Testing for differences in means within individuals over time

I'm currently looking at a paper involving panel data and intend to run a similar test on a different data set. However, I'm having a difficult time understanding exactly what test the authors performed to get their results. I have 12 years of panel data and want to compare the average of the first X years to the average of the last X years for each individual in the panel. I then plan on summarizing the data to show how many individuals fell above/below the relevant test statistic. The authors simply said they used a dependent t-test, but could anyone provide more detail on what test is appropriate here and what the relevant test statistic would be to compare against?

Note however that the t-test is a parametric test, as it assumes that the observed differences between the mean of the first $x$ years and the last $x$ years are a sample from normal distribution. The approximate validity of the confidence interval and hpypothesis test follows from the Central Limit Theorem. If the sample size is small and/or the true distribution of the differences is far from normal, the test may not be very reliable.