Frobisher, a sixteenth-century English explorer, had soil samples from

Let us read the question stem first. We are looking for an assumption. An assumption is a necessary missing premise. We are looking for the option that needs to be true for the conclusion to be true.

Premises:
Frobisher had soil samples from Canada’s Kodlunarn Island examined for gold content.
Because high gold content was reported, Elizabeth I funded two mining expeditions.
Neither expedition found any gold there.
Modern analysis of the island’s soil indicates a very low gold content.

Tell me, when you read the above premises, what possibilities come to mind? Frobisher had samples examined. High gold content was reported. No gold was actually found. Modern analysis show very low gold content.

The following possibilities come to my mind:
1. Either there was gold and before the expeditions were sent, it was mined (very unlikely!)
2. His methods were inaccurate.
3. Someone had added gold to the soil he tested.

Conclusion:
The methods used to determine the gold content of Frobisher’s samples must have been inaccurate.

If I am concluding that his methods were inaccurate, then I am assuming that no one added gold to his samples and gold was not mined before the expeditions were sent. (Technically, gold could have been added and his methods could have been inaccurate too but lets not mess with that.)
Hence option (E) is an assumption.
Also, use you can use assumption negation technique to see that it is the right answer.
I negate (E) : Gold was added to the soil samples collected by Frobisher before the samples were examined.

I can not conclude now that his methods were inaccurate.
Hence (E) is the correct answer.

Option (C) is not correct. We did not assume in the argument that his methods were different. They could have been the same ones generally used in the 16th century, It is possible that 16th century methods were not accurate.