Final answer:
The example of scientific thinking is the researcher analyzing motions of the Moon and Earth to predict lunar eclipses, as it involves data analysis and evidence-based predictions which are central to scientific methodology.
Step-by-step explanation:
The scenario that exemplifies a scientific way of thinking is B) A researcher analyzes data on the motions of the Moon and Earth and then makes a prediction about when the next lunar eclipse will occur. This process aligns with scientific methodologies, which involve observing, collecting data, analyzing that data, and making predictions based on the collected evidence.
For comparison, the other scenarios involve persuasion, subjective judgement, or personal methods of enjoyment, none of which are central to the scientific process. Scientific thinking often includes the formulation of hypotheses, which are then tested through experiments or further data analysis, as described in the referenced materials on the scientific method.