So I'm not sure if I 100% grasp confidence intervals. Say I have a huge data set of a bond prices from 1996 to present in MINUTES. Suppose I separate each data by day. If I were to use a Dickey Fuller Test (or any other t-test), on a particular day (thus a sample is drawn based on all points in one day) and the value I got rejected the null hypothesis, would it be conceptually right for me to conclude that because based on my sample I was able to reject the null, I am able to make a conclusion about the whole dataset (population)? In other words, would picking one day and not randomly picking datapoints throughout matter?
Yes, it matters. And no, you cannot generalize, unless the parameter you are testing is some structural parameter, in which case you are really just using a lower-powered sample to test your time series model parameter.