Hypothesis testing is a structured way to use sample data to make decisions or draw conclusions about a population.

It answers questions like:

• “Is this new teaching method better?”
• “Is the average weight really 500 g?”
• “Do two groups differ?”
• “Is this effect real or just random noise?”

It’s the backbone of inferential statistics.

🎯 The Core Idea

You start with a claim about a population — the null hypothesis — and then ask:

If this claim were true, how likely is it that we would see data like ours?

If the data look too unlikely under that claim, you reject it.

If the data look compatible with the claim, you keep it.

🧩 The Two Hypotheses

1. Null hypothesis ()

A statement of no effect, no difference, or status quo.
Examples:

• The mean is 50
• The new method has no effect
• Two groups are equal

2. Alternative hypothesis ()

What you want to test for — a difference, an effect, a change.
Examples:

• The mean is not 50
• The new method improves scores
• The groups differ

🔍 How Hypothesis Testing Works (The Steps)

1. State the hypotheses

and

2. Choose a significance level ()

Common choices: 0.05, 0.01, 0.10

3. Collect data and compute a test statistic

Examples: z‑score, t‑score, chi‑square statistic

4. Measure how extreme your result is

Two ways:
p‑value method
critical value method

5. Make a decision

• If evidence is strong → reject
• If evidence is weak → fail to reject

Notice: we never say “accept ” — we just say we don’t have enough evidence against it.

🧠 A Simple Analogy

Imagine the null hypothesis is:

“This coin is fair.”

You flip it 20 times and get 18 heads.

You ask:

“If the coin were fair, how likely is this result?”

If the answer is “very unlikely,” you reject the idea that the coin is fair.

That’s hypothesis testing.

⭐ Fresh, Simple Examples

Example 1: Testing a Mean

Claim: The average battery life is 10 hours.
Your sample mean is 8.9 hours.
The p‑value is 0.01.

If :

• p = 0.01 ≤ 0.05 → reject
  Evidence the true mean is not 10.

Example 2: Testing a Proportion

Claim: 60% of customers prefer the new design.
Your sample gives p‑value = 0.27.

If :

• p = 0.27 > 0.05 → fail to reject
  Not enough evidence to say the true proportion differs from 60%.

Example 3: Comparing Two Groups

Claim: Two fertilizers produce the same plant height.
Your t‑test gives p = 0.003.

If :

• p = 0.003 ≤ 0.01 → reject
  Evidence the fertilizers differ.

🎨 The Big Picture

Hypothesis testing is a decision‑making framework built on probability.
It helps you decide whether your data show a real effect or just random variation.

It’s not about proving things with certainty — it’s about weighing evidence.