Implement Python Function for Statistical Test P-Values

Scenario

You need a utility that calculates p-values for one-sided and two-sided statistical tests.

Question

Write a Python function compute_p_value(stat, dist='z', df=None, alternative='two-sided') that returns the p-value. Your code should support Z-tests and Student-t tests, and handle 'less', 'greater', and 'two-sided' alternatives.

Hints

Use the CDF of the chosen distribution; for two-sided tests return 2*min(CDF, 1-CDF). Libraries like scipy.stats are allowed.

Compute P-Values for Z and t Tests

Implement compute_p_value(stat, dist='z', df=None, alternative='two-sided') that returns the p-value for one-sided and two-sided tests. For dist='z', use the standard normal distribution. For dist='t', use the Student's t distribution with degrees of freedom df (a positive integer). The alternative can be 'less', 'greater', or 'two-sided'. Compute p-values using the CDF of the chosen distribution: for 'less' return CDF(stat); for 'greater' return 1 - CDF(stat); for 'two-sided' return 2 * min(CDF(stat), 1 - CDF(stat)). Do not use external libraries. Return a float in [0, 1].

Constraints

• dist is 'z' or 't'
• alternative is 'less', 'greater', or 'two-sided'
• For dist='t', df is a positive integer (1 <= df <= 10^6)
• stat is a finite float (|stat| <= 1e6)
• Use only the Python standard library
• Return value within absolute error 1e-9 of the true p-value

Examples

Example 1

Input: stat = 1.96, dist = 'z', alternative = 'two-sided'
Output: 0.049996

Explanation: For a Z-test, CDF(1.96) ≈ 0.9750. Two-sided p-value = 2*min(0.9750, 0.0250) ≈ 0.05.

Example 2

Input: stat = 1.0, dist = 't', df = 1, alternative = 'less'
Output: 0.75

Explanation: For t with df=1 (Cauchy), CDF(1) = 0.5 + arctan(1)/pi = 0.75. One-sided 'less' returns 0.75.

Hints

1. For the normal CDF, use erf: Phi(z) = 0.5 * (1 + erf(z / sqrt(2))).
2. For the t-distribution CDF, use the regularized incomplete beta function: if t >= 0, F(t) = 1 - 0.5 * I_{nu/(nu + t^2)}(nu/2, 1/2); else F(t) = 0.5 * I_{nu/(nu + t^2)}(nu/2, 1/2).
3. Compute the regularized incomplete beta via a continued fraction (Lentz's method).
4. For two-sided tests, use 2 * min(CDF, 1 - CDF) and clamp results to [0,1].

Approach

Compute the cumulative distribution function (CDF) of the specified distribution at the test statistic. For the normal distribution, use the error function. For the t-distribution with df degrees of freedom, use the regularized incomplete beta function via a stable continued fraction (Lentz's method). The one-sided p-values are directly CDF or its complement, and the two-sided p-value is 2 * min(CDF, 1 - CDF).

• Time complexity: O(1)
• Space complexity: O(1)