:1a. Solution. Since this problem involves classifying and
counting, we mark each ticket in the original box with a 0 or a
1. The tickets with a 4 on them are marked with a 1, the others
with a 0. The number of 4's in the sample from the original box
is like the sum of 100 draws from the new box. To find the SD of
the new box, we need to know the fraction of 4's in the original
box, but this is unknown. We estimate it as .2, the fraction of
4's in the sample. Similarly, the fraction of non-4's in the
original box is estimated as .8. The SD of the new box is
estimated by the bootstrap method as the square root of .2 x.8,
or .4. The SE for the number of 4's in a sample of 100 is
estimated as 10 x .4 = 4. This is 4% of 100, the size of the
sample, so the SE for the percentage of 4's in the sample is
estimated as 4.
The estimate of the percentage of 4's in the box is the
percentage of 4's in the sample, which is 20%. The lower
endpoint of the 99.7% confidence interval is three SE's - as just
computed - below this estimate, or 20% - (3 x 4%) = 20% - 12% =
8%.