Survivors in that dose range tended to be healthier than the unexposed Japanese. That is, their death from all causes was lower than for the unexposed Japanese. Nuclear shipyard workers were much healthier than non-nuclear shipyard workers. The low death rate from all causes for the nuclear workers was statistically very significant. This suggests that we need more radiation than we get from nature to be healthy.
People living in areas with high natural background generally have less cancer.
The amount of radiation from these last 2 sources varies with the geographical location and the material used in the buildings where you work and live. In addition, the contribution from radon varies depending on the construction of your home and the amount of uranium in the soil beneath it. If ionizing radiation is a significant cause of cancer, we would expect the millions of people who live in areas with high natural levels of radiation to have more cancer. However, that is not the case. Radon in mines increases lung cancer.