I have a few datasets of quality ratings for different producers (i.e. x1 ratings for producer y1, x2 ratings for producer y2 etc). The ratings are exponentially distributed (as shown by the histogram). The data sample sizes vary between 1000 and 100,000.

The differences between means of these data sets are relatively small, so I would like to test whether these differences are statistically significant. With my basic knowledge of statistics, I would assume that a t-test would be a good test statistic for that. However, most (but not all) texts on t-test usually assume normally distributed data. Is t-test applicable in my case, where i have exponentially distributed data? If not, what would be the appropriate test.