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I'm a TA in stat for med students and when discussing confidence intervals together with the one-sample t-test, I got doubts on the decimals that should be retained. Specifically, we tested a data set of the body height of 3400 people against the null hypothesis of the average being equal to 170 cm. R gave a sample mean of 172.3974 with a confidence interval from 172.0846 to 172.7102. Sample SD was 9.8, the input heights were given in whole numbers.

A participant noted that giving seven significant figures would equal faking precision. I agree, but with such a narrow CI, should it really only be three like in the input data?

Random generation of 3400 body heights up to microns with the same mean and SD indicated that the first five significant figures of the upper and lower bound remained the same after cropping the four simulated decimals.

What's more, if both bounds of the interval rounded to the same integer, as could easily have been the case, the CI would be a single number.

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  • $\begingroup$ Why do you have an arbitrary null hypothesis about the mean height? Is it a hypothesis made up in order to be able to use a hypothesis test? If so then consider whether you really need to test a hypothesis. Most likely the estimation provided by the confidence interval is all that you will need. $\endgroup$ – Michael Lew Jan 13 '18 at 20:40
  • $\begingroup$ The hypothesis was made up to demonstrate the relationship between p values and confidence intervals, and a point was made that the latter are at times more intuitive. Pedagogically it would certainly be preferable to replace it with an H0 that corresponds to some actual prior belief in the future, but the same question could still arise. $\endgroup$ – bolzep Jan 14 '18 at 12:19
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The standard error of the mean (to 1 or 2 significant digits, I chose 2) is 9.8/sqrt(3400) = 0.17. The mean should be reported with the number of significant digits that makes it consistent with its standard error, i.e., 172.40. The result should be reported as 172.40 +/- 0.17 and you should identify it as being the estimated value of the height and its standard uncertainty. The Guide to Uncertainty in Measurement (GUM ) takes us this far. I have not seen explicit guidance on the number of significant digits to give in the upper and lower coverage intervals that are constructed with the mean, standard uncertainty, and coverage factor, but is seems reasonable to express them to the same number as the standard uncertainty, i.e., (172.07, 172.73) if we assume normality (coverage factor = 1.96) and want a 95% coverage interval.

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To report the confidence interval I would use one extra digit of precision beyond the precision in the data. If the data are reported to full centimetres, use one decimal for the confidence interval.

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  • $\begingroup$ The first paragraph is good advice, but the second paragraph is very sloppily worded and would support the dangerous conflation of statistical significance within a model and a real world truth. $\endgroup$ – Michael Lew Jan 13 '18 at 20:41
  • $\begingroup$ Is there any authoritative statement (like the GUM above, which doesn't seem to cover CIs) to refer to? $\endgroup$ – bolzep Jan 14 '18 at 20:40

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