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WAMath MethodsSyllabus dot point

What controls the width of a confidence interval, and how large must a sample be to achieve a required precision?

Relate the margin of error to confidence level and sample size, and determine the sample size needed for a required margin of error

WACE Year 12 Mathematics Methods Unit 4 margin of error and sample size: the margin of error formula, how confidence level and sample size affect width, and solving for the sample size needed for a required precision, with worked examples.

Generated by Claude Opus 4.76 min answer

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  1. What this dot point is asking
  2. The margin of error
  3. The square-root relationship
  4. Determining the sample size

What this dot point is asking

SCSA Unit 4 develops the margin of error as the half-width of a confidence interval for a proportion. This dot point asks you to relate it to the confidence level and sample size, and to find the sample size required for a given precision. It is examined in both sections, with sample-size calculations common in the calculator-assumed section.

The margin of error

A confidence interval for a proportion has the form p^±E\hat{p}\pm E, where EE is the margin of error.

Two levers control EE: the confidence level through zz, and the sample size through nn. A higher confidence level needs a larger zz and so a wider interval. A larger sample shrinks the standard error and so narrows the interval.

The square-root relationship

Because EE depends on 1n\dfrac{1}{\sqrt{n}}, precision improves slowly with sample size. To halve the margin of error you must roughly quadruple nn, since 4=2\sqrt{4}=2. This is why large gains in precision become expensive.

Determining the sample size

Rearranging the margin-of-error formula for nn lets you find the sample size needed for a target margin EE.