Key takeaway: Hypothesis testing is a method for judging a claim about a population from sample data, by checking how surprising the sample would look if that claim were true.
Grade 12 – AP StatisticsAP Statistics Unit 3Statistics and Data
Definition of Hypothesis Testing
Hypothesis testing works like an indirect argument. Start by assuming the claim under test, the null hypothesis, is true. Then ask how likely it would be to see sample results as extreme as the ones observed. If such results would be very unlikely, the assumption looks doubtful.
The procedure never proves a claim true. It either finds convincing evidence against the null hypothesis or fails to find it. A significance level chosen in advance fixes what counts as unlikely enough.
Formula and notation
A standard sequence is State, Plan, Do, Conclude. For a proportion, the test statistic is
\[ SE_0=\sqrt{\frac{p_0(1-p_0)}{n}},\qquad z=\frac{\hat{p}-p_0}{SE_0}, \]
where \(p_0\) is the value claimed by the null hypothesis. The tail area beyond \(z\) on the normal curve gives the p-value.
Key facts
- The null hypothesis is assumed true while the sampling distribution is built.
- A tiny tail probability counts against the null claim; it does not prove the alternative.
- Hypothesis testing can fail to reject a false null, so not rejecting is not confirmation.
- Conditions such as randomness and sufficiently large counts must be checked first.
Hypothesis Testing example
The student council claims that 30% of the school rides the bus. Among 200 randomly chosen students, 74 ride it. Using the 5% level, does this sample show that the true rate is higher than 30%?
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Hypotheses: \(H_0: p=0.30\) versus \(H_a: p\gt 0.30\). Sample proportion: \(\hat{p}=\frac{74}{200}=0.37\).
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Assuming the claim, \(SE_0=\sqrt{\frac{0.30\cdot 0.70}{200}}\approx 0.0324\).
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Distance from the claim: \(z=\frac{0.07}{0.0324}\approx 2.16\). Upper-tail area: about 0.0154.
The tail area 0.0154 is below 0.05, so \(H_0\) is rejected: the sample gives convincing evidence of a rate above 30%.
Common mistake
Watch out
In a different test the p-value is 0.21, and a student concludes, “The null hypothesis is true.”
Why it is wrong
A p-value of 0.21 only says the data are not unusual enough under the null hypothesis. The correct wording is “fail to reject”. Hypothesis testing treats missing evidence as inconclusive, since a small sample may be too small to detect a real effect.
Related terms and lessons
- Null and Alternative Hypotheses
- P-Value
- Significance Level (Alpha)
- Type I and Type II Errors
- Z-Score
- AP Statistics · Inference for ProportionsLessonWorksheetQuiz
- AP Statistics · Inference for MeansLessonWorksheetQuiz
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