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Hypothesis testing with the binomial

A hypothesis test assesses a claim about a population proportion by calculating how unlikely the observed data would be if the claim were true. State the hypotheses, carry out one- and two-tailed binomial tests at a stated significance level, and find critical regions and actual significance levels.

Year 12-13EDEXCEL 9MA0 P3 5.1, 5.2

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.

COMMON MISCONCEPTION

Rejecting the null hypothesis proves the alternative is true.

A test rejects H₀ or does not reject it, and proves neither hypothesis. Everything is computed assuming H₀ is true, so rejection says only that data at least as extreme as those observed are less likely than the significance level under that assumption. Rejecting H₀ when H₀ is true has a probability of its own, the actual significance level of the critical region.

The logic of a test

The null hypothesis H₀ is the specified population claim being tested, always a particular value of the population proportion p. The alternative H₁ states the departure under investigation: p > 0.5 or p < 0.5 for a one-tailed test, p ≠ 0.5 for a two-tailed test. The significance level, usually 5%, is fixed before the data are collected and sets how unlikely the data must be for H₀ to be rejected. A test either rejects H₀ or does not reject it; it does not prove either hypothesis true.

Twenty-one bars of a fair-coin binomial, the bars from fifteen upwards coloured coral: together they carry about 0.021 of probability, small enough that observing a count out there discredits the null hypothesis at the five percent level. — FIG. 1Assume the coin is fair and the count of heads has a known distribution; if the observed count lands in a tail this small, the fairness hypothesis is rejected.

Everything is computed assuming H₀ is true: under H₀ the count follows a known binomial, so the probability of a result at least as extreme as the one observed, the p-value, can be found on a calculator.

A p-value is not the probability that H₀ is true, and it is not the probability that the result occurred by chance. It is the probability of data at least as extreme as those observed given that H₀ holds.

A one-tailed 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 at the 5% significance level reject H₀ and conclude that there is sufficient evidence that the coin is biased towards heads.

A complete conclusion has three parts: the comparison, the decision about H₀, and the interpretation in context.

The tail probability is P(X ≥ 15), not P(X = 15). The test uses the probability of a result at least as extreme as the one observed, and using the probability of the exact observed value can reverse the decision.

Two tails

The fair-coin binomial with both extremes coloured coral, counts of five or fewer and fifteen or more, each tail as unlikely as the other: a two-tailed test spends its significance level on both ends. — FIG. 2“Biased either way” divides the significance level between both tails.

A two-tailed test splits the significance level between the two tails, 2.5% each at the 5% level, and the observed tail probability is compared with that half. The wording determines which test applies: “has the proportion changed” is two-tailed, “has it increased” is one-tailed.

WORKED EXAMPLE

A spinner that lands red too rarely

A spinner is claimed to land on red 30% of the time. In 25 spins it lands red 3 times. Test at the 5% level whether the proportion differs from 0.3.

H₀: p = 0.3, H₁: p ≠ 0.3. Under H₀, X ~ B(25, 0.3).

3 is below the expected 7.5, so this is the lower tail: P(X ≤ 3) = 0.0332.

Compare with 0.025, not with 0.05. Since 0.0332 > 0.025, at the 5% significance level do not reject H₀; there is insufficient evidence that the proportion of reds differs from 0.3.

Comparing 0.0332 with 0.05 instead would reverse the decision. In a two-tailed test the comparison is always against half the significance level.

Critical regions

Building the critical region for B(20, 0.5) at the 5 percent level: the tail from 15 carries 0.0207 and fits, the tail from 14 carries 0.0577 and does not — FIG. 3P(X ≥ 14) = 0.0577 will not fit inside the five percent level and P(X ≥ 15) = 0.0207 will, so the critical region starts at fifteen.

A critical region is the set of values of the test statistic that lead to rejection of H₀, determined before the data are collected. For X ~ B(20, 0.5) against p > 0.5 at 5%, P(X ≥ 14) = 0.0577 exceeds 0.05 and P(X ≥ 15) = 0.0207 does not, so the critical region is X ≥ 15.

The actual significance level is the probability of the critical region under H₀, here 2.07%. Because the binomial takes only whole-number values, this probability is usually less than the nominal 5%. State its value and its meaning: it is the probability of rejecting H₀ when H₀ is true.

A two-tailed critical region is built one tail at a time, each against 2.5%. With X ~ B(25, 0.3), the region is X ≤ 2 or X ≥ 13, and its actual significance level is 0.00896 + 0.01747 = 0.02643, or 2.64%.

ASSESSMENT FOCUS

State the hypotheses in terms of p: write “H₀: p = 0.3” rather than “the coin is fair”.
Either compare a tail probability with the significance level or compare the observed value with a critical region, and state which method is being used.
Write the conclusion in one of two forms: “At the stated significance level, reject H₀ and conclude that there is sufficient evidence of …”, or “At the stated significance level, do not reject H₀. There is insufficient evidence of …”, naming the context in each case.
For a two-tailed test, halve the significance level before comparing.
The actual significance level is the probability of the critical region under H₀, and is usually smaller than the nominal level.
Do not use P(X = observed). The test uses the probability of a result at least as extreme, which is a tail probability.

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 when H₀ is true.

Assume H₀, calculate how extreme the data are under it, and compare with the stated significance level; then reject H₀ or do not reject it.

The critical region is decided before the data, and 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.

Or read them with their worked answers on the hypothesis testing with the binomial questions page.

USED LATER IN

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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.

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