Why does Jem feel optimistic about the verdict? How does his age play a role in these feelings?

Aslan has already answered this question for you.....

"Jem sees that there is no possible evidence that Tom could be guilty of the rape of Mayella Ewell. Atticus's case is airtight. The jury is even aware who really raped Mayella, her own father Bob Ewell. Unfortunately Jem underestimates the power of ingrained bigotry. Jem is still naive to the fact that insidious hatred still lingers in the hearts of many whites in the South."

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