Introduction to Hypothesis Testing with the Binomial Distribution (Year 12)

In this post, we introduce the basics of hypothesis testing using the binomial distribution — a key topic in A-Level Maths (Year 12 Statistics). You’ll learn how to structure hypotheses, decide which tail to check, and determine whether results are statistically significant.

What is Hypothesis Testing?

Hypothesis testing is a statistical method used to decide whether an observed result provides sufficient evidence to challenge an initial assumption.

For example, suppose you toss a coin 10 times and record the number of heads. Let

p = proportion of heads obtained.

If we assume the coin is fair, our starting point is:

• Null Hypothesis (H₀): p = 0.5

• Alternative Hypothesis (H₁): this depends on the claim we want to test.

One-Tailed and Two-Tailed Tests

Case 1 — More Heads than Expected (Right-Tailed Test)

Suppose the experiment produces 9 heads from the 10 tosses. We say that 9 is the Test Statistic. Could this be evidence that the coin is biased towards heads (p > 0.5),

or is it simply random variation?

For this case our hypotheses are:

• H₀: p = 0.5

• H₁: p > 0.5

Let X = number of heads from the 10 tosses.

Under H₀, X ~ B(10, 0.5)

We calculate the probability of obtaining the observed result or something more extreme:

P(X ≥ 9) = 0.0107

Since 0.0107 < 0.05, the result is unlikely under H₀.

We say that the test is significant → Reject H₀.

There is evidence that the coin is biased towards heads.

Case 2 — Fewer Heads than Expected (Left-Tailed Test)

Now suppose the experiment produces 3 heads from the 10 tosses, so the Test Statistic is 3.

Is this evidence that the coin is biased against heads (p < 0.5)?

Our hypotheses are:

• H₀: p = 0.5

• H₁: p < 0.5

Under H₀, X ~ B(10, 0.5)

As before, calculate the probability of obtaining the observed result or something more extreme:

P(X ≤ 3) = 0.1719

Since 0.1719 > 0.05, the result is not unusual.

We say that the test is not significant → Do not reject H₀.

There is insufficient evidence that the coin is biased.

Two-Tailed Tests

A two-tailed test checks for a difference in either direction.

Hypotheses:

• H₀: p = 0.5

• H₁: p ≠ 0.5

Because there are two tails, the 5% significance level splits into:

• 2.5% in the left tail

• 2.5% in the right tail

Worked Example: Two-Tailed Binomial Test

A manufacturer claims that 70% of its components are defect-free.

A random sample of 20 components contains 10 defect-free items.

Test at the 5% significance level whether the true proportion differs from 0.7.

Step 1: State the hypotheses

• H₀: p = 0.7

• H₁: p ≠ 0.7 (two-tailed)

Step 2: Model the test statistic

Let X = number of defect-free components.

Under H₀:

X ~ B(20, 0.7)

Expected value: np = 14

Observed value: X = 10

Step 3: Decide which tail to check

Since 10 < 14, the result lies below the expectation. So we check the left tail.

For a two-tailed test, compare with 0.025 (half of 5%).

We need:

P(X ≤ 10)

Step 4: Calculate the probability

From calculator/binomial tables:

P(X ≤ 10) = 0.0328

Step 5: Compare with the significance level

We compare against 0.025:

• 0.0328 > 0.025 → Not in the critical region

So:

Do not reject H₀.

Step 6: Conclusion (in context)

At the 5% significance level, there is insufficient evidence that the true proportion of defect-free components differs from 0.7.

The manufacturer’s claim cannot be rejected based on this sample.

Conclusion

Hypothesis testing allows you to use a binomial model to decide whether observed outcomes are consistent with an initial assumption. Using this method, A-level students can apply statistical reasoning to small sample experiments like coin tosses.

For a deeper understanding of normal approximation to the binomial, see my related post: Normal Approximation to the Binomial Distribution (Year 13)