Q: What does H2 Maths Notes (JC 1-2): 6.4) Sampling cover? A: Sampling methods, statistic distributions, confidence intervals, and variance formulas tuned to the H2 Maths 2026 syllabus.
Before you revise Recap descriptive statistics (mean, variance) so you can translate between population parameters and sample statistics. Keep a table of common critical values (z0.95=1.645, z0.975=1.96
, t-values for small
n
) beside you while practising.
Status: SEAB H2 Mathematics (9758, first exam 2026) syllabus last checked 2025-11-29 (PDF last modified 2024-10-16). Topic 6.4 scope unchanged; Section B still contributes 30 marks per paper (Probability & Statistics).
Sampling Language
A population has true mean μ and variance σ2; the sampling frame lists every eligible unit.
A sample of size n yields statistics Xˉ (sample mean) and S2 (sample variance). Sampling without replacement from a finite population of size N reduces variability via the finite population correction N−1N−n.
Core sampling methods:
Simple random sampling (SRS): every size-n subset is equally likely.
Systematic sampling: select every k-th unit after a random start; watch for periodicity bias.
Stratified sampling: divide the population into homogeneous strata, then run SRS within each stratum (often proportional to stratum size).
Cluster sampling: randomly select whole groups (clusters) and survey everyone inside.
Bias checkpoints: undercoverage (missing groups), non-response (selected units decline), measurement bias (faulty instruments or questions).
Example -- Proportional stratified sample
A school has 420 JC1s in the Science stream and 280 in the Arts stream. To survey n=70 students with proportional stratification:
nScience=70×700420=42,nArts=70×700280=28.
Select 42 Science students and 28 Arts students via SRS inside each stratum.
Distribution of the Sample Mean
If X1,X2,…,Xn are independent with mean μ and variance σ2, then Xˉ=n1∑i=1nXi has E(Xˉ)=μ and Var(Xˉ)=nσ2. The standard error is SE(Xˉ)=nσ.
For large n (Central Limit Theorem), Xˉ is approximately N(μ,σ2/n) even if the parent distribution is non-normal. Rule of thumb: n≥30
If the population is normal, Xˉ is exactly normal for any n.
With sampling without replacement from a finite N, adjust the variance by the correction factor N−1N−n when n exceeds roughly 10
Example -- Average revision hours
JC students have revision hours with μ=6.4, σ=1.8. For a random sample of n=36, find P(Xˉ>7).
Xˉ∼N(6.4,1.82/36)=N(6.4,0.09).
Standardise: z=0.37−6.4=2.0.
Probability P(Xˉ>7)=1−Φ(2.0)≈0.0228.
Example -- Sample size for desired precision
How large should n be so that P(∣Xˉ−μ∣<0.5)≥0.95 when σ=2.4 and sampling is i.i.d.?
Round up to the next whole number: take n=90 observations.
Confidence Intervals
For known σ or large n, a (1−α) confidence interval for μ is Xˉ±z1−α/2nσ.
When σ is unknown and n≤30, use the t-distribution with n−1 degrees of freedom: Xˉ±t1−α/2,n−1nS
Workflow checklist:
Identify the estimator (mean or proportion) and note whether σ is known.
Compute the standard error (σ/n or S/n
Desired half-width E for a mean with known σ: n=(Ez1−α/2σ)2
Example -- Interval for a mean
A pilot sample of n=25 chemistry students gives Xˉ=12.4 hours of weekly study with sample standard deviation s=2.1. Construct a 95% confidence interval for μ.
Small sample, unknown σ⇒ t-interval with 24 degrees of freedom.
t0.975,24=2.064.
Margin =2.064×252.1=0.867.
Interval: (11.5,13.3) hours.
Example -- Planning sample size
To estimate mean commuting time within ±5 minutes at 95 percent confidence when past data suggest σ=18 minutes:
n=(51.96×18)2=49.7⇒take n=50 commuters.
Sampling Proportions
For n large, p^=nX (sample proportion) satisfies p^≈N(p,np(1−p)).
Confidence interval: p^±z1−α/2np^(1−p^)
Check np and n(1−p)≥5 before using the approximation.
Sample size for margin E at confidence level 1−α: n=E2z1−α/22p∗(1−p∗)
When conditions fail (very small n or p^ near 0/1), flag that the normal approximation is suspect and consider exact binomial reasoning.
Example -- Device adoption
In a survey of 180 JC students, 126 use an iPad for note-taking. Estimate the true proportion with 90% confidence.
p^=180126=0.700.
Check approximation validity: np^=126 and n(1−p^)=54 are both well above 5.
z0.95=1.645.
Margin =1.6451800.7×0.3=0.056
Interval: (0.644,0.756).
Calculator Workflows
Use graphing calculator (GC) statistics mode (1-Var Stats) to obtain Xˉ and S from raw data.
TI: tInterval, zInterval, 1-PropZInt functions automate standard intervals; record the command, parameters, and output range explicitly.
Casio: Run STAT > DIST > NORM or t for tail probabilities; confidence intervals may need manual computation, so pre-program formulas if allowed.
Exam Watch Points
State sampling design in words (e.g. “select every 5th student in the register”) and comment on bias.
Identify whether σ is known and justify the use of z or t-intervals.
Always interpret the interval in context: “We are 95% confident the true mean weekly revision time lies between...”.
Quote degrees of freedom when using t and note when CLT assumptions are invoked.
Practice Quiz
Apply sampling design principles, standard error calculations, and confidence interval interpretations under exam pacing.
Quick Revision Checklist
Distinguish population vs sample parameters quickly (μ vs Xˉ).
Compute Var(Xˉ) and standard errors without mixing up S and σ.
Build both mean and proportion confidence intervals, choosing z or t correctly.
Explain the meaning of a confidence level in sentences, avoiding probability statements about μ.
Sources
SEAB H2 Mathematics syllabus (9758), examinations from 2026 - Topic 6 Probability and statistics sub-topic 6.4 Sampling (sample mean distribution, standard error, Central Limit Theorem conditions, confidence intervals for means/proportions, simple/stratified/systematic/cluster sampling design points): https://www.seab.gov.sg/files/A%20Level%20Syllabus%20Sch%20Cddts/2026/9758_y26_sy.pdf
