Introduction to Hypothesis Testing: quick revision, 15 bites

Hypothesis Testing: Deciding with Evidence

Data on trial: innocent until proven guilty.

Hypothesis testing is a formal procedure for deciding whether sample data provides **enough evidence** to reject a claim about a population.

Takeaway. Hypothesis testing never proves $H_0$ — it only checks whether the data is unlikely under $H_0$.

Null vs Alternative Hypothesis

Two claims walk into a courtroom.

Takeaway. Write $H_0$ and $H_a$ before touching the data — they must be mutually exclusive and exhaustive.

The Test Statistic: Measuring the Gap

How far is your sample from the claim?

The test statistic converts the gap between your sample and $H_0$ into a **standardized number** — how many standard errors away the sample sits.

Takeaway. A larger $|z|$ means the sample is further from $H_0$ — stronger evidence against it.

Significance Level α: The Risk You Accept

Set your error budget before you test.

Takeaway. $\alpha$ is chosen **before** the test — never after seeing the data.

What Is a p-value?

The probability that kills (or saves) a hypothesis.

Decision rule: if $p \leq \alpha$, **reject $H_0$**; if $p > \alpha$, **fail to reject $H_0$**.

Takeaway. A small p-value means the data is unlikely under $H_0$ — but it is NOT the probability that $H_0$ is true.

Never Say 'Accept H₀'

The most misquoted verdict in statistics.

Takeaway. Absence of evidence is not evidence of absence — failing to reject $H_0$ is not proof that $H_0$ is true.

Type I vs Type II Errors

Two ways to be wrong — pick your poison.

Takeaway. Lowering $\alpha$ reduces Type I errors but increases Type II errors — there is always a trade-off.

The 5-Step Hypothesis Testing Workflow

Five steps from question to verdict.

Takeaway. Always state the conclusion in the language of the original problem — not just 'reject' or 'fail to reject'.

Worked Example: Is the Machine Off-Target?

A factory, a sample, and a verdict.

Takeaway. A $z$-score of 2.0 exceeds the critical value 1.96 at $\alpha = 0.05$ (two-tailed) — evidence is significant.

One-Tailed vs Two-Tailed Tests

Which side of the curve are you watching?

Takeaway. Choose the tail direction from $H_a$ **before** looking at the data — never after.

Where Hypothesis Testing Runs the World

From medicine to your Netflix feed.

Takeaway. Anywhere a claim needs evidence, hypothesis testing is the referee.

Remember the p-value Rule

If p is low, H₀ must go.

**'If the p is low, the null must go. If the p is high, the null will fly.'**

Takeaway. p ≤ α → reject $H_0$; p > α → fail to reject $H_0$.

Exam Checklist for Hypothesis Testing

Score full marks — every time.

Takeaway. Examiners award marks for the **reasoning chain**, not just the final verdict.

Statistical ≠ Practical Significance

Significant doesn't always mean important.

Takeaway. Always ask: is the effect size large enough to matter in the real world?

What You Need Before You Test

Build the base before the test.

Takeaway. Hypothesis testing is built on sampling distributions — master those first.