Hypothesis Testing (t-test)

Hypothesis Testing: t-test

A t-test tests whether an observed difference is statistically significant or could be due to random chance.

One-sample t-statistic tests whether a sample mean differs from a hypothesized value $\mu_0$: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

where $s$ is the sample standard deviation and $n$ is the sample size. Large $|t|$ values indicate the sample mean is unlikely to come from a distribution with mean $\mu_0$.

Two-sample t-statistic (Welch's) tests whether two independent samples have different means: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Welch's version is preferred because it does not assume equal variances.

Your Task

Implement:

• t_statistic(sample, mu0) — one-sample t-statistic
• two_sample_t(xs, ys) — Welch's two-sample t-statistic