A survey team wants its approximate proportion margin to be at most 0.04, or four percentage points. It has not collected the sample, so it does not know p̂. Predict how a planning calculation can proceed without pretending the future sample proportion is already available. A conservative variance bound provides one answer.
Predict
Predict the effect of halving the desired margin. Since uncertainty scales with the inverse square root of n, twice as many observations will not halve it. Also decide whether a calculated requirement of 600.25 permits a sample size of 600. A minimum requirement must be rounded upward.
Experiment
Using supplied z=1.96 and planning proportion p=0.5, require 1.96√(0.25/n)≤0.04. Squaring and rearranging gives n≥1.96²·0.25/0.04²=600.25, so choose at least 601 completed independent observations under the planning model.
Desired margin
Formula requirement before rounding
Minimum n
0.04
600.25
601
0.02
2401
2401
The near-fourfold ratio after rounding reflects the exact fourfold relationship before integer adjustments. Mark completed observations separately from people contacted.
Understand
The approximate planning formula is n≥z²p(1−p)/E², where E is the desired margin and p is a planning value, not the unknown true proportion. The formula rearranges the normal standard-error expression. It assumes a suitable sampling design and enough effective observations for the intended approximation.
For any proportion between zero and one, p(1−p)=1/4−(p−1/2)²≤1/4. Therefore p*=0.5 gives the largest variance factor and a conservative n within this formula. It does not guarantee actual interval coverage, representative sampling or freedom from measurement bias. It protects against choosing too small a variance factor when the proportion itself is uncertain.
A credible pilot or prior study may justify another planning value. If p* is near zero, the resulting n can be much smaller, but the pilot may be noisy or concern a different population. Conduct sensitivity calculations across plausible values rather than presenting one optimistic number as inevitable. Check that planned success and failure counts are adequate for the proposed interval method.
Nonresponse affects recruitment planning. If 601 completed responses are needed and the anticipated response probability is 0.8, 601/0.8=751.25 suggests at least 752 contacts for that expected response count. It does not guarantee 601 responses: the realized count varies. Nor does increasing invitations repair systematic differences between responders and nonresponders. Follow-up and a justified design remain necessary.
Cluster sampling, unequal weights and dependence can change precision relative to the simple independent model. A sample-size formula cannot be transplanted to those designs without accounting for their variance. Likewise, a small finite population may call for an explicit finite-population correction. Label the scope of this introductory calculation rather than implying that every survey needs the same n.
Apply
Write a recruitment brief with desired confidence, target margin, planning proportion, rounded completed-sample requirement and assumptions. Add a nonresponse sensitivity calculation. Explain which additional uncertainties are outside the margin, especially if the sampling frame omits relevant residents or an outcome definition is ambiguous.
Continue
Next change from estimating a range to evaluating a specified model. Hypotheses and tail direction must be chosen before the result is inspected.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Use z*=2, desired margin 0.05 and planning p*=0.5. What minimum integer n satisfies the normal planning bound?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A planning formula gives n≥384.16. What minimum integer sample size is required?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Halving desired margin while keeping other planning assumptions fixed requires:
Why is p*=0.5 conservative for this planning formula?
You need 240 completed responses and anticipate a 60% response rate. Plan a contact count and explain what it does not guarantee.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
A pilot suggests p≈0.1. Explain the tradeoff between planning with 0.1 and 0.5.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
