H2 Maths Hypothesis Testing Formula Sheet | Test Statistics
H2 Maths Hypothesis Testing Formula Sheet | Test Statistics
Study guide/
H2 Maths hypothesis testing formula sheet: null and alternative hypotheses, z and t test statistics, one- and two-tailed tests, p-value and critical-region decision rules - ever...
Before you revise Prepare a quick-reference sheet: H0/H1 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 H0
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. The test-statistic formulas are provided in MF27, but you must know when to use each one, how to set up H0 and H1, and how to apply the decision rules. Worked examples for each appear in the sections below.
Hypotheses
Element
Form
Null hypothesis
H0:μ=μ0
Alternative hypothesis (right-tailed)
H1:μ>μ0
Alternative hypothesis (left-tailed)
H1:μ<μ0
Alternative hypothesis (two-tailed)
H1:μ=μ0
Test statistics
Scenario
Test statistic
Distribution under H0
Normal population, σ known
Z=σ/nXˉ−μ0
Z∼N(0,1)
Large sample (n≥30), σ unknown
Z≈s/nXˉ−μ0
Decision rules
Method
Reject H0 when
p-value
p-value <α
Critical region (right-tailed)
Z>zα
Critical region (left-tailed)
Z<−zα
Critical region (two-tailed)
∣Z∣>zα/2
Common critical values
Significance level α
One-tailed zα
Two-tailed zα/2
10%
1.282
1.645
5%
1.645
1.960
1%
2.326
2.576
Hypothesis Testing Framework
State hypotheses: H0 (status quo) and H1 (claim), referencing the parameter μ.
Choose test statistic and significance level α.
Calculate the observed statistic and either the p-value or compare with critical values.
Decision: reject or fail to reject H0 based on the evidence.
Conclusion: interpret in context, noting the significance level.
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.
Question wording
Alternative hypothesis
Tail to shade
Reject when using p-value
Conclusion stem
"has increased", "is greater than", "exceeds"
H1:μ>μ0
Right tail
p-value <α
There is sufficient evidence that the mean has increased
"has decreased", "is less than", "has fallen"
H1:μ<μ0
Left tail
p-value <α
"has changed", "is different", "is no longer"
H1:μ=μ0
Both tails
p-value <α
When the p-value is not smaller than α, 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:
Calculation result
Exam-safe wording
p-value =0.018<0.05 for H1:μ<520
Since the p-value is less than 0.05, reject H0. There is sufficient evidence at the 5% level that the mean attendance has decreased below 520.
p-value =0.118>0.05 for H1:μ<520
Since the p-value is greater than 0.05
Trap check: the direction of the sample mean alone does not decide the test. The wording of the claim decides H1; 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 H1.
Decision from comparison
Safe conclusion shape
Do not write
Why
Reject H0
There is sufficient evidence at the stated level that the mean has changed in the direction of H1.
H1 is true.
A test gives evidence, not proof.
Do not reject H0
There is insufficient evidence at the stated level that the mean has changed in the direction of H1.
H0
Two-tailed test
Say the mean differs from the claimed value only if you reject H0.
The mean is higher or lower unless the question asks for that direction.
A two-tailed test checks for any difference.
One-tailed test
Use the same direction word as the question, such as increased or decreased.
The mean has changed.
Direction is part of the alternative hypothesis.
Worked check: if H1:μ>50, α=0.05, and the p-value is 0.072, do not reject H0. A safe sentence is: "There is insufficient evidence at the 5% level that the mean has increased above 50."
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.
H0 (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, H0 always takes the form μ=μ0 for some stated value μ0.
H1 (alternative hypothesis) - what you suspect might be true instead. It reflects the direction of the question: "has it increased?", "has it decreased?", "has it changed at all?"
The p-value answers: "If H0 were true, how likely is data as extreme as what I observed?" A very small p-value means the data would be very unusual under H0, so you have reason to doubt H0
The conclusion is never "H1 is true." It is always one of these two sentence shapes: "There is sufficient evidence to reject H0 at the stated significance level" or "There is insufficient evidence to reject H0
Tests for a Population Mean μ
In the 2026 syllabus, hypothesis testing is for a population mean μ:
Normal population, known variance: Z=σ/nXˉ−μ0 and Z∼N(0,1) under H0.
Large sample from any population (often n≥30): by CLT, treat Z≈N(0,1). If σ is not provided, use s (sample standard deviation) as an approximation in s/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.
Question cue
Standard error to write
Distribution statement
Common trap
Normal population, σ known
σ/n
Z∼N(0,1) under H0
Using σ instead of σ/n.
Large sample, σ unknown
s/n
Z≈N(0,1)
Sample mean already standardised by GC output
Record the calculator's z and p-value
Match the tail to H1
Recomputing with a different tail from the one entered in the GC.
