Confidence intervals
Judging an estimator told you how your estimator sits
relative to the parameter: where it centres, and how far it spreads. Your estimator came from
a random sample, so it has a distribution and would have come out differently on a different
sample.
Now use that to say how far the parameter is from what you got.
θ^±c⋅se(θ^)
- Estimate — what you computed
- Standard error — the spread, from page 02, with sample quantities substituted for the
unknown population ones
- c — a percentile, looked up in whatever distribution the estimator follows
What you write: “We are 95% confident that the true value of θ is between ℓ and
u.”
Not “there is a 95% probability that θ lies in (ℓ,u).” θ is fixed, so it
has no probability. The 95% is a property of the procedure: 95% of all possible samples of
size n yield an interval containing θ.
Finding c for the mean
Finding c for other parameters
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[Judging an estimator](02%20Judging%20an%20estimator.md) told you how your estimator sits
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relative to the parameter: where it centres, and how far it spreads. Your estimator came from
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a random sample, so it has a distribution and would have come out differently on a different
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sample.
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Now use that to say how far the parameter is from what you got.
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$$\hat\theta \pm c \cdot \mathrm{se}(\hat\theta)$$
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- **Estimate** — what you computed
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- **Standard error** — the spread, from page 02, with sample quantities substituted for the
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unknown population ones
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- **$c$** — a percentile, looked up in whatever distribution the estimator follows
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What you write: **"We are 95% confident that the true value of $\theta$ is between $\ell$ and
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$u$."**
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Not "there is a 95% probability that $\theta$ lies in $(\ell, u)$." $\theta$ is fixed, so it
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has no probability. The 95% is a property of the procedure: 95% of all possible samples of
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size $n$ yield an interval containing $\theta$.
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## Finding $c$ for the mean
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| Data | Condition | Use |
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|---|---|---|
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| Normal | $\sigma$ known | $z$ |
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| Normal | $\sigma$ unknown | $t(n-1)$ |
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| Unknown | $n$ large | $z$, by the CLT |
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| Unknown | $n$ small | bootstrap |
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## Finding $c$ for other parameters
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| Parameter | Distribution |
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|---|---|
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| Mean | $z$ or $t(n-1)$, per above |
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| Variance or $\sigma$ | $\chi^2(n-1)$, not symmetric, so two separate lookups |
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| Proportion | $z$, by the CLT |