B. unnecessary because it does not relate to the topic of the study.
Since the researcher is studying the relationship between self-esteem and intelligence, he should not look at any variable that doesn't pertain to the study. So let's look at the available options as regards eye color.
A. necessary because it may be an influencing variable in the study.
It may be. After all, eye color may affect self-esteem. If the researcher was interested, he should be doing a study on the effect of eye color on self esteem. But he isn't, so this is the wrong answer.
B. unnecessary because it does not relate to the topic of the study.
This option hits the spot. Eye color is an unnecessary variable that's not needed for the topic of the study.
C. necessary because it may improve the validity of the study.
This falls into the "Hey, I've got this data available, why not use it?" category. And doing so, it just makes things more complicated and harder to understand. Keep things simple. This is the wrong answer.
D. unnecessary because eye color influences all variables being studied.
If eye color has such a large influence, then there ought to be a separate study about it. But since the study is about the effects of self-esteem on intelligence, then eye color should be ignored. This is the wrong answer.
