Hypothesis Testing

Hypothesis testing is a formal procedure for deciding whether sample data provide enough evidence to reject a claim about a population. It is one of the most widely used tools in statistics.

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The Logic of Hypothesis Testing

1. Start with a claim about a population parameter
2. Collect sample data
3. Determine how likely the observed data would be if the claim were true
4. Decide whether to reject or fail to reject the claim

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Step-by-Step Framework

Step 1: State the Hypotheses

Hypothesis	Symbol	Description
Null hypothesis	H₀	The default claim — typically "no effect" or "no difference"
Alternative hypothesis	H₁ (or Hₐ)	The claim you want to test — "there is an effect" or "there is a difference"

Example: A manufacturer claims the mean lifetime of a battery is 500 hours.

H₀: μ = 500
H₁: μ ≠ 500    (two-tailed test)

Step 2: Choose the Significance Level

The significance level (α) is the probability of rejecting H₀ when it is actually true (Type I error). Common choices:

α	Interpretation
0.10	Relaxed — more willing to reject H₀
0.05	Standard in most fields
0.01	Strict — e.g., particle physics uses even smaller values

Step 3: Compute the Test Statistic

The test statistic measures how far the sample result is from the null hypothesis value, in standardised units.

Z-test (σ known or large n):

z = (x̄ − μ₀) / (σ / √n)

t-test (σ unknown):

t = (x̄ − μ₀) / (s / √n)

Step 4: Determine the p-value

The p-value is the probability of observing a test statistic as extreme as (or more extreme than) the one calculated, assuming H₀ is true.

• Small p-value → strong evidence against H₀
• Large p-value → insufficient evidence against H₀

Step 5: Make a Decision

If p-value ≤ α  →  Reject H₀
If p-value > α  →  Fail to reject H₀

Important: "Failing to reject H₀" is not the same as proving H₀ is true. It means the data did not provide sufficient evidence against it.

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Types of Tests