Standard Error Statistics Example 4

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Example 4

hard
A polling company wants the standard error of a proportion to be no more than 0.02 (2%). If a preliminary estimate suggests p^≈0.5\hat{p} \approx 0.5, what minimum sample size is needed? Use SE=p^(1−p^)nSE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.

Solution

  1. 1
    Step 1: Set up: 0.02=0.5×0.5n=0.25n0.02 = \sqrt{\frac{0.5 \times 0.5}{n}} = \sqrt{\frac{0.25}{n}}. Square both sides: 0.0004=0.25n0.0004 = \frac{0.25}{n}.
  2. 2
    Step 2: Solve: n=0.250.0004=625n = \frac{0.25}{0.0004} = 625. A minimum sample size of 625 is needed.

Answer

A minimum sample size of n=625n = 625 is needed for a standard error of at most 2%.
Sample size determination is a practical application of standard error. Using p^=0.5\hat{p} = 0.5 gives the most conservative (largest) sample size estimate because p^(1−p^)\hat{p}(1-\hat{p}) is maximised at p=0.5p = 0.5. This ensures the precision goal is met regardless of the true proportion.

About Standard Error

The standard error (SE) is the standard deviation of a sampling distribution, measuring how much a sample statistic (like the sample mean) typically varies from the true population parameter across repeated samples. It decreases as sample size increases.

Learn more about Standard Error →

More Standard Error Examples

Example 1 easy

A population has a standard deviation of [formula]. If you take a random sample of [formula], what i

Example 2 medium

How does the standard error change when you quadruple the sample size from [formula] to [formula]? A

Example 3 medium

A researcher measures the reaction time of 64 participants and finds a sample standard deviation of

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