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Image a set of test statistics like:-

histogram

For the purposes of this question, can we assume that Z and p values couldn't be derived. (I know that p can via a cumulative distribution function).

As I understand it, a p value indicates the likelihood of the null hypothesis being correct within certain bounds. So p = 0.00000003 suggests the alternate hypothesis, whist p = 0.4 suggests the null hypothesis.

Q. What advantage does a p value give us, over stating that the test statistic was say 0.0001 within a possible range of -0.0006 to +0.0006?

Paul Uszak
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    p = 0.4 suggests the null hypothesis. Actually, p values like that mean there's not enough information to decide whether the null hypothesis is true or false. People do give statistics as "x plus or minus y" for laymen, but statisticians prefer to know how many standard deviations y is. Usually plus or minus means 2 standard deviations or a 95% confidence level. – Barry Carter Jul 21 '22 at 12:48
  • @barrycarter You comment ties in perfectly with my question. Since I can't fit a curve to the histogram (it's empirical), how can I calculate a std.dev? The above curve has no algebraic form, other than via gross approximation. That's why I ask whether if it's under the curve, Ho is true, and if it isn't, Ho is rejected. – Paul Uszak Jul 21 '22 at 13:51
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    Re " a p value indicates the likelihood of the null hypothesis being correct within certain bounds": reading some of our threads on p-values, such as https://stats.stackexchange.com/questions/31 , might help fix this misinterpretation and bring you closer to understanding your situation. – whuber Jul 21 '22 at 13:56
  • In theory you could compute an SD and mean for most probability distributions including this one, but you'd have to do some weird pixel stuff. Or you could just pixel measure your 95% interval. To be pedantic to the point of insanity, that function does have an algebraic definition because it's only defined on a finite number of pixels. – Barry Carter Jul 21 '22 at 14:49

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