The Toolkit

A t-interval from 10 observations is promised to cover the true mean 95% of the time. With normal data it does: 95.0% in 20,000 simulations. With skewed (lognormal) data it covers only 83.6%. The nominal level is a promise conditional on assumptions; small, skewed samples break it.

• Bootstrap intervals help with skew but need enough data themselves; BCa intervals correct for bias and skew (Efron 1987).
• An interval for the mean says nothing about where individual values will fall; that is a prediction interval.
• Overlapping intervals are not a significance test for a difference.

from scipy import stats
rng = np.random.default_rng(14)
def coverage(draw, true_mean, n=10, reps=20_000):
    x = draw((reps, n))
    half = stats.t.ppf(0.975, n - 1) * x.std(axis=1, ddof=1) / np.sqrt(n)
    return (np.abs(x.mean(axis=1) - true_mean) <= half).mean()
normal = coverage(lambda s: rng.normal(1, 1, s), 1)
skewed = coverage(lambda s: rng.lognormal(0, 1, s), np.exp(0.5))   # lognormal, mean e^0.5

Assumes import numpy as np and import pandas as pd (and SciPy where imported). Each snippet simulates or makes up its own data, as the comments say.

# Python — confidence intervals
import numpy as np
from scipy.stats import sem, t

# Parametric (t-distribution for small samples)
n = len(data)
ci = t.interval(0.95, df=n-1, loc=np.mean(data), scale=sem(data))

# Bootstrap
boots = [np.mean(np.random.choice(data, size=n, replace=True))
         for _ in range(10000)]
ci_boot = np.percentile(boots, [2.5, 97.5])