In a closed insulated container, there is no exchange of heat between the system and the surroundings. This makes it an adiabatic process, so $q = 0$. Stirring the liquid with a paddle means mechanical work is done on the system, which raises its temperature, so $W \neq 0$. According to the first law of thermodynamics, the change in internal energy is given by $\Delta E = q + W$. Since $q = 0$, the change in internal energy $\Delta E$ is exactly equal to the work done on the system. Therefore, $\Delta E = W \neq 0$.