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The latter would be employed, for example, if one wanted a BCa confidence interval for the difference in means of two samples. bca(boot) Stata Once the height data is entered, the following line of code invokes the bootstrap to produce a 95% confidence interval for the interquartile deviation, plus a point estimate of its bias. 4 The Bootstrap-t Suppose the hypothesis we wish to test is that θ = θ0 , and that θˆ is our estimator. The accuracy of bootstrap confidence intervals for testing purposes can be improved by using the distribution of the differences θˆ ∗ − θˆ rather than the distribution of the differences θˆ ∗ − θ0 according to Hall and Wilson [1991].

In this crossover trial, each of eight patients received in random order each of the following: • • • Patch containing hormone that was manufactured at the old site Patch containing hormone that was manufactured at the new site Patch without hormone (placebo) that was manufactured at the new site. 20 where θ = E(new) − E(old) and μ = E(old) − E(placebo). The natural estimate for θ is the average of the old-patch hormone level in patients minus the average of the new-patch hormone level in patients.

The result is a confidence interval whose lower bound is 52 cm2 and whose upper bound is 95 cm2 . 7 cm2 is neither more nor less likely than any other value in the interval [52,95]. 1 Limitations of the Percentile Bootstrap Confidence Interval Almost immediately, two questions arise about the confidence interval we just derived: 1. Is it accurate? That is, is it correct to say that 90% of the time we expect this interval to cover the true value of the population variance? Or is it really a smaller percentage?

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