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MathidiaMathematics Academy
Statistical foundations
Lesson 44 of 75
Challenge

Hypothesis tests

Objective. Use null models, p-values, power and error rates.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

A hypothesis test is a formal procedure for deciding between a null hypothesis H₀, representing no effect or a default assumption, and an alternative hypothesis H₁, using sample evidence. The p-value is the probability of observing data as extreme as, or more extreme than, what was found, assuming H₀ is true; a small p-value (typically below a significance level α = 0.05) leads to rejecting H₀. A Type I error occurs when a true H₀ is wrongly rejected, with probability α, while a Type II error occurs when a false H₀ is wrongly retained, with probability β; the power of a test is 1 − β, the probability of correctly rejecting a false H₀. Larger sample sizes generally increase power by reducing the standard error of the test statistic.

Worked example

A hypothesis test gives a p-value of 0.02 using significance level α = 0.05. State the conclusion of this hypothesis test.

0.02 … 0.05
Illustrated card showing 0.02 … 0.05
  1. 1Compare the p-value to the significance level: 0.02 versus 0.05.
  2. 2Note that the p-value is smaller than the significance level.
  3. 3Recall the decision rule: reject H₀ when the p-value is less than α.
  4. 4State the conclusion based on this comparison.

Answer: Reject H₀, since the p-value (0.02) is less than α (0.05)

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Common mistake. Students interpret a p-value of 0.03 as "a 3% chance the null hypothesis is true"; the p-value instead measures how extreme the observed data is assuming H₀ is true, not the probability of H₀ itself.

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