H2 Maths Hypothesis Testing Formula Sheet | Test Statistics

Q: What does H2 Maths Notes (JC 1-2): 6.5) Hypothesis Testing cover?
A: Test statistics, critical regions, p-values, and exam-ready interpretations for H2 Maths hypothesis tests.

Download: Get the H2 Maths Hypothesis Testing formula sheet (PDF) for quick revision, or the complete notes (PDF) for the full walkthrough.

Before you revise
Prepare a quick-reference sheet: $H_0/H_1$ templates for a population mean, common $z$-critical values, and graphing calculator (GC) commands for a mean test. Always practise writing the full conclusion sentence-MOE wants context, not just “reject $H_0$

A note on reasoning style
Hypothesis testing requires a different reasoning style from pure mathematics. In calculus, you find exact answers. In hypothesis testing, you make probabilistic judgments about evidence. This shift confuses many students because the questions feel “less mathematical” - but the logic is just as rigorous, only expressed in terms of probability rather than certainty.

• A hypothesis test asks whether sample data is surprising under a claim: State the null hypothesis.
• Direction decides the tail: Match "increased", "decreased", or "changed" to the alternative hypothesis.
• The conclusion must be about evidence, not proof: Write the decision in the context of the question.

Concrete example: If the school claims the mean is 520 and your sample mean is 508, the test asks whether 508 is unusually low if 520 were still true.

Status: SEAB's current H2 Mathematics (9758) syllabus PDF is labelled for 2026. Topic 6.5 is assessed in Paper 2 Section B (Probability and Statistics, 60 marks) and focuses on hypothesis tests for a population mean (no proportion tests; no correlation hypothesis tests).

Formulas at a glance

Every result the 9758 syllabus expects you to use, on one screen. MF27 does not provide the normal z-test statistics, normal critical values, hypothesis setup, or decision rules below. The shared booklet includes a Wilcoxon signed-rank critical-value table, which is not a substitute for the H2 normal-distribution workflow. Worked examples for each H2 method appear below.

Hypotheses

Test statistics

Decision rules

Common critical values

Hypothesis Testing Framework

1. State hypotheses: $H_0$ (status quo) and $H_1$ (claim), referencing the parameter $\mu$.
2. Choose test statistic and significance level $\alpha$

Tail, Decision, and Conclusion Map

Use this map after reading the question once. It separates the three decisions students often mix together: the direction of the alternative hypothesis, the comparison rule, and the wording of the final sentence.

When the p-value is not smaller than $\alpha$, switch the conclusion stem to "There is insufficient evidence that..." and keep the context phrase from the question. Do not write that the null hypothesis is proven true.

Worked decision sentence:

Trap check: the direction of the sample mean alone does not decide the test. The wording of the claim decides $H_1$; the sample then decides whether the evidence is strong enough.

Conclusion wording checkpoint

After the comparison step, translate the statistical decision into evidence language. The conclusion should name the context, the significance level, and the direction from $H_1$.

Worked check: if $H_1: \mu > 50$, $\alpha = 0.05$, and the p-value is $0.072$, do not reject $H_0$

Misconception check: "insufficient evidence" does not mean "no effect exists". It means this sample did not cross the rejection threshold for the test you set up.

What Am I Actually Testing?

This is the most common source of confusion. Here is the plain-language version.

You are testing whether observed data is consistent with a claimed population parameter. The sample gives you one window into reality; the hypothesis test asks how surprising that window is if the claim were true.

• $H_0$ (null hypothesis) - the "nothing special is happening" claim. It is what you assume is true unless the evidence is strong enough to overturn it. In H2 Maths, $H_0$ always takes the form $\mu = \mu_0$

Tests for a Population Mean $\mu$

In the 2026 syllabus, hypothesis testing is for a population mean $\mu$:

• Normal population, known variance: $Z = \dfrac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}$

Test-statistic setup checkpoint

Before pressing the calculator, write the denominator of the test statistic. This prevents mixing up the sample standard deviation, population standard deviation, and standard error.

Worked check: if $n=64$ and $\sigma=40$, the standard error is $40/\sqrt{64}=5$

Misconception check: the numerator measures how far the sample mean is from the claimed population mean; the denominator measures the typical spread of sample means, not the spread of individual observations.

Example -- JC lecture attendance

Historical mean attendance is 520 students. After a scheduling change, a sample of $n = 64$ lectures gives $\bar{X} = 508$, $\sigma = 40$. Test at 5% whether attendance decreased.

1. $H_0: \mu = 520$, $H_1: \mu \lt 520$

One-tailed vs Two-tailed Tests

• If $H_1: \mu > \mu_0$ (increase), the critical region is in the right tail.
• If $H_1: \mu < \mu_0$

Example -- Two-tailed mean test

A manufacturer claims the mean battery life is $\mu = 10.0$ hours with known $\sigma = 1.5$ hours. A sample of $n = 36$ batteries gives $\bar{x} = 9.6$

1. $H_0: \mu = 10.0$, $H_1: \mu \neq 10.0$

P-values vs Critical Regions

• P-value: probability, under $H_0$, of observing a statistic at least as extreme as the sample.
• If $\text{p-value} \lt \alpha$, reject $H_0$

Method Consistency Checkpoint

If you use both a p-value and a critical region as checks, they must lead to the same decision. Use this order to catch contradictions before writing the conclusion.

