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What is the reason behind the name "power of a test"? The use of the word "power" confuses me.

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    $\begingroup$ This may just be a matter of opinion. My take on this is that the higher 1-$\beta$ is the more powerful is the inference. $\endgroup$ – Michael R. Chernick Apr 28 '18 at 15:14
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    $\begingroup$ You probably need to read one of the early Neyman and Pearson papers to find out why they called it that. $\endgroup$ – mdewey Apr 28 '18 at 16:16
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    $\begingroup$ I read power in the sense of "power to distinguish" (...the actual situation from that under the null). $\endgroup$ – Glen_b -Reinstate Monica Apr 28 '18 at 22:49
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You should associate the "power" of a test with the test's ability to reject the null hypothesis. The quantity $1-\beta$ is the one minus the probability of a type II error, so $1-\beta$ is the probability that the test rejects the null hypothesis when in fact the alternative hypothesis is correct. The higher this number, the higher the test's "power" ... to reject the null.

Rejecting the null when the alternative is true is a desirable behavior for any hypothesis test. But if you haven't collected enough data, your test won't be powerful enough to reject the null, even if the state of the world is that the alternative is correct. When people mention that an experimental study is underpowered, it means that not enough data will have been collected to be able to decide between the null and alternative. Read more in this Wikipedia article.

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