Whenever you're asked to weaken a conclusion, you want to identify any logical gaps between the premises and the conclusion.

Premise:
- patients with body temperatures below 36C have a higher mortality rate than patients with body temps above 36C

Conclusion:
- mortality rates will be reduced if operating rooms are kept warmer

Logical Gaps:
- Body temp = room temp? Will changing the temperature of the room necessarily change the temperature of the bodies?
- Body temp causes mortality rate? Is there something else going on with these people that could lower body temperature and mortality rate? If so, raising body temp might not solve that underlying problem.
- Other effects of temperature on mortality? Can raising the room temperature have some other negative consequences?

I agree with most of the assessments of the answer choices above, but just to go through all of them...

A. Summer v. winter doesn't necessarily affect the temperature of the room.

B. One specific case (delivery rooms) won't speak to all cases (all operating rooms). Hypothermia is not the same thing as mortality rate. If anything, this one would seem to strengthen a little bit, by suggesting that higher temps are good.

C. Cost is not relevant to mortality rate (unless other information were added, such as "fewer patients will get necessary surgeries" or something).

D. Beware of any answer choices that talk about people's feelings or preferences. It usually won't be very relevant to outcomes. Do their preferences affect mortality rate? No.

E. Other commenters are right - this one requires a leap from "dexterity and alertness go down" to "surgeons are less competent." It's not a great answer choice. But correct answers on WEAKEN questions sometimes will require you to make a bit of a leap (unlike assumption questions, which should stick very close to scope). It's the least-bad answer choice, so we have to pick it.