enthalpy change

Frankly speaking, 001 answer is not true. What he had calculated was just the heat released by the experiment, without calculating the actual enthalpy change of the reaction.
In fact, the solution should be as follows:
Heat released = 11.3 * (27.5 - 20.2) = 82.49 kJ
(negative sign can be ignored because the term "heat released" already involved exothermic reaction.
In order words, no matter you are calculating heat released or heat absorbed, sign should not be added.)
Then, we can calculate the enthalpy change
Enthalpy change = - Heat released / no. of moles of methane
= - 82.49 / (1.5 / 16)
= -880 kJ mol^-1
(In this case, enthalpy change should be added because the sign actually express whether the reaction is exothermic or endothermic. Also, the unit is J mol^-1 / kJ mol^-1, not J / kJ.

In this case... You only need the heat capacity (C) and the change of temperature (T) to solve the question. (q = CxT)

Enthalpy change = q = CxT
= 11.3 kJ/℃ x (27.5-20.2 ℃ )
= 11.3 kJ/ ℃ x (7.3 ℃ )
= 82.49kJ

Since all combustion reactions are exothermic; therefore, the enthalpy change should be negative. Thus, the answer is -82.49 kJ.