# Hypothesis test between means of week and weekend patterns of the same time series

I have a data set with phone calls, that contains the caller id and call duration. I want to test the hypothesis that the call duration during the week is smaller than the call duration during the weekend. I was thinking about using the Welch t-test but as the same caller can do calls on week days and weekends, I believe that I can´t assume iid. The time series is autocorrelated, what complicates the matter. Would ANOVA help in this case?

I would not expect call length to be normally distributed, so I would not jump into a Welch's-t test. Try a non-parametric Mann-Whitney U-test. However, that latter test is less powerful than a paired test which goes to interpretation. That is, if the test is highly significant, then a paired test with the same $N$ would be more significant. Problem is, you have some $n<N$ that are paired, so your test will be less than optimally powerful. You could try a Wilcoxon signed-ranks-W test on the $n$ pairs if $n$ is a substantial fraction $N$, and of those two non-parametric tests one will be more significant than the other.