''As a result'' should be used as a transition when connecting phrases that show cause and effect because it would show that the effect is as a result of the cause/ scenario depicted first.
Here we see that older and wealthier Americans are more likely to vote and the effect of this is that they have a greater influence on government because they decide who gets political office. ''As a result'' would show the relationship between these two by showing that the effect is as a result of the cause.
