-013

Enter 1.

Maximum likelihood chooses the candidate that makes the observed data most plausible among the candidates being compared. It does not prove that the model family contains the true explanation.

MATH-C10-C-014

Enter 1.

A confidence interval reports sampling uncertainty in an estimate. It is not a promise about what will happen on one particular future example.

MATH-C10-C-015

Enter 1.

Cross-validation can guide model selection. Once it influences choices, it is development evidence. A separate untouched test set may still be needed for a final performance report.