Hypothesis Testing

CIE A-Level Mathematics· Unit 4: Probability & Statistics 1· 15 min read

1. Core Terminology & Hypothesis Setup★★☆☆☆⏱ 4 min

📘 Definition

Null Hypothesis

$H_{0}$

The statement of no effect or no difference from the hypothesized population value. We assume $H_{0}$ is true unless we find sufficient evidence to reject it.

Example:

For testing if a coin is biased, $H_{0} : p = 0.5$ where $p$ is the true probability of heads.

📘 Definition

Alternative Hypothesis

$H_{1}$

The alternative claim we accept if we reject $H_{0}$ . Can be one-tailed ( $p > p_{0}$ or $p < p_{0}$ ) or two-tailed ( $p \neq = p_{0}$ ).

Example:

For testing if the coin is biased towards heads, $H_{1} : p > 0.5$ .

The first step of any hypothesis test is to define the population parameter you are testing, then write $H_{0}$ and $H_{1}$ in terms of that parameter. Hypotheses are always about population parameters, never sample statistics.

✓ Quick check

Check your understanding:

A bakery claims 10% of their pastries are undercooked. An inspector thinks the proportion is higher. What is the correct $H_{1}$ ?
- $H_{1} : p > 0.1$
- $H_{1} : \overset{p}{^} > 0.1$
- $H_{1} : p < 0.1$

2. Hypothesis Tests for Proportions★★★☆☆⏱ 5 min

For large samples, we approximate the binomial distribution of the number of successes with a normal distribution, then calculate a p-value to compare to our significance level. We follow a standard 5-step process for every test:

1. Define the parameter and write hypotheses
1. State the significance level
1. State the distribution of the test statistic under $H_{0}$
1. Calculate the p-value (or compare test statistic to critical value)
1. Draw a conclusion in context

📐 Worked Example

A machine produces components, 10% of which are normally defective. After a service, a sample of 100 components finds 14 defective. Test at the 5% significance level if the proportion defective has increased.

1
1. Define parameter and hypotheses: Let $p =$ true proportion of defective components after the service.
2
$H_0: p = 0.10 \newline H_1: p > 0.10$
3
1. State significance level: $α = 0.05$
4
1. Under $H_{0}$ , the number of defective $X \sim B (100, 0.1)$ . Approximate with normal, mean = $n p = 10$ , variance = $n p (1 - p) = 9$ , so $X \sim N (10, 9)$ .
5
1. Calculate p-value for $X \geq 14$ , apply continuity correction: $P (X \geq 14) = P (X > 13.5)$
6
$z = \frac{13.5 - 10}{3} = 1.167, \quad p\text{-value} = 1 - \Phi(1.167) = 0.1216$
7
1. Compare: $0.1216 > 0.05$ , so we do not reject $H_{0}$ .
8
Conclusion: There is insufficient evidence at the 5% significance level to conclude the proportion defective has increased.

3. Hypothesis Tests for Population Means★★★☆☆⏱ 5 min

When the population is normally distributed with known variance, we use a z-test for the population mean $μ$ . The test statistic follows a standard normal distribution when $H_{0}$ is true.

📘 Definition

Z-test Statistic for Means

$z$

For a sample of size $n$ , sample mean $\overset{x}{ˉ}$ , hypothesized mean $μ_{0}$ and known population standard deviation $σ$ : $z = \frac{x ˉ - μ _{0}}{σ / n}$ . This measures how many standard errors the sample mean is from the hypothesized value.

Example:

Sample size 25, mean 12, $σ = 2$ , $μ_{0} = 11$ gives $z = 2.5$ .

📐 Worked Example

A bottling machine fills bottles with a known population standard deviation of 5 ml. The mean fill volume is supposed to be 500 ml. A sample of 10 bottles has a mean of 497 ml. Test at the 1% significance level if the mean fill volume is different from 500 ml.

1
1. Define parameter and hypotheses: Let $μ =$ true mean fill volume.
2
$H_0: \mu = 500 \newline H_1: \mu \neq 500$
3
1. This is a two-tailed test, total significance level $α = 0.01$ .
4
1. Calculate the z-test statistic:
5
$z = \frac{497 - 500}{5/\sqrt{10}} \approx -1.898$
6
1. Calculate two-tailed p-value:
7
$p\text{-value} = 2 \times P(Z < -1.898) = 2 \times 0.0287 = 0.0574$
8
1. Compare: $0.0574 > 0.01$ , so do not reject $H_{0}$ .
9
Conclusion: There is insufficient evidence at 1% significance that the mean fill volume is different from 500 ml.

