Comparing two populations - which approach to use? Can someone please guide me on the method that I need to use for following scenario:


*

*We have two systems via which customer report their payments every quarter.

*I have data from these two systems with customer & day of quarter they report their payment

*We want to compare and see if Any one of the system really results in customer reporting sooner or not? Or are both systems equal.


Would comparing these two samples via their mean would be the right approach here? What I mean here is:


*

*null hypothesis that both mean are equal to 0. 

*Alternative hypothesis that they are not equal to 0.

*I will set significance level of 5%, two tailed test with 2.5% on either end

*Calculate z-score for significance level

*Calculate z-score for the sample data using the formula (mean1 - mean2)/standard error

*Finally, compare the two z-scores


Does the above approach sound correct? I fairly new to the world of statistics, so appreciate your guidance here.
Thanks in advance,
Chintan
 A: If I understand correctly, what you are looking for is a two-sample test for equal means - which means you want the test if the mean if the first population, $\mu_1$, is significantly different from the mean of the second population, $\mu_2$.
Additionally, I understand your problem as being univariate (both variables have only one dimension, "1 data vector")
Lastly, the data is not "tied"/paired in any manner - you don't have observations for the same instance in both populations
If that is correct, you are looking for an unpaired two-sample test with the null Hypothesis $\mu_1 = \mu_2$ with alternative $\mu_1 \neq \mu_2$. Two are usually used for this scenario:
Mathematical setup, you also have:


*

*$\sigma_1$, $\sigma_2$: Estimated standard errors of the populations

*$n_1$, $n_2$: Sample sizes for each population
You have two options


*

*Case 1. Assumption: Your two populations are normally distributed. Approach: Two-sample t-test (which you are hinting at). Calculate all the above metrics and then calculate the test statistic $t = \frac{\mu_1 - \mu_2}{\sqrt{\sigma_1 / n_1 + \sigma_2 / n2}}$. This is then assumed to follow a t-distribution with degrees of freedom $n_1 + n_2 - 2$ (subject to some assumptions), or, simpler, if your N is reasonably large, a normal distribution (Wald test). That means, you just take your statistic t, and use the software package of your choice to calculate the p-value for this statistic. If you would like to conduct a two-sided test on a confidence level $\alpha$%, you would reject that they are equal if the p-value is below $\alpha/2$ or above 100 - $\alpha/2$

*Case 2. Your populations don't seem to follow a normal distribution: You are better off using a non-parametric test, such as the Wilcoxon rank sum test (NOT the Wilcoxon signed rank test, which is for paired samples). Without going into the assumptions or the calculation (which are tedious), every statistical software package should have an implementation for it
