Maths › Statistics › Hypothesis testing with the binomial
Hypothesis testing with the binomial
A claim about a proportion meets some data, and the question is whether the data embarrasses the claim. Hypothesis testing is that confrontation made procedural: assume the claim, compute how surprising the data would be, and let a pre-agreed threshold decide.
Builds on The binomial distribution.
IN THIS TOPIC
- Set up H₀ and H₁ for a claim about a binomial probability p.
- Carry out one- and two-tailed tests by comparing a tail probability with the significance level, concluding in context.
- Find a critical region and state the actual significance level of a test.
WHAT YOU PROBABLY THINK
Rejecting the null hypothesis proves the alternative is true.
The logic of a test
The null hypothesis H₀ is the boring claim, a specific value of p, and it stays innocent until the data makes it look silly. The alternative H₁ says which direction of silliness you are looking for: p > 0.5 or p < 0.5 for a one-tailed test, p ≠ 0.5 for two. The significance level, usually 5%, is agreed before the data arrives: it is how surprising the data must be before H₀ falls.
Everything is computed assuming H₀ is true. That is the whole trick: under H₀ the count is a known binomial, so the probability of a result at least as extreme as the one observed, the p-value, is a number a calculator can produce.
Running a test
WORKED EXAMPLE
Is the coin biased towards heads?
A coin is flipped 20 times and lands heads 15 times. Test at the 5% level whether the coin is biased towards heads.
H₀: p = 0.5, 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: there is significant evidence at the 5% level that the coin is biased towards heads.
The conclusion needs all three parts: the comparison, the decision about H₀, and the meaning in context.
A two-tailed test splits the significance level across both tails, 2.5% each at the 5% level, and the observed tail probability is compared against the half.
Critical regions
A critical region answers the question before the data: which counts would reject H₀? For X ~ B(20, 0.5) against p > 0.5 at 5%, P(X ≥ 14) = 0.0577 is too big and P(X ≥ 15) = 0.0207 fits, so the critical region is X ≥ 15.
The actual significance level is the probability of the critical region under H₀, here 2.07%: binomials are lumpy, so the real rejection probability rarely equals the advertised 5%. Quoting it, and saying it is the probability of wrongly rejecting a true H₀, is a standard mark.
THE EXAM BIT
- Hypotheses are about p, written with p: “H₀: p = 0.3” earns; “the coin is fair” alone does not.
- Compare a tail probability with the level, or the observed value with a critical region; either route works if stated cleanly.
- Conclusions come in the exam's dialect: “reject H₀; significant evidence that…” or “insufficient evidence that…”, always in context.
- Actual significance level = probability of the critical region under H₀; expect it to undershoot the nominal level.
CHECK YOURSELF
X ~ B(30, 0.2) under H₀, and the critical region for a one-tailed test is X ≥ 11 with P(X ≥ 11) = 0.0256. State the actual significance level and what it measures.
Show a hint
The region's probability, assuming the null.
Show the answer
The actual significance level is 2.56%.
It is the probability of rejecting H₀ even though H₀ is true, the test's built-in false alarm rate.
Assume H₀, measure how extreme the data is, compare with the agreed level.
The critical region is decided before the data; its probability under H₀ is the actual significance level.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Set up H₀ and H₁ for a claim about a binomial probability p.
- Carry out one- and two-tailed tests by comparing a tail probability with the significance level, concluding in context.
- Find a critical region and state the actual significance level of a test.
No animated video for this topic yet; these notes stand alone.