Hypothesis testing

The Nautilus.Testing module provides building blocks for classical hypothesis tests. All functions are pure f32 computations that compose normal_cdf, student_t_cdf, and chi_squared_cdf from the Distributions module.

Z-tests

Function	Signature	Notes
z_statistic	(sample_mean, pop_mean, pop_std, sample_n: f32) -> f32	(x_bar - mu) / (sigma / sqrt(n))
z_p_value_two_sided	(z: f32) -> f32	2 * (1 - Phi(|z|))
z_p_value_upper	(z: f32) -> f32	1 - Phi(z)
z_p_value_lower	(z: f32) -> f32	Phi(z)
normal_ci_half_width	(confidence, pop_std, sample_n: f32) -> f32	z_crit * sigma / sqrt(n)

T-tests

Function	Signature	Notes
t_statistic_one_sample	(sample_mean, sample_std, sample_n, pop_mean: f32) -> f32	One-sample t
t_statistic_two_sample_pooled	(mean1, std1, n1, mean2, std2, n2: f32) -> f32	Equal-variance pooled t
welch_t_statistic	(mean1, std1, n1, mean2, std2, n2: f32) -> f32	Unequal-variance Welch's t
welch_t_df	(std1, n1, std2, n2: f32) -> f32	Welch-Satterthwaite degrees of freedom

P-values

Function	Signature	Notes
t_p_value_two_sided	(t, df: f32) -> f32	Via student_t_cdf
t_p_value_upper	(t, df: f32) -> f32	Upper-tail
t_p_value_lower	(t, df: f32) -> f32	Lower-tail
chi_squared_p_value	(statistic, df: f32) -> f32	1 - chi_squared_cdf(statistic, df)

Example: two-sided z-test

import Nautilus.Testing (z_statistic, z_p_value_two_sided)

def z_test_demo() -> f32 = {
  z = z_statistic(cast(5.2, f32), cast(5.0, f32),
                  cast(1.5, f32), cast(36.0, f32))
  p = z_p_value_two_sided(z)
  p
}

This computes z = (5.2 - 5.0) / (1.5 / sqrt(36)) = 0.8 and the corresponding two-sided p-value of approximately 0.424.

Example: Welch's t-test

import Nautilus.Testing (welch_t_statistic, welch_t_df, t_p_value_two_sided)

def welch_demo() -> f32 = {
  t = welch_t_statistic(cast(12.0, f32), cast(2.0, f32), cast(30.0, f32),
                        cast(10.0, f32), cast(3.0, f32), cast(25.0, f32))
  df = welch_t_df(cast(2.0, f32), cast(30.0, f32),
                  cast(3.0, f32), cast(25.0, f32))
  t_p_value_two_sided(t, df)
}

Notes

• All parameters are f32, including sample sizes. Pass cast(n, f32).
• The p-value functions use the standard-normal or Student-t CDF internally. Accuracy depends on the underlying CDF approximation (erf-based for normal, betai-based for Student-t).
• chi_squared_p_value returns the upper-tail probability (the conventional test p-value for goodness-of-fit tests).