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A problem asked me to explain why a $99\%$ confidence interval of the difference of two sample means did not offer substantial evidence of a difference in the two populations. The interval was something like $(1,11)$ and clearly didn't include $0$.

I was under the impression that if $0$ is an unlikely difference value, then there is evidence that the two population means are different.

What am I not understanding? Did I possibly overlook some essential info?

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1 Answer 1

up vote 2 down vote accepted

Your concern is appropriate. A CI for the difference that does not include zero provides statistical evidence that the two means are different. This is true when some assumptions are verified, i.e. the two populations have homogeneous variances, the populations are normally distributed, and each value is sampled independently from each other value.

If no mention of these assumptions is specified in the problem, a possible justification for not having 'substantial evidence" of different means despite the lack of zero in the CI might depend on violations of these assumptions.

It should also be clarified whether the authors of the problem gave to the word "substantial" a particular meaning (e.g. in terms of difference magnitude, and so on).

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I don't think verification of assumptions is needed when drawing conclusions from the CI (such as that there is a significant difference between means), instead it is needed when calculating the CI. –  user133281 May 31 at 21:32
    
Verification of assumptions is needed for both: clearly when calculating the CI, but also to draw conclusions from CI. If the CI has been (inappropriately) calculated without verification, its meaning is questionable and no reliable conclusions can be drawn. –  Anatoly Jun 1 at 7:58
    
That you cannot draw conclusions when the CI is calculated without verification, means that you need verification when calculating the CI. It is a subtle point, but given a CI (which we of course assume is calculated correctly) conclusions can always be drawn. –  user133281 Jun 1 at 8:10

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