Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman

14-12 CHAPTER 14 Bootstrap Methods and Permutation Tests
SECTION 14.1
Exercises
Unless an exercise instructs you otherwise, use 1000 resamples for all bootstrap
exercises. S-PLUS uses 1000 resamples unless you ask for a different number.
Always save your bootstrap results so that you can use them again later.
14.1 To illustrate the bootstrap procedure, let’s bootstrap a small random subset of
the Verizon data:
3.12 0.00 1.57 19.67 0.22 2.20
(a) Sample with replacement from this initial SRS by rolling a die. Rolling a
1 means select the first member of the SRS, a 2 means select the second
member, and so on. (You can also use Table B of random digits, responding
only to digits 1 to 6.) Create 20 resamples of size n = 6.
(b) Calculate the sample mean for each of the resamples.
(c) Make a stemplot of the means of the 20 resamples. This is the bootstrap
distribution.
(d) Calculate the bootstrap standard error.
Inspecting the bootstrap distribution of a statistic helps us judge whether the
sampling distribution of the statistic is close to normal. Bootstrap the sample
mean x for each of the data sets in Exercises 14.2 to 14.5. Use a histogram and
normal quantile plot to assess normality of the bootstrap distribution. On the
basis of your work, do you expect the sampling distribution of xtobecloseto
normal? Save your bootstrap results for later analysis.
14.2 The distribution of the 60 IQ test scores in Table 1.3 (page 14) is roughly normal
(see Figure 1.5) and the sample size is large enough that we expect a normal
sampling distribution.
14.3 The distribution of the 64 amounts of oil in Exercise 1.33 (page 37) is strongly
skewed, but the sample size is large enough that the central limit theorem may
(or may not) result in a roughly normal sampling distribution.
14.4 The amounts of vitamin C in a random sample of 8 lots of corn soy blend
(Example 7.1, page 453) are
26 31 23 22 11 22 14 31
The distribution has no outliers, but we cannot assess normality from so small
asample.
14.5 The measurements of C-reactive protein in 40 children (Exercise 7.2, page
472) are very strongly skewed. We were hesitant to use t procedures for inference
from these data.
14.6 The “survival times” of machines before a breakdown and of cancer patients
after treatment are typically strongly right-skewed. Table 1.8 (page 38) gives
the survival times (in days) of 72 guinea pigs in a medical trial. 5