Sequential binary hypothesis testing from temporally heterogeneously generated random samples with an active decision maker under budget constraints is considered. The problem is motivated from applications in crowdsourced classification and sequential detection from sensory data in IoT networks. In such applications, at each time slot, the source of data may vary from time to time, and the decision on the two possible hypotheses is to be made in a reliable, fast, and cost effective manner. In particular, the active decision maker either takes the current source and collects a sample, or skips the current source and waits for the next time slot. At the end of each time slot, the decision maker either decides to claim the decision on the two hypotheses, or continues to observe the data source for the next time slot. The goal is to design action taking and decision making policies so that the probability of error is minimized under two constraints: one on the total number of samples collected by the decision maker, and the other is on the total number of time slots. In this work, the available data source changes among n possible ones i.i.d. over time, and the two constraints are in expectation. We establish the optimal error exponents of the two types of error probabilities as the two constraints tend to infinity with a fixed proportion. For achievability, a scheme that combines a sequential probability ratio test and an adaptive randomized policy that dynamically switches between two sets of accepting probabilities of the current source, according to the observed samples collected so far, is proposed. Matching upper bounds on the error exponents are developed using data processing inequality and Doob’s Optional Stopping Theorem.