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

10.5. GENERAL FORMATION OF CONFIDENCE INTERVALS FROM APPROXIMATELY NORMAL ESTIM
equivalent:
• “µ is in the interval”
• “the interval contains µ”
So it is ridiculous to say that the first is incorrect. Yet many instructors of statistics say so.
Where did this craziness come from? Well, way back in the early days of statistics, some instructor
was afraid that a statement like “The probability is 95% that µ is in the interval” would make
it sound like µ is a random variable. Granted, that was a legitimate fear, because µ is not a
random variable, and without proper warning, some learners of statistics might think incorrectly.
The random entity is the interval (both its center and radius), not µ. This is clear in our program
above—the 10 is constant, while wbar and s vary from interval to interval.
So, it was reasonable for teachers to warn students not to think µ is a random variable. But later
on, some idiot must have then decided that it is incorrect to say “µ is in the interval,” and other
idiots then followed suit. They continue to this day, sadly.
10.5 General Formation of Confidence Intervals from Approximately
Normal Estimators
Recall that the idea of a confidence interval is really simple: We report our estimate, plus or minus
a margin of error. In (10.22),
margin of error = 1.96× estimated standard deviation of W = 1.96 × s
√ n
Remember, W is a random variable. In our Davis people example, each line of the notebook would
correspond to a different sample of 1000 people, and thus each line would have a different value for
W . Thus it makes sense to talk about V ar(W ), and to refer to the square root of that quantity,
i.e. the standard deviation of W .
In (10.13), we found the latter to be σ/ √ n and decided to estimate it by s/ √ n. The latter is called
the standard error of the estimate (or just standard error, s.e.), meaning the estimate of
the standard deviation of the estimate W . (The word estimate was used twice in the preceding
sentence. Make sure to understand the two different settings that they apply to.)
That gives us a general way to form confidence intervals, as long as we use approximately normally
distributed estimators: