Answer:
Water is a poor conductor of heat: Most of the heat will move in a convection current within the water at the top of the test tube, only a small fraction of it will conduct down to the ice cube.
Water doesn't need to have a large amount of impurities to act as a good conductor of electricity; even a small amount of ions can enable a source of water to conduct electric current. In a nutshell, water is able to conduct electricity due to the dissolved ions and impurities within it.
