Margin of Error and Sample Size

12.3: Margin of Error and Sample Size. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 3 of 63 parts Practice Flashcards

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Margin-of-error interval

Objective: Form an interval around a stated estimate.

Prerequisite skills: percentages, interval interpretation, and random samples. An estimate reported as e ± m gives an interval from e−m to e+m in the same units. A percentage-point margin is added to a percentage directly; it is not a percent multiplication of that estimate. Common error: treating percentage points as a percentage of the estimate. Restrictions and interpretation: A reported margin of error describes sampling uncertainty under stated conditions, not every source of error or an individual-outcome range.

Formulas

$interval=[estimate−margin, estimate+margin]$

Worked examples

Try itA survey reports support of 61% with a margin of error of 3 percentage points. What is the lower endpoint in percent?Show answer

The lower endpoint is 61−3=58%. The margin is in percentage points, so do not multiply 61 by 0.97. Therefore the requested answer is 58.

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Margin-of-error interval

Objective: Form an interval around a stated estimate.

Formulas

$interval=[estimate−margin, estimate+margin]$

Worked examples

Try itA survey reports support of 61% with a margin of error of 3 percentage points. What is the lower endpoint in percent?Show answer

The lower endpoint is 61−3=58%. The margin is in percentage points, so do not multiply 61 by 0.97. Therefore the requested answer is 58.

Sample size and precision

Objective: Identify the effect of a larger random sample under similar conditions.

For comparable random samples and the same estimation procedure, larger sample size generally reduces sampling variability and margin of error. It does not guarantee a closer individual estimate or eliminate bias. Common error: claiming a larger sample removes nonresponse or measurement bias. Restrictions and interpretation: A reported margin of error describes sampling uncertainty under stated conditions, not every source of error or an individual-outcome range.

Formulas

rough scaling: margin ∝ 1/√n under comparable conditions

Worked examples

Try itTwo otherwise comparable random surveys use the same method and confidence level. Survey B has 2600 respondents, more than survey A’s 100. Which is generally expected?Show answer

A larger random sample generally reduces sampling variability and thus margin of error under comparable conditions. It cannot guarantee any particular realized error or remove non-sampling bias. Therefore the requested answer is Survey B has a smaller margin of error.

Compare estimates with uncertainty

Objective: Use reported intervals to judge whether a claimed difference is supported by the supplied information.

Reported intervals show the range of values compatible with each estimate under its method. Overlap alone is not a universal significance test, but it shows that simply comparing point estimates can overstate what the supplied intervals establish. Common error: asserting a population ordering solely from two overlapping point estimates. Restrictions and interpretation: A reported margin of error describes sampling uncertainty under stated conditions, not every source of error or an individual-outcome range. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

compatible interval endpoint = estimate ± reported margin

Worked examples

Try itA poll estimates option A support at 68% ± 4 points and option B at 70% ± 4 points. Based only on those overlapping reported intervals, which statement is best?Show answer

The intervals overlap: A ranges from 64% to 72%, while B ranges from 66% to 74%. The point estimates alone do not prove a definite ordering. A formal difference analysis would require additional information. Therefore the requested answer is The point estimates differ, but the intervals alone do not establish a definite population ordering.

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