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Explain what a confidence interval around a population mean tells you

CFA® Level I · Quantitative Methods · Estimation and Hypothesis Testing · QM.5.a

The idea

A confidence interval for a population mean is the sample mean plus or minus a reliability factor times the standard error, where the standard error is the sample standard deviation divided by the square root of n. The interval widens with dispersion and narrows with sample size, but only with the square root of it: quadrupling the sample halves the interval. Use z when the population variance is known and t when it is estimated, and t is always wider because estimating the variance costs precision. A 95% interval does not mean a 95% chance the mean sits inside this one; it means 95 of every 100 such intervals would contain it.

Formulas

Confidence interval for a population mean

x̄ ± z × (s ÷ √n)

x̄
the sample mean
z
the reliability factor — 1.96 for 95%, 2.58 for 99%
s ÷ √n
the standard error of the mean

Use t rather than z when the population variance is unknown and the sample is small. The interval narrows with √n, so four times the data halves the width.

A practice question

From the question bank, with the reason each wrong option is wrong.

The standard error of the sample mean is the sample standard deviation divided by:

  1. the square root of the sample size — correct
  2. the sample sizeWhy not: Dividing by n rather than √n would make precision improve far faster than it does — a sample of 100 would be ten times better than it is.
  3. the population meanWhy not: A location, not a measure of sample size, and generally unknown. The standard error does not involve it.

The square root is why precision improves slowly: quadrupling the sample only halves the standard error.

A harder one

A sample of 100 has a mean of 8% and a standard deviation of 20%. The standard error is:

  1. 0.2%Why not: 20% ÷ 100 — divided by n rather than √n. The square root of 100 is 10, giving 2%.
  2. 2% — correct
  3. 20%Why not: The standard deviation itself, which describes the spread of the observations. The standard error describes the spread of the *mean* and is always smaller.

Twenty per cent divided by the square root of 100 is 2%, which is the dispersion of the mean rather than of the data.

The hardest tier

A researcher keeps sampling until an interval excludes zero, then reports that interval. The stated confidence level is:

  1. correctWhy not: It would be correct for a sample size fixed in advance. Stopping when the result looks right selects the favourable samples, so the procedure being run is not the one the 95% describes.
  2. understatedWhy not: The wrong direction. Selecting on the outcome makes the interval look more reliable than it is, not less.
  3. overstated — correct

Stopping on a result changes the procedure, and the confidence level only describes the procedure that was specified in advance.

In the app: 18 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.

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