Start by clarifying the requirements: one-sided vs two-sided tests, z vs t distributions, and the constraint of no external libraries for the z CDF. Then outline a modular design with separate functions for the z CDF (using math.erf), t CDF (using the regularized incomplete beta function), and p-value calculation, and discuss trade-offs like accuracy vs simplicity.
Pro tip: Mention that for large sample sizes the t-distribution converges to the z-distribution, so you can use the z-approximation as a fallback if implementing the t CDF is too complex, but always prefer the exact t when possible.
Confirm the exact input parameters (test statistic, degrees of freedom, test type) and the constraint of not using external libraries for the z CDF. Discuss whether the t CDF can use libraries or must also be implemented from scratch.
Use the math.erf function to compute the standard normal CDF: Φ(x) = 0.5 * (1 + erf(x / sqrt(2))). Explain that this is accurate and avoids external dependencies.
For the t CDF, use the relationship with the regularized incomplete beta function. Implement it using continued fractions or series expansions, or use a library like scipy if allowed. Discuss numerical stability and edge cases.
For a given test statistic and distribution, compute the p-value: one-sided as the tail probability in the direction of the alternative, and two-sided as twice the smaller tail (or 2 * min(cdf, 1-cdf)). Ensure correct handling of negative statistics.
Validate the implementation against known values (e.g., from scipy or statistical tables) and discuss potential numerical issues. Mention performance considerations for large-scale A/B testing at Roblox.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
They asked this as a follow-up while I was still coding.
Start by acknowledging the importance of robust input validation and graceful error handling in statistical functions. Then, outline a systematic approach: validate inputs, handle edge cases explicitly, and document behavior. Emphasize the trade-offs between returning NaN, raising exceptions, or using approximations, and how you would choose based on context.
Pro tip: Mention that you would write unit tests for these edge cases and consider using established libraries like SciPy that already handle them, but also understand the underlying logic to customize if needed.
Check for NaN, infinite, or invalid inputs (e.g., degrees of freedom < 1) at the beginning of the function and handle them appropriately, such as raising a ValueError or returning a sentinel value.
For degrees of freedom < 1, decide whether to return NaN, raise an error, or use a different distribution; for extreme test statistics, ensure numerical stability by using log-space computations or asymptotic approximations.
Clearly document the function's behavior for edge cases and ensure that errors propagate meaningfully to the caller, possibly with custom exceptions or warnings.
Write comprehensive unit tests covering edge cases, including NaN, inf, df < 1, and extreme values, to verify correct behavior and prevent regressions.
Discuss trade-offs between returning NaN, raising exceptions, or using approximations, and justify your choice based on the application's requirements and user expectations.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by outlining a layered testing strategy: unit tests for core functions with known inputs/outputs, then property-based tests for monotonicity and edge cases. Emphasize how these tests validate both correctness and robustness, especially for algorithms used in A/B testing or data processing at Roblox scale.
Pro tip: Mention that you also test for numerical stability and performance characteristics, as data science code often deals with large datasets and floating-point precision. This shows you think beyond basic correctness.
Break down the implementation into testable units (e.g., functions, classes) and define expected behavior, including invariants like monotonicity or symmetry.
For each component, create test cases with hand-computed expected outputs, covering typical, edge, and error cases.
Use frameworks like Hypothesis to generate random inputs and verify properties such as monotonicity, idempotence, or commutativity.
Ensure components work together correctly, especially when chaining operations or handling real-world data distributions.
Integrate tests into CI/CD, track coverage, and add regression tests for bugs found in production.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.