From Algorithms to Z-Scores - matloff - University of California, Davis

204 CHAPTER 10. INTRODUCTION TO CONFIDENCE INTERVALS
Definition 28 Suppose θ is a sample-based estimator of a population quantity θ. 6 The samplebased
estimate of the standard deviation of θ is called the standard error of θ.
We can see from (10.22) what to do in general:
Suppose θ is a sample-based estimator of a population quantity θ, and that, due to
being composed of sums or some other reason, θ is approximately normally distributed.
Then the quantity
has an approximate N(0,1) distribution. 7
θ − θ
s.e.( θ)
(10.23)
That means we can mimic the derivation that led to (10.22), showing that an approximate
95% confidence interval for θ is
In other words, the margin of error is 1.96 s.e.( θ).
θ ± 1.96 · s.e.( θ) (10.24)
The standard error of the estimate is one of the most commonly-used quantities in
statistical applications. You will encounter it frequently in the output of R, for instance,
and in the subsequent portions of this book. Make sure you understand what
it means and how it is used.
10.6 Example: Standard Errors of Combined Estimators
Here is further chance to exercise your skills in the mailing tubes regarding variance.
Suppose we have two population values to estimate, ω and γ, and that we are also interested in
the quantity ω + 2γ. We’ll estimate the latter with ˆω + 2ˆγ. Suppose the standard errors of ˆω and
ˆγ turn out to be 3.2 and 8.8, respectively, and that the two estimators or independent. Let’s find
the standard error of ˆω + 2ˆγ.
We have (make sure you can supply the reasons)
6 The quantity is pronounced “theta-hat.” The “hat” symbol is traditional for “estimate of.”
7 This also presumes that θ is a consistent estimator of θ, meaning that θ converges to θ as n → ∞. There are
some other technical issues at work here, but they are beyond the scope of this book.