Worked check: if n=64 and σ=40, the standard error is 40/64=5, not 40. A sample mean that is 12 below μ0 gives z=−12/5=−2.40.
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 Xˉ=508, σ=40. Test at 5% whether attendance decreased.
H0:μ=520, H1:μ<520.
Z=40/64508−520=−2.40
Critical value: z0.05=−1.645. Since −2.40<−1.645, reject H0
Conclusion: There is sufficient evidence at the 5% level that average attendance fell below 520.
One-tailed vs Two-tailed Tests
If H1:μ>μ0 (increase), the critical region is in the right tail.
If H1:μ<μ0 (decrease), the critical region is in the left tail.
If H1:μ=μ0 (different), it is two-tailed: reject if ∣Z∣>z1−α/2
Example -- Two-tailed mean test
A manufacturer claims the mean battery life is μ=10.0 hours with known σ=1.5 hours. A sample of n=36 batteries gives xˉ=9.6. Test at 5% if the mean differs from 10.0 hours.
H0:μ=10.0, H1:μ=10.0.
Z=1.5/369.6−10.0=0.25−0.4=−1.60
Critical values: ±z0.975=±1.96. Since ∣−1.60∣<1.96, fail to reject H0
Conclusion: At the 5% level, there is insufficient evidence that the mean battery life differs from 10.0 hours.
P-values vs Critical Regions
P-value: probability, under H0, of observing a statistic at least as extreme as the sample.
If p-value<α, reject H0; otherwise do not reject.
Mention explicitly when reporting: “p-value=0.032<0.05, so reject H0.”
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.
Fix the tail from H1: left, right, or two-tailed.
Compute the observed test statistic z.
Compare either the p-value with α, or compare z with the critical region.
Write one decision only: reject H0 or do not reject H0.
Convert that decision into a contextual conclusion.
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. Since both methods agree, the conclusion should say there is sufficient evidence for the decrease claimed in H1.
Common trap: computing a two-tailed p-value after writing a one-tailed H1. 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 (μ), not statistics (Xˉ).
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), state that Xˉ is approximately normal and show the standard error used.
Common Mistakes
1. Confusing "reject H0" with "accept H1"
You never accept H1. Failing to reject H0 does not prove H0 is true either - it only means the data did not provide enough evidence against it. Use the exact phrasing "reject H0" or "do not reject H0".
2. Omitting context in the conclusion
A bare "reject H0" will lose marks. SEAB expects the conclusion to reflect the real-world scenario. Compare:
Incomplete: "Since p-value <0.05, reject H0."
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 H1 specifies a direction: μ>μ0 or μ<μ0. This is signalled by words such as "increased", "exceeded", "fallen".
Use a two-tailed test when H1:μ=μ0. This is signalled by words such as "changed", "different", "no longer equal to".
Setting the wrong tail produces the wrong critical value and p-value, even if all arithmetic is correct.
Practice Quiz
Rehearse full hypothesis-test writeups, from hypotheses to p-values and contextual conclusions.
Quick Revision Checklist
Set up H0, H1 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 H0 as "the mean weight is 50 kg" rather than H0:μ=50. SEAB expects symbolic form referencing the population parameter μ - always write H0:μ=μ0 and H1:μ=μ0 (or </> for one-tailed).
Using xˉ instead of μ in the hypotheses: Hypotheses are claims about the population mean μ, not the sample mean xˉ. Writing H0:xˉ=520
Confusing one-tailed and two-tailed critical regions: The critical region for a one-tailed test at the 5 percent level uses z0.05=1.645, while a two-tailed test at 5 percent uses z0.025=1.96 on each side. Applying 1.645
Using the wrong tail direction for H1: If a question states "a manager claims production has increased", H1 must be μ>μ0
Forgetting to state the conclusion in context: Writing only "reject H0" is incomplete. SEAB expects a sentence that references the real-world scenario and the significance level, e.g. "There is sufficient evidence at the 5% significance level that the mean daily output has increased from 420 units."
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 every result you need: hypothesis forms for H0 and H1, the z test-statistic formulas for both known-σ and large-sample scenarios, p-value and critical-region decision rules, and common critical values at the 10%, 5%, and 1% significance levels. MF27 provides the test-statistic formulas (including the z formula for a population mean), but you must still know when to apply each scenario, how to set up the hypotheses, and how to write the contextual conclusion - those elements are not in MF27.
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 μ. [1] This covers two scenarios: a normal population with known variance σ2, and a large sample from any distribution where the Central Limit Theorem applies so Xˉ≈N!(μ,σ2/n). Chi-squared goodness-of-fit tests, t-tests, and proportion tests are all excluded - do not introduce them in your answers.
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 α) 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 zcalc against zcrit) 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:
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://www.seab.gov.sg/files/A%20Level%20Syllabus%20Sch%20Cddts/2026/9758_y26_sy.pdf
