Key Takeaways: Inference for Means

One-Sentence Summary

The one-sample t-test — the workhorse of statistical inference — tests claims about a population mean using the t-distribution (which honestly accounts for our uncertainty about $\sigma$), is robust to non-normality for moderate-to-large samples, and should be your default whenever $\sigma$ is unknown (which is almost always).

Core Concepts at a Glance

The One-Sample t-Test Formula

$$\boxed{t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}}$$

where: - $\bar{x}$ = sample mean - $\mu_0$ = hypothesized population mean (from $H_0$) - $s$ = sample standard deviation - $n$ = sample size - $df = n - 1$ (degrees of freedom)

In plain English: The t-statistic measures how many standard errors the sample mean is from the hypothesized value. Large values (positive or negative) mean the data are far from $H_0$; values near zero mean the data are consistent with $H_0$.

The Five-Step Procedure

Three Conditions for the t-Test

The Normality Condition: Quick Guide

Robustness Summary

z-Test vs. t-Test

Rule of thumb: Default to the t-test. You'll be right 99% of the time.

Confidence Interval for a Mean

$$\boxed{\bar{x} \pm t^*_{n-1} \cdot \frac{s}{\sqrt{n}}}$$

CI-Test Duality:

$$\text{Reject } H_0: \mu = \mu_0 \text{ at } \alpha = 0.05 \iff \mu_0 \text{ is NOT in the 95\% CI}$$

Always report the CI alongside the test — the CI tells you how large the effect might be, not just whether it exists.

P-Value Calculation

Python Quick Reference

import numpy as np
from scipy import stats

# --- One-sample t-test (raw data) ---
data = np.array([...])  # your data
result = stats.ttest_1samp(data, popmean=mu_0)
# result.statistic = t-value
# result.pvalue = two-tailed p-value

# For one-tailed (SciPy ≥ 1.7):
result = stats.ttest_1samp(data, popmean=mu_0, alternative='greater')
result = stats.ttest_1samp(data, popmean=mu_0, alternative='less')

# --- t-test from summary statistics ---
t_stat = (x_bar - mu_0) / (s / np.sqrt(n))
p_two = 2 * stats.t.sf(abs(t_stat), df=n-1)    # two-tailed
p_right = stats.t.sf(t_stat, df=n-1)             # right-tailed
p_left = stats.t.cdf(t_stat, df=n-1)             # left-tailed

# --- Confidence interval ---
t_star = stats.t.ppf(0.975, df=n-1)              # 95% CI
margin = t_star * s / np.sqrt(n)
ci = (x_bar - margin, x_bar + margin)

# --- Normality check ---
stat, p = stats.shapiro(data)                     # Shapiro-Wilk test
stats.probplot(data, dist="norm", plot=plt)       # QQ-plot

Excel Quick Reference

Common Misconceptions

How This Chapter Connects

The Key Theme

The t-distribution embodies a fundamental statistical virtue: honesty about uncertainty. When $\sigma$ is unknown and must be estimated from data, the t-distribution widens its tails to say: "We're less certain than we'd be if we knew $\sigma$." As $n$ grows and our estimate improves, the t-distribution relaxes toward the normal. This isn't a weakness — it's intellectual integrity. The t-distribution doesn't pretend to know more than it does.

The One Thing to Remember

If you forget everything else from this chapter, remember this:

When you want to test whether a population mean equals a specific value, use the one-sample t-test: $t = (\bar{x} - \mu_0) / (s / \sqrt{n})$, with $df = n - 1$. Use it whenever $\sigma$ is unknown, which is almost always. Check three conditions (random, independent, normal enough), and remember that the t-test is robust to non-normality for $n \geq 30$. Always pair the hypothesis test with a confidence interval — the test tells you WHETHER an effect exists; the CI tells you HOW LARGE it might be. The t-distribution's wider tails aren't a limitation — they're the honest price of admitting we don't know $\sigma$.

Key Terms