STAT 101
M06 · L02
Module 6 · Lesson 2

Common Statistical Tests

You know the logic. Now meet the toolkit. Seven tests that cover most real-world hypothesis testing scenarios — from comparing a single mean to finding associations in categorical data.

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STAT 101
M06 · L02
Known population SD

Z-Test for Means

Z-Statistic
Z = \dfrac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}

Use when σ is known (rare in practice). Critical values: ±1.96 for α = 0.05 two-tailed.

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STAT 101
M06 · L02
Unknown population SD

One-Sample t-Test

t-Statistic
t = \dfrac{\bar{x} - \mu_0}{s / \sqrt{n}}

s = sample SD, df = n − 1. The t-distribution has heavier tails than Z — extra uncertainty from estimating σ. As n → ∞, t → Z.

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STAT 101
M06 · L02
Two independent groups

Two-Sample t-Test

Pooled variance
s_p^2 = \dfrac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2}

df = n₁ + n₂ − 2. H₀: μ₁ = μ₂. Prefer Welch’s t-test when variances may differ — it’s safer and nearly as powerful when they’re equal.

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STAT 101
M06 · L02
Before/After · Matched samples

Paired t-Test

Paired t-Statistic
t = \dfrac{\bar{d}}{s_d / \sqrt{n}}

d̄ = mean of differences, df = n − 1 pairs. More powerful than two-sample t when pairs are strongly correlated — eliminates between-subject noise.

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STAT 101
M06 · L02
Binary outcomes

One-Proportion Z-Test

Proportion Z-Statistic
Z = \dfrac{\hat{p} - p_0}{\sqrt{p_0(1-p_0)/n}}

p̂ = sample proportion. Valid when np₀ ≥ 10 and n(1−p₀) ≥ 10.

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STAT 101
M06 · L02
Comparing two groups

Two-Proportion Z-Test

Two-Prop Z-Statistic
Z = \dfrac{\hat{p}_1 - \hat{p}_2}{\sqrt{\bar{p}(1-\bar{p})(1/n_1+1/n_2)}}

p̄ = pooled proportion. The backbone of A/B testing in industry — comparing conversion rates, click-through rates, etc.

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STAT 101
M06 · L02
Categorical associations

Chi-Squared Test

Chi-Squared Statistic
\chi^2 = \sum \dfrac{(O - E)^2}{E}

O = observed, E = expected under independence. df = (r−1)(c−1). Requires E ≥ 5 in every cell.

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STAT 101
M06 · L02
How big is the effect?

Effect Size

Cohen’s d
|d| ≈ 0.2 small
|d| ≈ 0.5 medium
|d| ≈ 0.8 large
Odds Ratio
OR = 1 → no effect
OR > 1 → higher odds
OR < 1 → lower odds

A tiny p-value doesn’t mean a big effect. Always report both.

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STAT 101
M06 · L02
Decision guide

Which Test?

  • One mean, σ known → Z-test
  • One mean, σ unknown → one-sample t
  • Two independent means → Welch’s t
  • Matched/before-after → paired t
  • One proportion → proportion Z
  • Two proportions → two-prop Z
  • Categorical association → chi-squared
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STAT 101
Knowledge Check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
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STAT 101
Summary
Recap

What you learned

  • Z-test: use only when σ is known; ±1.96 critical values
  • t-tests: one-sample, two-sample (Welch), paired — match to your data structure
  • Proportion Z-tests: for binary outcomes, one or two groups
  • Chi-squared: independence between two categorical variables
  • Effect size (Cohen’s d, OR) completes the picture p-values can’t alone
Up next
Lesson 3 — ANOVA (comparing 3+ groups) →
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