Implement in Python a function p_value(stat, alternative, dist, df=None) that returns the p-value for one-sided and two-sided tests. Requirements: (1) alternative ∈ {'less','greater','two-sided'}; (2) dist ∈ {'z','t'} where 'z' uses the standard normal and 't' uses Student's t with degrees of freedom df; (3) for 'z', do not use external libraries—implement the CDF via math.erf/erfc to achieve at least 1e-9 relative error for |z| ≤ 8 and use numerically stable tails; (4) for 't', you may use scipy.stats.t.cdf if available, otherwise implement a reasonable approximation (e.g., incomplete beta/continued fraction) and document error bounds; (5) handle edge cases: NaN/inf inputs, df<1, and extreme |stat|; (6) include minimal unit tests: z=0 two-sided→1.0; z=1.96 two-sided≈0.0500; z=5 greater≈2.87e-7; t=2.0 with df=10 two-sided≈0.070; monotonicity checks; (7) explain how you ensure numerical stability for very small p-values.

This question evaluates proficiency in statistical hypothesis testing, numerical computing, and numerical stability when implementing p-value calculations for z and t distributions, and it targets the Statistics & Math domain for a data scientist role.

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Implement robust one/two-sided p-value function

Q: Implement robust one/two-sided p-value function

This question evaluates proficiency in statistical hypothesis testing, numerical computing, and numerical stability when implementing p-value calculations for z and t distributions, and it targets the Statistics & Math domain for a data scientist role.

Q: How do I approach Statistics & Math interview questions?

Statistics & Math questions require understanding of core concepts and practice. PracHub provides solutions with explanations to help you master statistics & math interviews.

Q: What difficulty level is this interview question?

This is a hard difficulty Statistics & Math question, commonly asked during Technical Screen rounds at Roblox.

Q: What role is this question designed for?

This question is commonly asked for Data Scientist candidates at Roblox during technical interviews.

Implement p_value(stat, alternative, dist, df=None)

Context: You're building a small, production-quality helper to compute p-values for common one- and two-sided hypothesis tests. The function must be numerically stable in the tails and handle edge cases cleanly.

Requirements

Signature and options
- Implement in Python: p_value(stat, alternative, dist, df=None) .
- alternative ∈ {'less', 'greater', 'two-sided'}.
- dist ∈ {'z', 't'}.
Distributions
- 'z': Use the standard normal distribution. Do not use external libraries. Implement the CDF via math.erf/erfc with at least 1e-9 relative error for |z| ≤ 8, and use numerically stable tails.
- 't': Use Student's t with df degrees of freedom. You may use scipy.stats.t.cdf (and sf ) if available. Otherwise, implement a reasonable approximation (e.g., via the regularized incomplete beta using a continued fraction), and document error bounds.
Edge cases
- Handle NaN/inf inputs.
- For t-tests, reject invalid df (e.g., df < 1 ).
- Handle extreme |stat| without catastrophic cancellation.
Tests (minimal)
- z=0, two-sided → 1.0
- z=1.96, two-sided ≈ 0.0500
- z=5, greater ≈ 2.87e-7
- t=2.0 with df=10, two-sided ≈ 0.070
- Monotonicity checks: for 'greater' p-value decreases as stat increases; two-sided p-value decreases as |stat| increases.
Explain briefly how your implementation is numerically stable for very small p-values.

Requirements

Signature and options

Implement in Python: p_value(stat, alternative, dist, df=None) .
alternative ∈ {'less', 'greater', 'two-sided'}.
dist ∈ {'z', 't'}.

Distributions

'z': Use the standard normal distribution. Do not use external libraries. Implement the CDF via math.erf/erfc with at least 1e-9 relative error for |z| ≤ 8, and use numerically stable tails.
't': Use Student's t with df degrees of freedom. You may use scipy.stats.t.cdf (and sf ) if available. Otherwise, implement a reasonable approximation (e.g., via the regularized incomplete beta using a continued fraction), and document error bounds.

Edge cases

Handle NaN/inf inputs.
For t-tests, reject invalid df (e.g., df < 1 ).
Handle extreme |stat| without catastrophic cancellation.

Tests (minimal)

z=0, two-sided → 1.0
z=1.96, two-sided ≈ 0.0500
z=5, greater ≈ 2.87e-7
t=2.0 with df=10, two-sided ≈ 0.070
Monotonicity checks: for 'greater' p-value decreases as stat increases; two-sided p-value decreases as |stat| increases.

Explain briefly how your implementation is numerically stable for very small p-values.

Implement robust one/two-sided p-value function

Quick Overview

Implement p_value(stat, alternative, dist, df=None)

Requirements

Solution

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Implement robust one/two-sided p-value function

Quick Overview

Implement p_value(stat, alternative, dist, df=None)

Requirements

Solution

Comments (0)