4. Type I and Type II Errors★★★★☆⏱ 4 min

📘 Definition

Type I Error

Rejecting $H_{0}$ when $H_{0}$ is actually true. The probability of a Type I error is always equal to the significance level $α$ of the test.

Example:

Concluding a coin is biased when it is actually fair.

📘 Definition

Type II Error

Failing to reject $H_{0}$ when $H_{0}$ is actually false. The probability of a Type II error depends on the true value of the population parameter.

Example:

Concluding a coin is fair when it is actually biased.

📐 Worked Example

For a hypothesis test $H_{0} : p = 0.1$ , $H_{1} : p > 0.1$ , carried out at 5% significance level, what is the probability of a Type I error?

1
By definition, the probability of a Type I error is the probability that we reject $H_{0}$ when $H_{0}$ is true. This equals the significance level of the test:
2
$P(\text{Type I Error}) = 0.05$

5. Common Pitfalls

Wrong move:

Writing hypotheses in terms of sample statistics (e.g. $H_{0} : \overset{x}{ˉ} = 500$ instead of $H_{0} : μ = 500$ )

Why:

Hypotheses are always claims about unknown population parameters, not observed sample values

Correct move:

Always write hypotheses in terms of the population parameter ( $p$ for proportion, $μ$ for mean)

Wrong move:

Forgetting continuity correction when approximating binomial with normal for proportion tests

Why:

Binomial is discrete and normal is continuous, so the p-value will be incorrect without the correction

Correct move:

Always apply continuity correction when working with counts of successes in proportion tests

Wrong move:

Not doubling the p-value for two-tailed tests

Why:

Two-tailed tests account for difference in either direction, so a one-tailed p-value is half the correct value

Correct move:

Calculate the one-tailed p-value, then double it for two-tailed tests, or use a two-tailed critical value

Wrong move:

Saying 'we accept $H_{0}$ ' when p-value > significance level

Why:

We never prove $H_{0}$ is true, we only fail to find sufficient evidence to reject it

Correct move:

Always state 'we do not reject $H_{0}$ ' or 'there is insufficient evidence to reject $H_{0}$ '

Wrong move:

Confusing Type I and Type II errors in exam questions

Why:

The order of errors is easy to mix up under pressure

Correct move:

Use the mnemonic First you Reject, Second you Stay to recall which error is which

6. Quick Reference Cheatsheet

Term	Key Fact/Formula	Notes
$H_{0}$	Null hypothesis: no effect	Assumed true unless proven otherwise
$H_{1}$	Alternative hypothesis: effect/difference	One-tailed = specific direction; two-tailed = any difference
Proportion test statistic	$z = \frac{p ^ - p _{0}}{\frac{p _{0} ( 1 - p _{0} )}{n}}$	Always apply continuity correction to count $X$
Mean test statistic (known $σ$ )	$z = \frac{x ˉ - μ _{0}}{σ / n}$	Requires normal population distribution
P(Type I error)	$= α$ (significance level)	Rejecting true $H_{0}$
P(Type II error)	$P (not reject H_{0} ∣ H_{0} false)$	Depends on the true parameter value

7. Frequently Asked

Do I need to define the population parameter in my answer?

Yes, CIE always awards a mark for explicitly defining the population parameter (e.g. Let $p$ = true proportion of defective components) so never skip this step.

What is the difference between one-tailed and two-tailed tests?

One-tailed tests check for a difference in a specific direction (increase or decrease only), while two-tailed tests check for any difference from the null value, regardless of direction. You must double the one-tailed p-value for two-tailed tests.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · 4
Proportion test, Type I error calculation
2022 · 4
Two-tailed test for population mean
2021 · 4
Interpretation of hypothesis errors

What's Next

Hypothesis testing is the foundation of all statistical inference for CIE A-Level Mathematics, and it is heavily tested in Paper 4, usually as an 8-10 mark longer question. Mastering the structured approach and common exam conventions (like defining parameters and correct conclusion wording) will earn you full marks on these predictable questions. This topic leads directly into more advanced inference topics including Chi-squared tests for categorical data, t-tests for means with unknown variance, and hypothesis testing for correlation and regression coefficients, all of which are core topics for full A-Level Mathematics and Further Mathematics.