The groups are ((16 high improvement+16 low improvement)+30 control), so why is it (15), t(15), t(30), and then later t(16)? Does t(n) not mean that it's a t statistic over a population of n?
Not usually. Numbers in brackets after a well-known statistic normally represent parameters for that statistic's distribution; in the case of a t-test the bracketed number would be the number of degrees of freedom, which might be one less than the sample size (for a one-sample t-test) or two less than the sum of sample sizes (for an equal variances two-sample t-test).
(Disclaimer: I haven't read the paper.)
[Edited for unambiguity.]
Yes, that sounds familiar. Thanks.
Following up on the 2010 study, Jaeggi and University of Michigan people have run a Single N-back study on 60 or so children.
The abstract is confident and the mainstream coverage unquestioning of the basic claim. But reading it, the data did not seem very solid at all - I will forbear from describing my reservations exactly; I have been accused of being biased against n-backing, however, and I'd appreciate outside opinions, especially from people with expertise in the area.
(Background: Jaeggi 2011 in my DNB FAQ. Don't read it unless you can't render the above requested opinion, since it includes my criticisms.)