here, and the reason why Macauley wanted these limits in the first place, is that for this
memory test, the lower bound of what is classed as “normal functioning” is a score of 10.
The confidence interval does include 10 as its upper limit, and so we cannot reject the
null hypothesis that people in this age group, on average, fall in the normal range. An
examination of the sampling distribution reinforces this view, and perhaps gives us a
more complete understanding of the performance of this age cohort. The fact that there
are a number of individuals whose scores are well below 10 might lead us to seek a dif-
ferent confidence interval, that being limits on the proportionof people in that age
group who fall below 10. While that would be a perfectly legitimate use of bootstrap-
ping for these data, we will not pursue that question here.
This may not seem like the most inspiring example of bootstrapping, because it makes
bootstrapping look rather imprecise. It is a good example nonetheless, because it reflects
the sometimes awkward nature of real data. As we will see, however, not all data lead to
such discrete distributions. In addition, the discreteness of the result is inherent in the data,
not merely in the process itself. If we drew 10,000 samples from this population and calcu-
lated tvalues, the resulting tdistribution would be almost as discrete. The problem comes
from drawing samples from a distribution with a limited number of different values, in-
stead of modeling the results of drawing from continuous (e.g., normal) distributions. If it
is not reasonable to assume normality, it is not reasonable to draw from normal distribu-
tions just to get a prettier graph.
18.3 Resampling with Two Paired Samples
We will now move from the bootstrap, where we drew large numbers of samples from a
pseudo-population using sampling with replacement, to randomization, or permutation,
procedures that involve taking the full set of observations and randomly shuffling them and
assigning them to conditions randomly.
Section 18.3 Bootstrapping with Two Paired Sample 665
Figure 18.3 Histogram of the results displayed in Figure 18.2
