Here we will demonstrate bootstrap confidence intervals for the bootstrap percentile method.

**BOOTSTRAP PERCENTILE METHOD TEST**

Previously, we considered one of the simplest forms
for approximate bootstrap con-fidence intervals, namely, Efron’s percentile
method. Although there are many oth-er ways to generate bootstrap type
confidence intervals, such methods are beyond the scope of this text. Some
methods have better properties than the percentile method. To learn more about
them, see Chernick (1999), Efron and Tibshirani (1993), or Carpenter and
Bithell (2000). However, the relationship given in the previous section tells
us that for any such confidence interval we can construct a hy-pothesis test
through the one-to-one correspondence principle. Here we will demonstrate
bootstrap confidence intervals for the bootstrap percentile method.

Recall that in Section 8.9 we had the following ten
values for blood loss for the pigs in the treatment group: 543, 666, 455, 823,
1716, 797, 2828, 1251, 702, and 1078. The sample mean was 1085.9. Using the
Resampling Stats software, we found (based on 10,000 bootstrap samples) that an
approximate two-sided per centile method 95% confidence interval for the
population mean μ was [727.1, 1558.9].

From this information, we can construct a bootstrap
hypothesis test of the null hypothesis that the mean μ = μ_{0}, versus the two-sided alternative that μ_{ }≠ μ_{0}. The test rejects the null hypothesis if the
hypothesized μ_{0} < 727.1 or if the hypothesized μ_{0} > 1558.9. We will know μ_{0} and the result depends on whether or not μ_{0} is in the confidence interval. Recall we reject *H*_{0} if μ_{0} is outside the interval.

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