Hypothesis testing with the binomial distribution
A hypothesis test asks whether a result is surprising enough to doubt what you assumed. You assume the usual value of a probability, see how likely your data would be if that were true, and decide using a fixed cut-off.
Part 1 of 3: Learn it
In short
- H₀ states the parameter's assumed value; H₁ states how you suspect it has changed.
- If the probability of a result at least as extreme as yours is below the significance level, reject H₀.
- In a two-tailed test, split the significance level between the two tails.
Where this is in your specification
Spec points: DfE O1 and O2 (AQA 7357 O1 and O2, Edexcel 9MA0 Statistics topic 5, OCR A H240 2.05)
| Board | Topic: Hypothesis testing |
|---|---|
| DfE content | O1-O4 |
| AQA 7357 | O1-O4 |
| Edexcel 9MA0 | Statistics topic 5 |
| OCR H240 | 2.05 |
The language
| Term | Meaning |
|---|---|
| Null hypothesis H₀ | the assumption being tested, such as p = 0.5 |
| Alternative hypothesis H₁ | p > 0.5, p < 0.5 (one-tailed) or p ≠ 0.5 (two-tailed) |
| Significance level | the cut-off probability, often 5% or 1% |
| Test statistic | the value from the sample, such as the number of heads |
| Critical region | the values of the test statistic that would lead you to reject H₀ |
| Actual significance level | the true probability of landing in the critical region if H₀ is true |
The steps
- Define the parameter and write H₀ and H₁.
- Write the distribution of the test statistic assuming H₀ is true, such as X ~ B(20, 0.5).
- Find the probability of the observed value or anything more extreme in the direction of H₁.
- Compare it with the significance level (halved in each tail for a two-tailed test).
- Conclude in context, without claiming certainty.
Critical regions
Instead of a single probability, you can find every value that would cause H₀ to be rejected. List cumulative probabilities near the tail and choose the boundary where the tail probability first drops below the significance level. Because X is discrete, the actual significance level is usually a bit below the nominal one.
What does a two-tailed test at the 10% level put in each tail?
Show the answer
5% in each tail.
Part 2 of 3: See it worked
Worked examples
Example 1
A coin is suspected of landing heads more often than tails. In 20 flips it shows 15 heads. Test at the 5% level.
- Let p be the probability of heads. Test H₀: p = 0.5 against H₁: p > 0.5
- Under H₀, X ~ B(20, 0.5)
- P(X ≥ 15) = 1 − P(X ≤ 14) = 0.0207
- 0.0207 < 0.05, so reject H₀
Answer: There is sufficient evidence at the 5% level to suggest the coin is biased towards heads.
Example 2
For the same test, find the critical region and the actual significance level.
- P(X ≥ 14) = 0.0577, which is more than 0.05
- P(X ≥ 15) = 0.0207, which is less than 0.05
- So the critical region starts at 15
Answer: X ≥ 15, with an actual significance level of 2.07%.
Common mistakes
- Writing hypotheses about the sample (X) instead of the parameter (p).
- Finding P(X = 15) instead of P(X ≥ 15).
- Forgetting to halve the significance level for a two-tailed test.
- Concluding that H₀ has been proved true or false. A test only gives evidence.
In a test of H₀: p = 0.3 against H₁: p < 0.3, which values of X form the critical region: high or low?
Show the answer
Low values, in the lower tail.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
What does a two-tailed test at the 10% level put in each tail?
5% in each tail.
In a test of H₀: p = 0.3 against H₁: p < 0.3, which values of X form the critical region: high or low?
Low values, in the lower tail.
Why is the actual significance level usually less than the stated one?
X only takes whole numbers, so the tail probabilities jump in steps and rarely hit the stated level exactly.
Jobs that use this
- Data scientist (opens a new tab)
- Data analyst-statistician (opens a new tab)
- Pharmacologist (opens a new tab)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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