# Hypothesis testing on dataset with probability values

I have two samples at time $t-1$ and $t$ s.t:

Sample1: $P_1 , P_2 , .... , P_n$, where $P_n$ is the probability of a disaster to occur in 5 years from now, as estimated at time $t-1$

Sample2: $P'_1, P'_2 , ... ,P'_n$ , where $P'_n$ the probability of a disaster to occur in 5 years from now, as estimated at time $t$

I want to test the whether the mean probability has increased statistically significant from $t-1$ to $t$.

$H_0 = \bar{P'}-\bar{P} =0$

$H_1= \bar{P'}-\bar{P} > 0$

Where, $\bar{P'} = \frac{1}{n}$$\sum_{i=1}^{n} P'_i and \bar{P} = \frac{1}{n}$$\sum_{i=1}^{n} P_i$

Is there a specific test to determine the above? My question is whether the upper test embodies the assymetry across values. For example, some probabilities are close to 0 and other close to 1. On the other hand, taking the percentage changes yields to extreme values along the dataset.

• Could you please add some context? What catasthrophies? How was the probabilities estimated/evaluated? Do you know the eventual truth? Jul 10, 2018 at 13:21
• The probabilities are estimated and not evaluated. I want to check the impact of earthquakes in probability of default for some firms. I have a sample before the event and one after. I do not care about the realized impact as I examine only the estimate of the model ( a structural model in Credit Risk management). One thought passed by is to normalize the data to [0,1] and conduct a one sample t-test for the log change or a one sample t test between the mean of the probabilities after normalization. Jul 10, 2018 at 18:55