Statistical Inference

Estimates, margin of error, and plausible values

A sample statistic, like a poll's reported percentage, estimates a true population value but is almost never exactly equal to it. A margin of error expresses how far the estimate might reasonably be from the truth. The interval of plausible values runs from the estimate minus the margin of error to the estimate plus the margin of error.

A poll of 500 voters showing 52% support with a margin of error of $\pm 4\%$ means the true population support plausibly falls between 48% and 56%. It does not mean 52% is the exact value. Use the margin of error and confidence conditions stated or implied by the problem; the interval is interpreted under that survey or sampling procedure.

Worked example. A poll of 600 people finds that 41% support a ballot measure, with a margin of error of $\pm 4\%$. Find the interval of plausible values, and decide whether 44% support is plausible.

$41\% - 4\% = 37\%$ and $41\% + 4\% = 45\%$, so the interval is 37% to 45%. Since $37\% \le 44\% \le 45\%$, 44% is plausible.

Checking a claimed value

Checking whether a claimed value is plausible means checking whether it falls inside the computed interval, nothing more. A value just outside the interval is not ruled out in a strict mathematical sense, but for this kind of question, "plausible" means inside the stated interval and a value outside it is treated as not supported by the survey.

The routine is always the same: compute both endpoints, then compare. Suppose a poll of 250 people finds 46% support with a margin of error of $\pm 5\%$, and someone claims 53% support. The interval is 41% to 51%, and 53% exceeds the upper bound, so the claim is not supported. The common error is to decide that 53% is "close enough" because it is only two points past the boundary. Apply the interval as a hard cutoff.

The same check answers "could a majority support it?" Compute the interval and see whether it contains values above 50%. If the upper endpoint is 49%, a majority is not plausible; if it is 54%, it is.

Sample size and margin of error

Margin of error shrinks as sample size grows: a larger random sample reduces sampling variability, producing a more precise estimate and a narrower interval, all else being equal. The relationship is inverse proportionality to the square root of the sample size. Multiplying the sample size by a given factor divides the margin of error by the square root of that factor. Quadrupling a sample cuts the margin of error in half; a ninefold increase cuts it to a third. Going the other direction, to cut a margin of error to one-fifth of its size, you need 25 times as many people.

Sample size and representativeness are not the same thing. A bigger sample does not automatically fix bias or make a sample representative. Representativeness depends on how the sample was selected, and a very large but poorly chosen sample can still give a confidently wrong picture of the population.

Sampling variability

Different random samples from the same population naturally produce different sample statistics, purely because of which individuals happened to be selected. This is sampling variability, not a mistake or a flaw in the method. A simulation that repeatedly draws random samples from a population with a known true proportion will show sample proportions clustering around that true value, with some natural spread. Larger samples give a tighter cluster. If a second poll of the same population gives 36% when the first gave 35%, nothing went wrong; that is exactly what randomness predicts.

Testing a claim with simulation

A simulation can also test whether a population claim is consistent with observed data. Start by assuming the claim is true, simulate many random samples under that assumption, and record how often a result at least as extreme as the observed one occurs.

• If such results occur only rarely, the observed result is evidence against the claim, and the claim can be rejected under the decision rule given in the problem.
• If the observed result is not unusual under the claim, the correct conclusion is to fail to reject the claim. Failing to reject does not prove the claim true; it means the sample did not give strong enough evidence against it.

Worked example. A claim states that 40% of a population favors a proposal. In 1,000 simulated samples built from that claim, a result at least as extreme as the observed sample result occurred only 12 times. What conclusion is best supported?

12 out of 1,000 is rare under the claim, so the observed data is evidence against it and the claim should be rejected under a reasonable decision rule. This does not prove the claim false; it says the data is not well explained by it.

Traps to avoid

• Subtracting the margin of error twice, or applying it on only one side.
• Calling a value plausible because it is "close" instead of checking the interval.
• Reversing the sample size relationship: bigger samples shrink the margin of error.
• Blaming ordinary sample-to-sample differences on mistakes instead of sampling variability.
• Treating a rare simulation result as support for the claim, or treating "fail to reject" as proof.