What is Hypothesis Testing?
Hypothesis testing is a statistical method for deciding whether evidence from a sample supports or contradicts a claim about a population. It compares a null hypothesis (no effect) against an alternative hypothesis using a test statistic and p-value.
Hypothesis testing evaluates whether sample data provides enough evidence to reject a null hypothesis (H₀) in favor of an alternative hypothesis (H₁), based on a chosen significance level (commonly α = 0.05).
- 1↓State H₀ and H₁Define the null (no effect) and alternative hypotheses.
- 2↓Choose significance level αCommonly 0.05 — the risk of a false positive you'll accept.
- 3↓Collect data & compute test statistice.g., a z or t value from the sample.
- 4↓Find the p-valueProbability of observing data this extreme if H₀ is true.
- 5Decide: reject or fail to reject H₀Reject H₀ if the p-value is less than α.
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Step-by-step worked examples
A company claims its light bulbs last μ₀ = 1000 hours. A sample of n=36 bulbs has mean x̄=980 hours, σ=60 hours. Test at α=0.05 (two-tailed).
z = (x̄ − μ₀)/(σ/√n) = (980 − 1000)/(60/√36) = −20/10 = −2.0 Critical z at α=0.05 (two-tailed) = ±1.96 |−2.0| > 1.96 → reject H₀: bulb lifespan is significantly different from 1000 hours
A cereal box is labeled 500 g. A sample of n=25 boxes has mean x̄=495 g, σ=15 g. Test H₀: μ=500 at α=0.05.
z = (495 − 500)/(15/√25) = −5/3 ≈ −1.67 Critical z (two-tailed, α=0.05) = ±1.96 |−1.67| < 1.96 → fail to reject H₀: not enough evidence boxes are underfilled
A new teaching method is tested: a sample of n=49 students scores x̄=78, historical μ₀=75, σ=14. Test at α=0.05 one-tailed (H₁: μ>75).
z = (78 − 75)/(14/√49) = 3/2 = 1.5 Critical z (one-tailed, α=0.05) = 1.645 1.5 < 1.645 → fail to reject H₀: not enough evidence the method improves scores
Flashcards
Quick quiz
Q1.The null hypothesis (H₀) typically represents…
Q2.If the p-value < α, you should…
Q3.Rejecting a true H₀ is called a…
Q4.A common significance level (α) used in research is…
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Common mistakes
Believing a p-value tells you the probability H₀ is true. — Correct: A p-value is the probability of the data given H₀ is true, not the reverse.
Concluding 'fail to reject H₀' means H₀ is proven true. — Correct: It only means there isn't enough evidence to reject it — not proof of truth.
Using α=0.05 blindly for every context. — Correct: The significance level should reflect the cost of errors in that specific context.
Confusing statistical significance with practical importance. — Correct: A result can be statistically significant yet have a tiny, unimportant effect size.
FAQ
What is hypothesis testing?
A statistical method to decide, using sample data, whether to reject a null hypothesis in favor of an alternative.
What is the formula used in hypothesis testing?
For known population variance: z = (x̄ − μ₀)/(σ/√n); other tests use t, chi-square, or F statistics.
What are examples of hypothesis testing?
Testing whether a new drug lowers blood pressure, whether a product's average weight matches its label, or whether a teaching method improves scores.
How do you calculate a hypothesis test step by step?
State H₀/H₁, choose α, compute the test statistic, find the p-value, then compare it to α to decide.