Test for significance of variation in time series data Suppose a DMV office runs driving tests. Every week a number of people take the driving test, some pass and some fail. Looking at this data set, we notice that the pass rate varies from week to week. 
How do I test whether this is random variation, or whether there is some change over time in the population of drivers who are taking the test? 
It seems that if I suggest a specific hypothesis (drivers who take the test on weekends are better than drivers who take the test on weekdays), then I can use a $t$-test (assuming driving skill is normally distributed). 
But what if I don't have a specific hypothesis, but just want to know the probability that chance alone explains the variation in the rate over time? Is this a well-formed question? What approach makes the most sense here?
 A: You should form a robust Transfer Function model to predict/analyze the daily number who pass as a function of the number who are tested. Tests for daily effects can be based upon a model that includes day-of-the-week effects, month-of-the-year ; week-of-the-month ; week-of-the-year ; day-of-the-month ; holiday effects both pre and post ; long weekend effects ; level shifts ; trend shifts ; changes in daily effects over time ; weather effects ( rain no rain ) , pulse effects reflecting unusual values , ARIMA effects reflecting unspecified/omitted perhaps unknown causal variables. All of this is done while ensuring that both the error variance and the model's parameters are invariant over time. You should be quite careful (as usual) in selecting an application. There are always many easy approaches to performing incorrect analysis.
Finally one should studiously avoid converting the Y and the X to a percentage as this is an anachronistic approach based upon an inability to correctly construct a multivariate model and thus has consequences.
