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

1. Signature and options
   • Implement in Python: p_value(stat, alternative, dist, df=None) .
   • alternative ∈ {'less', 'greater', 'two-sided'}.
   • dist ∈ {'z', 't'}.
2. 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.
3. Edge cases
   • Handle NaN/inf inputs.
   • For t-tests, reject invalid df (e.g., df < 1 ).
   • Handle extreme |stat| without catastrophic cancellation.
4. 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.
5. Explain briefly how your implementation is numerically stable for very small p-values.