T-tests are for normal or close-to-normal distributions. You need a nonparametric test. Since you have the same two groups being assessed twice, I don't think you can use the McNemar test for dependent proportions. However, maybe you could do a chi-square test on the increases within each group as fractions of the sample sizes. E.g., if group A had 5/100 develop the illness and group B had 12/100, you would run the chi-square test using the numbers 5, 95, 12, and 88.
The p-value of a hypothesis test or a corresponding confidence interval depends on the treatment or choice of 2 issues:
1. Treatment of nuisance parameter
To preserve the size at the exact level, the type 1 error needs to be less than or equal to alpha for all possible values of the nuisance parameter.
The null hypothesis that the rates are equal, does not constrain the value itself, so it is a nuisance parameter.
Conditional tests like the exact conditional Poisson test or Fisher's exact test remove the nuisance parameter by conditioning on a summary statistic.
Unconditional exact test need to assert that the size is correct by using the max or sup over all possible values of the nuisance parameter.
Berger-Boos test limits the space of nuisance parameter for the max but adds a factor to make it exact, i.e. preserves the size alpha.
The Poisson E-test is not an exact test in this sense. It uses the "exact" distribution but it uses the estimated value of the nuisance parameter.
2. Location of two-sided rejection region
A two-sided test has rejection region in both lower and upper tail. The requirement that the sixe of the test is at most alpha, is a requirement on the probability to be in one of the tails, but it does not pin down the probability in each tail separately.
"central" or equal tail methods limit the probability of the each tail to be less than or equal to half the size, alpha / 2.
"minlike" uses the likelihood value (based on likelihood ratio test) to find the non-rejection region. The corresponding profile confidence interval will not have equal tails in skewed distributions like Poisson or Binomial.
One point that Michael P. Fay points out and emphasizes is the hypothesis tests and confidence interval are often not consistent with each other.
For example the exact poisson test in R uses "minlike" hypothesis test and exact pvalue, but reports exact "central" or equal-tail confidence intervals.
In one-sided test, the location is fixed and this distinction between "minlike" and "central" becomes irrelevant. Because there is only one tail, an exact test needs to preserve the size for that tail at level alpha.
Best Answer
I very much doubt that the differences will be statistically significant, as there is a lot of inter-annual variability in temperatures (especially for station level data), so you will need large deviations for a statistically significant difference from a small sample of measurements. Essentially the statistical power of the test is likely to be so low that the test is not useful.
Can you give some explanation of the purpose of the study? What hypothesis are you investigating?