T-tests imply converting the raw mean difference to a "t" value, so that this can be compared to a pre-specified criterion for rejecting (or not) the null hypothesis. Isn't it basically the same thing that we do when obtaining a confidence interval (except that, in such cases, we try to estimate the raw data equivalents of pre-specified "z" scores, typically -1.96 and +1.96)? Isn't it more straightforward to work in the same metric as the raw data (as we do with confidence intervals) rather than converting the raw data to a non-intuitive "t" score? Furthermore, isn't it redundant to present both a t-test (with its associated p-value) and a confidence interval of the mean difference? What are the advantages of doing such a thing?
If you are just using the p-value to make a binary decision, then it offers no further information than whether the 1-alpha/2 CI overlaps with H_0, providing they are calculated in the same way.
That said, there is a lot of discussion and debate about the meaning of p-values, CIs, and how to use each of these (but I'll let someone else open that can of tarantula hawks).
t test give you the result of all data. While the confidence interval let you know which of the Raw data is not laying within the confidence interval.
The t-distribution approaches the z (normal distribution) with a greater sample size. If you have a good sample size that fullfills the practical assumptions of parametric estimation, then there are no differences.
The !New Statistics! of confidence interval estimation is, to a certain extent exactly that. Not so new, then.
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