Chapter 14 - Bootstrap Methods and Permutation Tests - WH Freeman

14-24 CHAPTER 14 Bootstrap Methods and Permutation Tests
(a) What is the bootstrap estimate of the bias? Verify from the graphs of the
bootstrap distribution that the distribution is reasonably normal (some
right skew remains) and that the bias is small relative to the observed x.
The bootstrap t confidence interval for the population mean µ is therefore
justified.
(b) Give the 95% bootstrap t confidence interval for µ.
(c) The only difference between the bootstrap t and usual one-sample t confidence
intervals is that the bootstrap interval uses SE boot in place of the
formula-based standard error s/ √ n. What are the values of the two standard
errors? Give the usual t 95% interval and compare it with your interval
from (b).
14.10 Bootstrap distributions and quantities based on them differ randomly when
we repeat the resampling process. A key fact is that they do not differ very
much if we use a large number of resamples. Figure 14.7 shows one bootstrap
distribution for the trimmed mean selling price for Seattle real estate. Repeat
the resampling of the data in Table 14.1 to get another bootstrap distribution
for the trimmed mean.
(a) Plot the bootstrap distribution and compare it with Figure 14.7. Are the
two bootstrap distributions similar?
(b) What are the values of the mean statistic, bias, and bootstrap standard
error for your new bootstrap distribution? How do they compare with the
previous values given on page 14-16?
(c) Find the 95% bootstrap t confidence interval based on your bootstrap distribution.
Compare it with the previous result in Example 14.5.
14.11 For Example 14.5 we bootstrapped the 25% trimmed mean of the 50 selling
prices in Table 14.1. Another statistic whose sampling distribution is unfamiliar
to us is the standard deviation s. Bootstrap s for these data. Discuss
the shape and bias of the bootstrap distribution. Is the bootstrap t confidence
interval for the population standard deviation σ justified? If it is, give a 95%
confidence interval.
14.12 We will see in Section 14.3 that bootstrap methods often work poorly for the
median. To illustrate this, bootstrap the sample median of the 50 selling prices
in Table 14.1. Why is the bootstrap t confidence interval not justified?
14.13 We have a formula (page 488) for the standard error of x 1 − x 2 . This formula
does not depend on normality. How does this formula-based standard error
for the data of Example 14.6 compare with the bootstrap standard error?
14.14 Table 7.4 (page 491) gives the scores on a test of reading ability for two groups
of third-grade students. The treatment group used “directed reading activities”
and the control group followed the same curriculum without the
activities.
(a) Bootstrap the difference in means ¯x 1 − ¯x 2 and report the bootstrap standard
error.