Statistical inference—either confidence
intervals or hypothesis testing. Both try to infer what
is in the box (population) when have only information
from a sample (drawings from the box).
Steps in deriving a confidence interval.

1. Draw the sample, compute the sample mean and
standard deviation. (A percentage is a special sort of
mean.)

2. Use the mean of the sample to estimate the mean of the
population.

3. Use the standard deviation of the sample to estimate
the standard deviation of the population. (Textbook's
bootstrap.)

4. Use the estimate of the standard deviation of the
population to compute an estimated standard error of the
mean (or sum or percentage).

5. Decide how confident we want to be in the estimate.
You cannot be 100% confident—it must be less. Use
this confidence as an area on the normal curve and find a
z-score.

6. Your confidence interval is:

Sample average plus or minus z-score times
standard error of the mean.

(We will alter 5 eventually and use the t-tables
instead. It corrects for the error we introduce when we
estimate the standard deviation of the population from
the standard deviation of the sample. As the sample gets
bigger, the error should get less, and the t-table gets
increasingly close to the normal curve.)