Separation between two Gaussian distributions

First item: Statistical relevance is not an absolute but a relative statement. The proper way of stating it is "this is stat. relevant on a $2sigma$ level ($approx 5%$). Therefore, there is nothing wrong in using a $4 sigma$ level. So it would help if you could state the exact sentence and context.

You second question I don't really understand.

Is the resolution considered a maximum error, even if we can plot the Gaussian distribution?

In your example the resolution is the smallest time duration between two packages, which allows to detect them as separated packages. I reckon the Rayleigh criterion could be used in your example. However, there are other definitions of resolution as well.

Assuming that we measure $N$ differences $Delta t = t_1-t_2$. Does this allow us to consider $4 sigma_T / sqrt{N}$, as the error on the mean?

Second item: I guess you are mixing two concepts here.

• statistical significant: The difference $Delta t$ is taken to be statistical significant on the $4sigma_T$ level.
• error of the mean: If the error of a single datapoint is $sigma_t$ then the error of the mean of $N$ datapoints is $sigma_t/sqrt{N}$. Be aware, that the resolution $sigma_T$ doesn't have to be equal to the error $sigma_t$: E.g. the resolution could be the resolution of the sensor (= a single component of the hole experiment), while the error could is the error made including all components of the experiment.

So, let's assume that $sigma_t = sigma_T$. Then,

• if you take $N$ (independent and identical distributed) datapoints $(Delta t_1, Delta t_2, ... Delta t_N)$,
• you can calc their mean value $langle Delta trangle = frac{1}{n} sum_j Delta t_j$
• The standard deviation of the mean value is $sigma_{langle Delta trangle} = sigma_t/sqrt{N}$
• Now you compare the mean value with its standard deviation. E.g. if the time differences are normally distributed you could calc the $z-value$ $$z = frac{langle Delta trangle - t_0}{sigma_{langle Delta trangle}} = frac{langle Delta trangle - t_0}{sigma_t/sqrt{N}}$$ and calc the significance level for $langle Delta trangle$ be to a given value $t_0$, by using the standardized normal distribution $N(0,1)$. You are probably interested in $t_0 = 0s$.