Confidence intervals, and the interval you would trade

STAT · Chapter 311 min readAsked at Two Sigma, Citadel, Optiver, SIG

Assumes The law of large numbers and the central limit theorem.

After this lesson you should be able to

  • Build an interval for a mean and for a proportion.
  • State what a confidence interval means, and what it does not.
  • Convert an interval into a market you would actually quote.

A confidence interval is an estimate with its uncertainty attached. The construction is mechanical; the interpretation is where candidates lose marks, and the connection to quoting a market is where trading firms are really going with the question.

Equation 3.1

An interval for a mean

Point estimate, plus or minus a multiple of the standard error. Use tt rather than zz when nn is small and the variance is estimated.

xˉ±zα/2 sn\bar{x} \pm z_{\alpha/2}\,\frac{s}{\sqrt{n}}
s/ns/\sqrt{n}
The standard error of the mean.
zα/2z_{\alpha/2}
The normal quantile: 1.645 for 90%, 1.96 for 95%, 2.576 for 99%.
Two-sided levelzOne-sided tailz
50%0.67425%0.674
80%1.28210%1.282
90%1.6455%1.645
95%1.9602.5%1.960
99%2.5760.5%2.576
Table 3.2 · The quantiles to know cold. Note that the two-sided 90% and the one-sided 5% share a number — the commonest place to slip.
-3-2-10123−1.961.96Standard errors from the estimate
Figure 3.3 · The interval you would actually quote. The shaded middle holds 95%95\%. Note what the picture does not say: the parameter is not random and does not live in a distribution. The interval is, and 95%95\% of the intervals built this way cover the truth.

Proposition 3.4

Intervals for a proportion

For a proportion the standard error is p(1−p)/n\sqrt{p(1-p)/n}, which is largest at p=0.5p = 0.5 and small near the extremes. A useful field approximation is that the 95% margin of error on a proportion is about 1/n1/\sqrt{n}, since 1.960.25≈0.981.96\sqrt{0.25} \approx 0.98.

Holds when

  • A poll of 1,000 has a margin of error of about ±3%\pm 3\%, which is where that familiar figure comes from.
  • The normal approximation breaks down when npnp or n(1−p)n(1-p) is below about 10; use an exact or Wilson interval there.

Common trap. Saying "there is a 95% probability the true mean lies in this interval". Under the frequentist reading the true mean is fixed and the interval is random, so the probability is either 0 or 1 once you have computed it. Instead. Say: "if I repeated this procedure many times, 95% of the intervals it produces would contain the true mean". If you want the first statement, you want a Bayesian credible interval, and saying so is a strong answer.

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