Confidence Intervals

A confidence interval combines a sample estimate with a margin of error to describe plausible values for a population parameter.

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Statistical inference compares an estimate’s uncertainty with a stated claim. Four interval estimates appear against a vertical reference claim. Intervals that cross the line and intervals that miss it make compatibility with the reference visible without reducing uncertainty to a point estimate.

An estimate without uncertainty is incomplete; a confidence interval reports both a centre and a defensible range.

General structure

Many confidence intervals have form

\text{estimate}\pm\text{margin of error},

where

\text{margin of error}=\text{critical value}\times\text{standard error}.

The standard error measures how much the estimator would vary across repeated random samples.

Correct confidence interpretation

A $95\%$ confidence method is designed so that about $95\%$ of intervals produced over many repeated samples would contain the true parameter.

After one interval is computed, the parameter is fixed and the interval either contains it or does not. In introductory language, say “we are $95\%$ confident that the interval captures the population parameter,” not that $95\%$ of population values lie inside it.

Worked example

Statistical versus practical significance

An interval can show whether a null value is plausible and also reveal effect-size precision. A very small effect may be statistically distinguishable but practically unimportant; a wide interval may include both meaningful benefit and harm.

Interpret values in context, not only whether a threshold is crossed.

Common mistakes

Saying there is a $95\%$ chance the fixed parameter lies in this already-computed interval. Confidence belongs to the repeated method.

Saying $95\%$ of data lie between the bounds. The interval estimates a parameter.

Believing higher confidence narrows the interval. It widens it.

Believing large $n$ removes bias. It reduces sampling variability only.

Reporting bounds without population or conditions. Context defines the parameter.

Quick self-check

What population parameter is being estimated?
Was the sample collected in a way that supports inference?
Do method-specific independence, success/failure, or shape conditions hold?
Are estimate, standard error, critical value, and margin of error correct?
Is the confidence interpretation about the method and parameter—not individual data?
Are bias, practical importance, units, and population scope reported?