Cambridge International AS & A Level Mathematics 9709 (2026-2027) · Paper 6: Probability & Statistics 2 · official syllabus
SAMPLING AND ESTIMATION LAB
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Four short missions on unbiased estimates, sampling spread and a mean confidence interval. Separate a sample statistic from a fixed but unknown population parameter.
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Mission 1 of 4
Calculate a sample mean
An independent random sample contains values 8, 10 and 12. Set the sample mean, an unbiased estimate of the population mean under this sampling model. This is an estimate, not proof of the unknown population mean.
An unbiased estimate must equal the true parameter in every sample.
Teal dots are the three sample values. Move the gold candidate-mean line to their arithmetic average. The unknown population mean is not shown.
Your mean is 8. Divide the sample total 30 by 3.
Study notes
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Separate population, sample and sampling method 6.4
A population is the collection about which a question is asked; a sample is the part observed. A parameter, such as population mean mu or variance sigma squared, describes the population. A statistic, such as a sample mean, is calculated from the sample and varies between random samples.
Random selection helps avoid systematic selection bias. A large convenience sample can remain biased: people who volunteer, are easiest to contact, or appear at one time or location may differ from the target population. Explain a problem in terms of who is over- or under-represented, not only by naming bias.
A simple random selection can be produced by labelling the sampling frame and using random numbers, rejecting out-of-range labels and duplicates when sampling without replacement. A good random selection from an incomplete frame still omits people missing from the frame; randomness does not repair that omission.
Treat the sample mean as a random variable 6.4
For an independent random sample of n observations from the same population with mean mu and finite variance sigma squared, E(sample mean)=mu and Var(sample mean)=sigma squared/n. The standard deviation of the sample mean, its standard error, is sigma/sqrt(n).
If the population is normal, the sample mean is exactly normal with these parameters. For a suitably large sample, the Central Limit Theorem gives an approximate normal distribution for the sample mean even when the population is not normal. It does not say the original observations become normally distributed.
With population standard deviation 12 and sample size 36, the standard error is 2 and variance of the sample mean is 4. Quadrupling n halves the standard error. Increasing n reduces random sampling spread, not systematic selection bias.
Calculate unbiased estimates from sample data 6.4
The sample mean, sum of observations divided by n, is an unbiased estimate of the population mean under the independent random-sample model. Unbiased means the estimator has the correct expectation over repeated samples; an individual estimate need not equal the unknown parameter.
An unbiased estimate of population variance is s squared=[sum of x squared minus (sum of x)^2/n]/(n-1), for n>1. Equivalently divide the sum of squared deviations from the sample mean by n-1. Distinguish this from a descriptive variance using divisor n.
For sample 8,10,12, the total is 30, the sum of squares is 308 and the mean is 10. The squared-deviation sum is 308-30^2/3=8, so s squared=8/2=4. The corresponding standard deviation estimate is 2; this square root is not itself generally an unbiased estimator of population standard deviation.
Form a confidence interval for a population mean 6.4
For a normal population with known variance, a two-sided interval for mu is sample mean plus or minus z times sigma/sqrt(n), where the central standard normal area is the confidence level. For a large sample with unknown variance, an approximate interval uses the sample variance estimate in the standard error.
With normal population standard deviation known to be 12, n=36 and sample mean 50, use the supplied rounded 95% critical value z=1.96. The margin is 1.96(12/6)=3.92, giving (46.08,53.92). The endpoints use that rounded critical value and are not an exact expression for an unrounded normal quantile.
Do not silently use a small-sample unknown-variance normal interval. The stated syllabus mean intervals cover known-variance normal populations or large-sample cases. Check those conditions, independence and the sampling method before using the formula.
Interpret confidence and interval width 6.4
The population mean is fixed but unknown. Before sampling, the interval endpoints are random. A 95% procedure covers that fixed parameter in about 95% of repeated samples under its assumptions. After a particular interval is calculated, it either contains the parameter or does not; confidence describes the procedure.
A confidence interval for the mean is not an interval containing 95% of observations, nor a guarantee that the true mean is near its centre. The normal assumptions or approximation determine coverage, and bias in sampling can undermine it.
Holding other quantities fixed, a higher confidence level requires a larger critical value and a wider interval. A larger sample reduces the standard error and narrows the interval. To halve the margin at the same confidence level and population variance, multiply the sample size by 4.
Estimate a population proportion from a large sample 6.4
For a large independent sample, estimate a population proportion p by p-hat=x/n. A normal approximate confidence interval is p-hat plus or minus z sqrt[p-hat(1-p-hat)/n], when success and failure counts are sufficiently large for the approximation to be suitable.
Worked example: 60 successes in 200 observations gives p-hat=0.3. With z=1.96 the approximate margin is 1.96 sqrt(0.3 times 0.7/200), about 0.0635 to 4 d.p., giving an approximate interval (0.2365,0.3635). Both observed counts, 60 and 140, are large.
The simple normal interval can be poor near 0 or 1 or with a small sample, and can even give endpoints outside [0,1]. Treat that as a warning about the approximation rather than silently asserting valid coverage. Keep proportions distinct from percentages when substituting into the standard error.
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