Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman

14-18 CHAPTER 14 Bootstrap Methods and Permutation Tests
where SE boot is the bootstrap standard error for this statistic and t ∗ is
the critical value of the t(n − 1) distribution with area C between −t ∗
and t ∗ .
We want to estimate the 25% trimmed mean of the population of all
EXAMPLE 14.5
2002 Seattle real estate selling prices. Table 14.1 gives an SRS of size
n = 50. The software output above shows that the trimmed mean of this sample is
x 25% = 244 and that the bootstrap standard error of this statistic is SE boot = 16.83. A
95% confidence interval for the population trimmed mean is therefore
x 25% ± t ∗ SE boot = 244 ± (2.009)(16.83)
= 244 ± 33.81
= (210.19, 277.81)
Because Table D does not have entries for n − 1 = 49 degrees of freedom, we used
t ∗ = 2.009, the entry for 50 degrees of freedom.
We are 95% confident that the 25% trimmed mean (the mean of the middle 50%)
for the population of real estate sales in Seattle in 2002 is between $210,190 and
$277,810.
Bootstrapping to compare two groups
Two-sample problems (Section 7.2) are among the most common statistical
settings. In a two-sample problem, we wish to compare two populations, such
as male and female college students, based on separate samples from each
population. When both populations are roughly normal, the two-sample t procedures
compare the two population means. The bootstrap can also compare
two populations, without the normality condition and without the restriction
to comparison of means. The most important new idea is that bootstrap resampling
must mimic the “separate samples” design that produced the original
data.
BOOTSTRAP FOR COMPARING TWO POPULATIONS
Given independent SRSs of sizes n and m from two populations:
1. Draw a resample of size n with replacement from the first sample
and a separate resample of size m from the second sample. Compute a
statistic that compares the two groups, such as the difference between
the two sample means.
2. Repeat this resampling process hundreds of times.
3. Construct the bootstrap distribution of the statistic. Inspect its
shape, bias, and bootstrap standard error in the usual way.