Analyze the Argument
- Premise 1 (Cause): Spike in sea temperatures → Snow crab eggs hatch earlier (e.g., in the Fall instead of Winter).
- Premise 2 (Context): Deep sea fishing boats are more numerous in Arctic waters during the Fall season than the Winter season.
- Conclusion (Effect): Baby snow crabs are more likely to be caught by deep sea fishing boats.

The argument assumes that the crabs hatching earlier (in the Fall) directly increases their exposure to the fishing boats, leading to a higher likelihood of being caught.

Identify the Missing Link/Assumption
The core assumption is that the new, earlier hatching period (Fall) overlaps significantly with the period when fishing boats are present and fishing, while the old hatching period (Winter) did not.

To evaluate the conclusion, we need to know the specific timing of both events: hatching and fishing. The argument only gives the seasons (Fall/Winter) but not the overlap within those seasons.
- If the original winter hatch (e.g., February) was safe, and the new fall hatch (e.g., September) is during the main fishing season, the conclusion is strong.
- If the fishing boats leave in early September and the crabs now hatch in late November (still Fall), the overlap might be minimal, weakening the conclusion.

Evaluate the Options
- A) The total number of snow crabs that hatch versus the total number that had hatched before the increase in temperature: This is about the total population, not the likelihood of being caught. If more hatch, the overall number caught might increase, but the likelihood (probability/percentage) of any individual crab being caught is determined by the overlap in time/location, not the total number.
- B) The time of year the deep sea fishing boats arrive to the waters around Prudhoe Bay: This provides one half of the timing information (the start of the fishing season). This is crucial for determining the overlap.
- C) The month in which the boats leave, and the month in which the snow crabs eggs hatch: This gives the most precise data needed.
- Knowing the hatching month (e.g., September 15th) and the month the boats leave (e.g., October 30th) allows us to quantify the period of overlap (45 days of exposure).
- This is the most direct way to evaluate the change in exposure based on the shift from Winter to Fall hatching.
- D) The amount of area the fisherman's nets can cover in one day: This is about fishing efficiency, which is assumed to be constant. It does not help evaluate the change caused by the timing shift.
- E) The amount of time the deep sea fishing boats spend in water directly above where the snow crabs are likely to hatch: This is useful information, but ( C ) is more fundamental. If the boats leave before the crabs hatch, the amount of time they spend above the spot is irrelevant. (C) establishes the critical dates.

Conclusion: Statement (C) provides the specific temporal boundaries (when the vulnerable crabs are present and when the hazard/boats are absent) necessary to confirm if the earlier hatching truly increases the exposure risk.

The necessary piece of information is $\(\mathbf{C}\)$.