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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Count the observed result plus anything more extreme, compare with the level, and write the verdict in context.
The rewritten readings were written for this dataset.
More to read
Writing the hypotheses first
Begin by specifying H0, the null hypothesis, and H1, the alternative. For a coin, H0 might state that it is fair. The test asks how unusual the data would be under H0. Calculate its p-value using that assumption. Defining both hypotheses before examining the numbers keeps the test tied to a clear question.
Understanding testing errors
Rejecting H0 when it is true is a Type I error. A Type II error occurs when a false H0 is not rejected. The chosen significance level, often 5 percent, bounds the Type I error probability for a valid test. Testing balances false-alarm risk with power to detect effects. Failure to reject H0 does not establish that it is true.
Following a binomial test
Set out H0 and H1, including whether the alternative is one-sided or two-sided. Under H0, calculate the probability of a count at least as extreme as the observed count in the specified direction or directions. This is the p-value. Compare it with the significance level, then explain the decision using the original question. Follow this order so the test's direction is settled before calculation.
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2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.