Sampling and Estimation
Statistics
Sampling and Estimation
Syllabus tag: KASNEB CPA | Foundation Level | CA15 Quantitative Analysis | Topic 8 Sampling and Estimation
Lesson objectives
By the end of this topic, you will be able to:
- Distinguish the main sampling methods
- Explain the central limit theorem and the standard error
- Construct a confidence interval for a mean
- Compute the sample size needed for a required precision
- Carry out a hypothesis test and state its conclusion properly
Why this matters
Almost every conclusion drawn from data comes from a sample rather than a whole population. This topic is about how much confidence that permits — and the answer is never "certainty".
Sampling methods
Probability methods, where every item has a known chance of selection:
| Method | How it works |
|---|---|
| Simple random | Every item equally likely |
| Systematic | Every nth item after a random start |
| Stratified | The population is divided into groups and sampled in proportion |
| Cluster | Whole groups are selected and all members surveyed |
| Multi-stage | Successive rounds of selection |
Non-probability methods — quota, judgemental, convenience. Cheaper and quicker, and the sampling error cannot be measured, so no valid statistical inference follows.
Stratified sampling is the one worth understanding. Where a population contains distinct groups — branches, age bands, product lines — sampling each in proportion guarantees representation that simple random sampling may miss by chance.
Sampling error and bias
Sampling error is the difference between a sample result and the true population value, arising purely from chance. It is unavoidable and is reduced by a larger sample.
Bias is systematic error from a flawed method — surveying only customers who complain, or only those reachable by telephone. A larger sample does not reduce bias; it merely produces a more precise wrong answer.
That distinction is examinable and is the same one the CA24 auditing paper draws between sampling and non-sampling risk.
The central limit theorem
Whatever the shape of the population, the distribution of sample means is approximately normal for a sufficiently large sample — conventionally n of 30 or more.
Standard error of the mean = σ / √n
With σ = 60 and n = 36: 60 / 6 = 10.0
The √n matters. To halve the standard error you must quadruple the sample. Precision is bought at a rapidly rising price, which is why surveys do not simply keep expanding.
Confidence intervals
CI = x̄ ± z × (σ / √n)
| Confidence | z |
|---|---|
| 90% | 1.645 |
| 95% | 1.96 |
| 99% | 2.576 |
A sample mean of 500, σ = 60, n = 36:
Half-width = 1.96 × 10 = 19.6 95% confidence interval = 480.4 to 519.6
What it means, stated carefully. If the sampling were repeated many times, about 95% of the intervals constructed this way would contain the true population mean. It does not mean there is a 95% probability that the true mean lies in this particular interval — the true mean is a fixed number, and either it is in there or it is not.
Examiners accept the looser wording, but the precise version earns more marks.
Where the population standard deviation is unknown and the sample is small, the t-distribution replaces z, with n − 1 degrees of freedom.
Sample size
Rearranging the interval formula for a required precision e:
n = (z σ / e)²
For a margin of ±5 at 95% confidence with σ = 60:
n = (1.96 × 60 / 5)² = 553.19, so 554 items — always round up, since rounding down would fail to meet the required precision.
:::checkpoint A researcher wants to halve the margin of error from ±5 to ±2.5. Compute the new sample size and state the general relationship between precision and sample size. :::
Hypothesis testing
The procedure, in five steps:
- State the null hypothesis H₀ and the alternative H₁
- Choose the significance level, usually 5% or 1%
- Compute the test statistic
- Compare with the critical value
- State the conclusion in the context of the problem
Test statistic for a mean: z = (x̄ − μ) / (σ / √n)
One-tailed or two-tailed. "Has the mean changed?" is two-tailed, and the 5% is split between both tails, giving critical values of ±1.96. "Has the mean increased?" is one-tailed, with the whole 5% in one tail and a critical value of 1.645.
Two errors are possible:
| H₀ true | H₀ false | |
|---|---|---|
| Reject H₀ | Type I error | Correct |
| Do not reject | Correct | Type II error |
The significance level is the probability of a Type I error. Setting it at 1% rather than 5% reduces the risk of rejecting a true null — and increases the risk of a Type II error. The two cannot both be reduced except by increasing the sample size.
On wording. The correct conclusion is "there is insufficient evidence to reject H₀", not "H₀ is proved true". A test can fail to find evidence without establishing the opposite, and examiners mark that distinction.
:::checkpoint A test at the 5% level does not reject the null hypothesis that a machine fills to 500ml. A manager concludes the machine is proved accurate. Explain what is wrong with the conclusion. :::