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I have read that a bootstrap distribution does not emulate the distribution under the null, and as such, we need to recenter our means if we want to do hypothesis testing using bootstrap (i.e. by subtracting out the sample means and then adding back in the mean under the null). However, do we need to do this (bootstrapping under the null) also if we are calculating confidence intervals?

I generally do not see this done on posted examples showing bootstrapped confidence intervals, but my knowledge is not yet deep enough to ascertain if this is valid. So, also, if we don't have to bootstrap under the null for confidence intervals, why not?

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    $\begingroup$ Null hypotheses occur in hypothesis testing. There is no null for a confidence interval. There's a model of some kind, and some data. $\endgroup$
    – Glen_b
    Sep 25, 2019 at 0:13
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    $\begingroup$ @Glen_b I suppose I'm not able to write off the relationship so easily in my head. Could one not hypothesis test by seeing if the mean under the null fits in the confidence interval? Also don't we use null distributions (like t) to calculate critical values for confidence intervals (such as in a t test)? $\endgroup$
    – Josh
    Sep 25, 2019 at 2:21
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    $\begingroup$ @ Glen_b Ok I'm almost with you, but calculating confidence intervals off the t-distribution is throwing me off. Seems to me there that you use t-critical values to calculate a conf interval form the t-distribution, which I thought was emulating the distribution under the null. $\endgroup$
    – Josh
    Sep 25, 2019 at 13:12
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    $\begingroup$ @ Glen_b What's hanging me up is the t-test case I was mentioning and calculating confidence intervals there. In that you use a t-critical value, so a value from the t-distribution (CI = mean +- tcrit * standard error). My understanding was that the t-distribution was reflective of the t-statistic under the null. $\endgroup$
    – Josh
    Sep 25, 2019 at 15:17
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    $\begingroup$ @Glen_b I understand that I believe, but say I calculated a confidence interval in the context of a t-test-- those t-critical values calculated wouldn't just be from any old t-distribution, they would be taken specifically from the same one used to estimate the distribution under the null, symmetric and centered at 0 (my understanding). Reviewing though, I note the formula includes tcrit * standard error. Maybe I should interpret it as that takes it out of the realm of the null distribution and instead towards that of the sample/true pop (vs in bootstrapping, you're already there)? $\endgroup$
    – Josh
    Sep 26, 2019 at 12:38

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