1. Fix the tail from $H_1$: left, right, or two-tailed.
2. Compute the observed test statistic $z$.
3. Compare either the p-value with $\alpha$, or compare $z$ with the critical region.
4. Write one decision only: reject $H_0$

Worked check: for a left-tailed test at the $5\%$ level, suppose $z = -2.10$. The critical-value method rejects because $-2.10 < -1.645$. The p-value method also rejects because the left-tail p-value is about $0.018 < 0.05$

Common trap: computing a two-tailed p-value after writing a one-tailed $H_1$. That doubles the wrong tail area and can make the p-value decision contradict the critical-value decision.

Calculator Workflows

• TI: ZTest (or 1-Var Stats + normalcdf) for mean tests; record the inputs and outputs (test statistic, p-value).
• Casio: Use STAT > DIST > NORM for tail probabilities and Z Test where available; always confirm the tail (left/right/two) before pressing ENTER.
• Always double-check the tail (left/right/two) before pressing ENTER.

Exam Watch Points

• Hypotheses must refer to parameters ($\mu$), not statistics ($\bar{X}$).
• Quote the significance level and justify one- vs two-tailed tests from the context wording (“increase”, “different”).
• Round the test statistic to 3 decimal places and p-values to 3 significant figures.
• Ensure the final conclusion mentions the context and the significance level (e.g. “At the $5\%$ level…”).
• SEAB excludes the term “Type I error”, the concept of Type II error, and tests comparing two population means-avoid introducing them in your write-up.
• When using CLT (large $n$

Common Mistakes

1. Confusing "reject $H_0$" with "accept $H_1$"

You never accept $H_1$. Failing to reject $H_0$ does not prove $H_0$

2. Omitting context in the conclusion

A bare "reject $H_0$" will lose marks. SEAB expects the conclusion to reflect the real-world scenario. Compare:

• Incomplete: "Since p-value $< 0.05$, reject $H_0$."
• Complete: "Since p-value $< 0.05$, there is sufficient evidence at the 5% significance level that the mean mass has increased."

Always echo the context word from the question (e.g., "increased", "changed", "differs from").

3. Choosing the wrong tail

• Use a one-tailed test when $H_1$ specifies a direction: $\mu > \mu_0$ or $\mu < \mu_0$

Practice Quiz

Rehearse full hypothesis-test writeups, from hypotheses to p-values and contextual conclusions.

Quick Revision Checklist

• Set up $H_0$, $H_1$ correctly and select the appropriate test statistic.
• Carry out mean $z$-tests with calculator support while writing full working.
• Interpret p-values versus critical regions and articulate conclusions in context.
• Write conclusions in plain English without introducing excluded error terminology.

Want weekly guided practice on Hypothesis Testing? Our H2 Maths tuition programme builds fluency in this topic through structured problem sets and exam-style drills.

Common exam mistakes

• Writing hypotheses in words instead of symbols: A common error is stating $H_0$ as "the mean weight is 50 kg" rather than $H_0: \mu = 50$. SEAB expects symbolic form referencing the population parameter $\mu$

Frequently asked questions

Is there a formula sheet for H2 Maths hypothesis testing?
Yes - the "Formulas at a glance" section near the top of this page collects the hypothesis forms, normal z-test statistics, p-value and critical-region decision rules, and common significance levels used in H2 questions. These are study notes, not entries reproduced from MF27. MF27 does not provide the normal z-test statistics or normal critical values, so you must know the setup and approved graphing-calculator workflow.

Which paper and section does Hypothesis Testing appear in?
Hypothesis Testing is examined in Paper 2, Section B (Probability and Statistics), which carries 60 marks. [1] The section is compulsory, so every candidate attempts it. You can expect one full hypothesis-testing question, typically worth 8-12 marks, that requires setting up hypotheses, computing a test statistic, and writing a contextual conclusion.

Are chi-squared tests or t-tests examinable in H2 Maths (9758)?
No. The 9758 syllabus (examinations from 2026 onwards) restricts hypothesis testing to z-tests for a population mean $\mu$. [1] This covers two scenarios: a normal population with known variance $\sigma^2$, and a large sample from any distribution where the Central Limit Theorem applies so $\bar{X} \approx \operatorname{N}!\left(\mu, \sigma^2/n\right)$

Should I use the p-value method or the critical value method?
Both methods are accepted by SEAB and will earn full marks if applied correctly. The p-value method (compare the computed p-value against $\alpha$) is generally faster on a GC because you read the p-value directly from the test output. The critical value method (compare the test statistic $z_{\text{calc}}$ against $z_{\text{crit}}$) is easier to present clearly in written working. In exams where the question says "use an appropriate test", state which method you are using, show the comparison explicitly, and ensure the conclusion matches - a contradiction between the comparison and the stated conclusion will lose marks regardless of which method you chose.

Other H2 Maths formula sheets

Revising more than one topic? Grab the matching one-page formula sheet:

• Functions & graphs: Functions · Graphs & Transformations · Equations & Inequalities
• Sequences & series: Sequences & Series
• Vectors: Vectors
• Calculus: Differentiation · Maclaurin Series · Integration Techniques · Definite Integrals · Differential Equations
• Statistics: Probability · Discrete Random Variables · Normal Distribution · Sampling · Hypothesis Testing (this page) · Correlation & Regression

For the official SEAB reference booklet, see the H2 Maths MF27 formula list.

Sources

• SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 6 Probability and statistics sub-topic 6.5 Hypothesis testing (tests for a population mean; critical regions, critical values, level of significance, p-values; excludes Type I/II error terminology and tests comparing two means): https://isomer-user-content.by.gov.sg/334/f27e37f7-f0ec-4a35-b1e8-3a5e88ae2f81/9758_y26_sy.pdf

Next steps: Stay on the H2 Maths notes hub and pair these workflows with Topic 6.6 - Correlation & regression plus mixed sampling questions.