Class summary: Friday, 2/21
I finished the confidence interval example from last time, working out
a 98 percent confidence interval for the number of sixes in n rolls of
a die, for n=600 and for n=6,000,000. These numerical examples
illustrate a general phenomenon, the square
root law. This "law" is a rule of thumb saying that, as the value of
n increases, the following occur:
- The absolute width of a confidence interval
(corresponding to a fixed level of confidence, e.g., 95 %)
for the number of successes in n trials
increases at a
rate proportional to sqrt{n}.
- The relative width of such a
confidence interval (expressed as a percentage of n, or of its center,
mu) decreases at a rate proportional to 1/sqrt{n}.
-
The probability for
getting exactly [mu] successes (equivalently,
the maximal "height" of the binomial distribution) descreases at a rate
proportional to 1/sqrt{n}.
The square root law is a rule of thumb that enables you to do quick
calculations in your head and ballpark estimates. However, since it
is not a well-defined recipe or specific formula, it cannot be
used to solve concrete problems; for that, you still need to use the
formulas for normal approximation.
In the final part of the hour, I discussed an application of normal
approximation
to opinion polls
involving a single yes/no question. Such a poll can be modelled by
assuming each voter in the polling sample represents
a trial in an S/F trial sequence, with success meaning a "yes" vote,
failure a "no" vote, and the success probability equal to the (unknown)
proportion p of "yes" votes among the entire population. From the polling
results, one knows the proportion "p-hat" of "yes" votes among the polling
sample,
and the goal
is to determine an interval for the unknown proportion p,
with a prescribed level of confidence.
Since p-hat is the number of successes in the model divided by n, this
can be rephrased in terms of confidence intervals for the number of
successes.
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