Hypothesis Testing I: The Framework, p-values, z-tests and t-tests

print("critical values: two-tailed z ±", round(stats.norm.ppf(0.975), 3), "| one-tailed z", round(stats.norm.ppf(0.95), 3), "| two-tailed t(df=59) ±", round(stats.t.ppf(0.975, 59), 3))
for z in (1.2, 1.7, 2.06, 3.1): print(f"z = {z}: two-tailed p = {2 * stats.norm.sf(abs(z)):.4f} | one-tailed p = {stats.norm.sf(z):.4f}")

import pandas as pd
d = pd.read_csv("delivery_times.csv")
pune, hyd = d[d.city == "Pune"].delivery_minutes, d[d.city == "Hyderabad"].delivery_minutes
print(f"Pune: n={len(pune)}, mean {pune.mean():.2f}, sd {pune.std():.2f}")
t_two = stats.ttest_1samp(pune, 30)
t_one = stats.ttest_1samp(pune, 30, alternative="greater")
se = pune.std() / np.sqrt(len(pune))
print(f"t = {t_two.statistic:.3f}, df = {len(pune) - 1} | two-tailed p = {t_two.pvalue:.4f} | one-tailed (greater) p = {t_one.pvalue:.4f}")
print(f"by hand: t = ({pune.mean():.2f} − 30) / {se:.3f} = {(pune.mean() - 30) / se:.3f}")
ci = stats.t.interval(0.95, len(pune) - 1, loc=pune.mean(), scale=se)
print(f"95% CI for mean: {ci[0]:.2f} to {ci[1]:.2f} minutes (30 is {'outside' if not ci[0] <= 30 <= ci[1] else 'inside'})")
print(f"effect size (Cohen's d) = {(pune.mean() - 30) / pune.std():.2f}")

print(f"Hyderabad: n={len(hyd)}, mean {hyd.mean():.2f}, sd {hyd.std():.2f}")
lev = stats.levene(pune, hyd); print(f"Levene test for equal variances p = {lev.pvalue:.3f} → {'unequal' if lev.pvalue < 0.05 else 'equal'} variances")
w = stats.ttest_ind(hyd, pune, equal_var=False)             # Welch's t-test: safe default
print(f"Welch t = {w.statistic:.3f}, p = {w.pvalue:.4f}, difference = {hyd.mean() - pune.mean():.2f} min")
pooled = np.sqrt(((len(pune) - 1) * pune.var() + (len(hyd) - 1) * hyd.var()) / (len(pune) + len(hyd) - 2))
print(f"Cohen's d = {(hyd.mean() - pune.mean()) / pooled:.2f}")

rng = np.random.default_rng(5)
before = rng.normal(34, 5, 30).round(1)
after = (before - rng.normal(1.5, 1.5, 30)).round(1)          # simulated route-training effect
pt = stats.ttest_rel(before, after)
it = stats.ttest_ind(before, after, equal_var=False)
print(f"mean improvement {np.mean(before - after):.2f} min | paired p = {pt.pvalue:.4f} | (wrong) independent p = {it.pvalue:.4f}")

from statsmodels.stats.proportion import proportions_ztest
hr = pd.read_csv("hr_employees.csv")
ot = hr[hr.overtime == "Yes"].attrition.eq("Yes"); no = hr[hr.overtime == "No"].attrition.eq("Yes")
stat, p = proportions_ztest([ot.sum(), no.sum()], [len(ot), len(no)])
print(f"attrition: overtime {ot.mean():.1%} (n={len(ot)}) vs no overtime {no.mean():.1%} (n={len(no)}) | z = {stat:.2f}, p = {p:.2e}")
stat1, p1 = proportions_ztest(hr.attrition.eq("Yes").sum(), len(hr), value=0.20, alternative="larger")
print(f"is company attrition above the 20% industry benchmark? z = {stat1:.2f}, p = {p1:.2e}")