- Define and explain Hypothesis testing in your own words
- Use key terms such as Null hypothesis (H₀) accurately
- Apply what you have learned to new examples and questions
- Avoid the common mistakes learners make with this topic
This lesson focuses on Hypothesis testing: null and alternative hypotheses, significance levels, and deciding whether to reject H₀.
Null and alternative hypotheses, significance levels, and deciding whether to reject H₀.
Key ideas
The logic of hypothesis testing
Assume H₀ is true, compute the probability of results at least as extreme as observed (the p-value), and reject H₀ if this is below the significance level. Failing to reject H₀ is not proof it is true — only that the evidence is insufficient.
Binomial probabilities
P(X = r) = C(n, r) pʳ (1 − p)ⁿ⁻ʳ. The conditions matter: fixed n, two outcomes, constant p, independent trials. For n = 5, p = 0.3: P(X = 2) = 10 × 0.09 × 0.343 = 0.3087.
Key term — Null hypothesis (H₀): The default claim in a hypothesis test — typically 'no effect' or 'no change' — which is rejected only if the evidence is strong enough.
A test at the 5% level gives p-value 0.032. What is the conclusion?
0.032 is less than 0.05, so reject H₀; there is sufficient evidence against the null hypothesis.
Answer: 0.032 is less than 0.05, so reject H₀; there is sufficient evidence against the null hypothesis.
- Confusing one-tailed and two-tailed tests A two-tailed alternative ('not equal to') splits the significance level across both tails; using one tail doubles your chance of a false rejection.
- Using the binomial model when trials are not independent Check the conditions: without replacement from a small population, p changes each trial and the binomial does not apply.
Practice
It specifies an increase only, so the critical region lies in a single (upper) tail of the distribution.
P = C(4,0) × 0.5⁰ × 0.5⁴ = 0.0625.
P = 6 × 0.2 × 0.8⁵ = 6 × 0.2 × 0.32768 = 0.393216.
z = (130 − 100)/15 = 2.
Quick check
Which of these best defines "Null hypothesis (H₀)"?
For X ~ N(μ, σ²), what is P(X greater than μ)?
- Hypothesis testing: null and alternative hypotheses, significance levels, and deciding whether to reject H₀.
- The logic of hypothesis testing: Assume H₀ is true, compute the probability of results at least as extreme as observed (the p-value), and reject H₀ if this is below the significance level.
- Significance level: The probability threshold (often 5%) below which observed results are deemed too unlikely under H₀, so H₀ is rejected.
- Watch out for: confusing one-tailed and two-tailed tests