H1 MATHEMATICS · HYPOTHESIS TESTING (SYLLABUS 8865)

Test the mean.
Quantify the evidence.

A dedicated guide for Singapore A-Level H1 Mathematics. Formulate hypotheses for a population mean $μ$ , apply the Central Limit Theorem when needed, calculate the p-value using your graphing calculator, and write a rigorous contextual conclusion.

Study the worked example Try a hypothesis-test question

Population mean only
Known variance or large sample
SEAB-style conclusion

01Evidence under the null model The p-value is the probability, assuming

H_{0}

is true, of a result at least as extreme in the direction of

H_{1}

Aligned to 8865 H1 MathematicsOne-sample tests for a population mean, including one- and two-tailed tests

Not a binomial hypothesis testTests for proportions and binomial hypothesis tests are outside this syllabus

THE TESTING LOGIC

Assume first. Measure second.

The null hypothesis is the reference model. Under $H_{0}$ , calculate how unusual the observed sample mean would be.

If the p-value is no greater than the significance level, reject $H_{0}$ . Otherwise, do not reject $H_{0}$ ; a test does not prove that the null hypothesis is true.

SYLLABUS MODELS

Know which normal model applies.

Normal population, known variance

For a random sample of size $n$ from a normal population with known variance $σ^{2}$ , use $\bar{X} \sim N (μ, \frac{σ^{2}}{n})$ .

Large sample from any population

When the parent distribution is not known to be normal and $n \geq 30$ , invoke the Central Limit Theorem and use the unbiased estimate $s^{2}$ of the population variance.

Unbiased variance estimate

If the given sample variance $s_{n}^{2}$ uses divisor $n$ , then $s^{2} = \frac{n}{n - 1} s_{n}^{2}$ .

Outside H1 hypothesis testing

Do not use tests for a population proportion, binomial hypothesis tests, two-sample tests or t-tests for an 8865 H1 hypothesis-testing question.

THE FOUR-PART CHECK

Keep the parameter, model and conclusion aligned.

Define the parameter

Define $μ$ fully in context, including the quantity, units and population.

μ = population mean

State the hypotheses

Use the population mean—not the observed sample mean—in $H_{0}$ and $H_{1}$ .

H_{0} : μ = μ_{0}

Model under $H_{0}$

State the normal model and justify the CLT when the population is not known to be normal.

Z = \frac{\bar{X} - μ_{0}}{σ / \sqrt{n}} \sim N (0, 1)

Compare and conclude

Compare the labelled p-value with $α$ , state the decision, then conclude about the population mean in context.

p -value \leq α \Rightarrow reject H_{0}

WORKED EXAMPLE

Test a claimed population mean.

An energy drink manufacturer claims that each can contains a mean caffeine content of 120 mg. A consumer group suspects that the true mean is greater than 120 mg. A random sample of 40 cans has mean 121.8 mg and standard deviation 5.2 mg calculated using divisor 40. Test at the 5% level of significance.

EXAM HABIT

The divisor matters. Here 5.2 mg is $s_{n}$ , so convert it to the unbiased estimate before using it in the large-sample test.

1
Define the parameter
Let $μ$ be the population mean caffeine content, in mg, per can of this energy drink.
2
State the hypotheses
$H_{0} : μ = 120 against H_{1} : μ > 120$
3
State the model under $H_{0}$
Since the population distribution is unknown but $n = 40 \geq 30$ , by the Central Limit Theorem,
$\bar{X} \sim N (120, \frac{s^{2}}{40}) approximately under H_{0}$ $s^{2} = \frac{40}{39} (5.2)^{2} \approx 27.733, s \approx 5.266$
4
Calculate the test statistic and p-value
$z_{calc} = \frac{121.8 - 120}{5.266 / \sqrt{40}} \approx 2.162$ $p -value = P (Z \geq 2.162) \approx 0.0153$
On a graphing calculator, use a one-sample Z-Test with alternative $μ > 120$ .
5
Compare and decide
Since p-value = 0.0153 < 0.05, reject $H_{0}$ at the 5% level of significance. Equivalently, $z_{calc} = 2.162 > 1.645$ , the right-tail critical value.
6
Conclude in context
There is sufficient evidence at the 5% level of significance to conclude that the population mean caffeine content per can is greater than 120 mg.

DecisionReject $H_{0}$ ; the sample supports the group’s suspicion at the 5% level.

COMMON MISTAKES

What to catch before the examiner does.

Using $\bar{x}$ in the hypotheses

Hypotheses concern the population mean $μ$ , never the observed sample mean.

Omitting the CLT

If the population is not stated to be normal, justify the approximate normal model using the large sample.

Omitting “under $H_{0}$ ”

The stated distribution and test statistic are based on the assumption that the null hypothesis is true.

Saying “accept $H_{0}$ ”

When the result is not significant, write “do not reject” or “fail to reject” $H_{0}$ .

Giving a vague conclusion

Name the population mean, its context and the significance level; do not write only “the claim is supported”.

PRACTISE WITH FEEDBACK

Get the reasoning checked, not just the calculator value.

Ask Integrand to check your parameter, hypotheses, model, calculator result and contextual conclusion.

Try Integrand

1
Formulate
Define $μ$ and state $H_{0}$ and $H_{1}$ .
2
Calculate
Choose the correct normal model and obtain the p-value or critical value.
3
Interpret
Compare, decide and conclude about the population mean in context.

QUICK QUESTIONS

H1 hypothesis testing FAQ

What hypothesis tests are in Singapore H1 Maths 8865?

One-sample tests for a population mean $μ$ : either a sample from a normal population of known variance or a large sample from any population. One- and two-tailed tests are included. Tests for proportions, binomial hypothesis tests, two-sample tests and t-tests are not included.

When must I use the Central Limit Theorem?

Use the CLT when the parent population is not known to be normal and the sample is sufficiently large, such as $n \geq 30$ . If the population is already stated to be normal, a CLT justification is unnecessary.

Should I use the p-value or critical-value method?

Both are within the syllabus. The p-value method using a graphing calculator Z-Test is often faster, while a question may explicitly ask for a critical value or critical region.

What does a p-value measure?

It is the probability, assuming $H_{0}$ is true, of obtaining a result at least as extreme as the observation in the direction of $H_{1}$ .

Syllabus reference: SEAB 2026 H1 Mathematics 8865, sections 3.4–3.5.

READY TO PRACTISE?

Test a mean. Get feedback on every decision.

Open Integrand

Test the mean.Quantify the evidence.

Assume first. Measure second.

Know which normal model applies.

Normal population, known variance

Large sample from any population

Unbiased variance estimate

Outside H1 hypothesis testing

Keep the parameter, model and conclusion aligned.

Define the parameter

State the hypotheses

Model under H0

Compare and conclude

Test a claimed population mean.

Define the parameter

State the hypotheses

State the model under H0

Calculate the test statistic and p-value

Compare and decide

Conclude in context

What to catch before the examiner does.

Using x¯ in the hypotheses

Omitting the CLT

Omitting “under H0”

Saying “accept H0”

Giving a vague conclusion

Get the reasoning checked, not just the calculator value.

Formulate

Calculate

Interpret

H1 hypothesis testing FAQ

Explore another A-Level Maths guide

Conditional probability

Integration techniques

Mathematical induction

Test a mean. Get feedback on every decision.

Test the mean.
Quantify the evidence.

Model under $H_{0}$

State the model under $H_{0}$

Using $\bar{x}$ in the hypotheses

Omitting “under $H_{0}$ ”

Saying “accept $H_{0}$ ”