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May I ask why $95\%$ confidence is so commonly used? Does it have anything to do with $\frac{d}{d\alpha}e_n(\alpha)$, where $e_n(\alpha) = Z_{\alpha/2}\frac{S_n}{\sqrt n}$? (My professor asks me to evaluate this derivative at $\alpha = 0.05$, given $S_n = 4.7, n = 100$.)

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    $\begingroup$ It's arbitrary. $\endgroup$ – Mark Viola Mar 17 '18 at 20:11
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    $\begingroup$ I am 95% sure that the number is arbitrary :-). Maybe a vigesimal hangover? $\endgroup$ – copper.hat Mar 17 '18 at 20:13
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    $\begingroup$ There is a debate currently going on in various communities as to whether a lower $P$-value should become the tacit standard. $\endgroup$ – amd Mar 17 '18 at 22:57
  • $\begingroup$ A related CV question. $\endgroup$ – J. M. is a poor mathematician Mar 18 '18 at 9:29
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    $\begingroup$ Obligatory comic to show why it is stupid to just rely on 95% confidence in a world with publishing pressure! $\endgroup$ – user21820 Mar 18 '18 at 14:09
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From Wikipedia article 1.96 :

The use of this number in applied statistics can be traced to the influence of Ronald Fisher's classic textbook, Statistical Methods for Research Workers, first published in 1925:

"The value for which P = .05, or 1 in 20, is 1.96 or nearly 2 ; it is convenient to take this point as a limit in judging whether a deviation is to be considered significant or not."

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    $\begingroup$ It is worth noting that Fisher was working at the Rothamsted Experimental Station on crop research, so this $2$ standard deviation rule-of-thumb is likely to be something he had found practically useful $\endgroup$ – Henry Mar 17 '18 at 21:45
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$95\%$ is just the conventionally accepted boundary for "reasonably certain" in general cases. It has nothing to do with any specific formulas, and is rather an arbitrary choice that statisticians have agreed is a good compromise between getting results at all and getting results we can trust.

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    $\begingroup$ Though en.wikipedia.org/wiki/Replication_crisis indicates there might be good reason to shift to higher confidence intervals. Also: xkcd.com/882 $\endgroup$ – Daniel Schepler Mar 17 '18 at 20:45
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    $\begingroup$ @DanielSchepler "5% of all the research is actually wrong" is a mind-boglingly huge number, given how many papers are published every day. Imagine an enterprise that has the 1-to-20 ratio of rejects in its production. $\endgroup$ – Joker_vD Mar 18 '18 at 0:18
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    $\begingroup$ @Joker_vD: I think its even worse than that -- I imagine wrong results are much more likely to be publishable than right ones. $\endgroup$ – Hurkyl Mar 18 '18 at 1:42
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    $\begingroup$ @Joker_vD : Veritasium did a great YouTube video on this entitled "Why most research is wrong." Worth watching $\endgroup$ – DreamConspiracy Mar 18 '18 at 8:56
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    $\begingroup$ @JackAidley There isn't enough incentive in attempting to replicate other studies compared to making new science. $\endgroup$ – Arthur Mar 18 '18 at 14:00
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I don't think it is arbitrary because given a normal distribution

68.27% of all values lie within 1 standard deviation
95.45% of all values lie within 2 standard deviations and
99.73% of all values lie within 3 standard deviations
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  • $\begingroup$ But the multiples of standard deviation are themselves arbitrary. $\endgroup$ – JeremyC Mar 18 '18 at 16:12
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    $\begingroup$ @JeremyC: Indeed. Why not consider the percent of values within $\frac{2154}{1099}$ many standard deviations? $\endgroup$ – Dave L. Renfro Mar 18 '18 at 16:29

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