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I have a negative confidence interval ((-35.346,-8.570) for an independent samples t-test.

I understand confidence intervals for the mean. However, I am uncertain about the interpretation of the confidence interval for a difference in means.

Is this an accurate interpretation? The confidence interval for mean difference in life satisfaction for the two groups is (-35.346,-8.570); as this interval does not contain zero, I can be confident that I have used a method that that will produce significantly different or unequal population means 95% of the time.

Or should I interpret the CI as follows: The confidence interval for mean difference in life satisfaction for the two groups is (-35.346,-8.570); as this interval does not contain zero, I can be 95% confident that the population means are significantly different or unequal.

Is this splitting hairs?

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  • $\begingroup$ I would interpret it as 'The population estimate for the difference of the means is statistically significant at the 5% level' $\endgroup$
    – Conta
    Commented Feb 1, 2016 at 14:10
  • $\begingroup$ Thank you. I would interpret this similarly. However, I need to address the CI specifically. $\endgroup$
    – Katja Buckley
    Commented Feb 4, 2016 at 1:20
  • $\begingroup$ @KatjaBuckley why do you think you’re not? $\endgroup$
    – Dave
    Commented Apr 8, 2020 at 23:03

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The first interpretation is incorrect, but the second one is correct.

What will happen in multiple studies is dependent on the power and the true difference, not the confidence interval, so saying that something will happen 95% of the time based on this interval is incorrect.

The second paragraph is a correct interpretation, further you can conclude that the 2nd group has a higher mean than the first group (the entire interval is negative).

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By this confidence interval you are confident that if you draw random samples 100 times that in 95 cases (95%) the mean of the second sample is statistically signifanct larger than the mean of the first sample. Hence, you are 95% confident that the mean of the second sample's population is larger than the mean of the first sample's population. This is, since the test adresses to draw inference from the samples to the population. Therefore you can also interpret that the two samples do not come from the same population since they have different means.

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  • $\begingroup$ So you think that if the true difference in the means is around -35 (a value in the interval) that you would still see 5 out of 100 new samples with mean 1 higher than (or at least not statistically significantly lower than) mean 2? You also think that if the true difference in means is -10 (also in the interval, but much closer to 0), that you will only see 5 out of 100 new samples with mean 1 not significantly lower than mean 2? Your interpretation is based on power (and the true mean difference), not the confidence interval. $\endgroup$
    – Greg Snow
    Commented Apr 8, 2020 at 23:43
  • $\begingroup$ The idea of confidence intervals is to say P(C_l <= theta <= C_u) >= 1-alpha. In other words, in 1-alpha * 100 % of the cases the confidence interval encloses the true parameter. However, since we draw random samples, there is a probability of getting confidence intervals which do not enclose the true paramter (e.g.: [5, 10] or whatsoever). We do not know how far or close these are, but the probability of getting such sample mean/differences s is there. $\endgroup$
    – Erin Sprünken
    Commented Apr 9, 2020 at 0:00
  • $\begingroup$ But, indeed, I should have written my answer in another form, I see that there's a problem there. $\endgroup$
    – Erin Sprünken
    Commented Apr 9, 2020 at 0:02
  • $\begingroup$ Einstein said something about how things should be explained as simply as possible, but no simpler. Interpreting confidence intervals is one thing that is very hard to find that balance. It is very easy in the pursuit of making things simple enough to understand to make the explanation too simple and therefore potentially misleading. $\endgroup$
    – Greg Snow
    Commented Apr 9, 2020 at 17:48

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