--- title: "Testing a Binomial Proportion" description: "Test a claim about a proportion by finding the probability of a result at least as extreme under the null model, comparing it with the significance level, and writing the conclusion in the words of th" canonical: https://lightmysky.com/learn/mathematics/testing-a-binomial-proportion-mt_ZJ8MTLYYKE source: https://lightmysky.com/learn/mathematics/testing-a-binomial-proportion-mt_ZJ8MTLYYKE.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Testing a Binomial Proportion Test a claim about a proportion by finding the probability of a result at least as extreme under the null model, comparing it with the significance level, and writing the conclusion in the words of the situation. Subject: Mathematics · Area: Data & Statistics · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/testing-a-binomial-proportion-mt_ZJ8MTLYYKE ## Ready when they can - Calculate the probability of the observed count or anything more extreme - Compare that probability with a 5 percent level and state the decision - Find the critical region for a stated test and sample size ## Lesson: How surprising is this many heads Sam wonders if a coin favours heads after 4 heads in 4 tosses. The p-value is the probability, under H0, of that count or anything even less compatible with H0. For a fair coin only HHHH of 16 strings qualifies, so p = 1/16 = 0.0625. Jo is wrong to count the exact result alone: the tail belongs too. **Example.** Reject H0 only when the p-value sits at or below the level. With p = 0.0625 against 5%, 0.0625 is above 0.05, so do not reject: the coin looks fair enough. With p = 0.03 the result is significant and H0 goes. Never say H0 is proven; the evidence just fell short. The critical region holds the counts that would reject H0, fixed before seeing data. For n = 5 one-sided at 5%, P(X = 5) = 1/32 = 0.03125 fits under, but P(X >= 4) = 6/32 = 0.1875 does not, so only {5} rejects. Five heads in five tosses has chance 0.5^5 = 0.03125. **Example.** After 9 heads in 12 tosses, test at 5% whether the coin is biased and write the verdict in context. P(X >= 9) is about 0.073, above 0.05, so do not reject: the tosses give no convincing evidence of bias. Context turns numbers into an answer. **Recap.** Count the observed result plus anything more extreme, compare with the level, and write the verdict in context. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Setting Up a Hypothesis Test](https://lightmysky.com/learn/mathematics/setting-up-a-hypothesis-test-mt_JmpLNHK0Nf) - [The Binomial Distribution](https://lightmysky.com/learn/mathematics/the-binomial-distribution-mt_X2kVGK67Md) ## Opens up - [Correlation and the Product-Moment Coefficient](https://lightmysky.com/learn/mathematics/correlation-and-the-product-moment-coefficient-mt_Rs_Z0P7ELJ)