The Atlantic provides a clearly written summary of a recent longitudinal study. The headline reads:

Study: Math Skills at Age 7 Predict How Much Money You'll Make

According to the summary, the researchers measured kids' IQ, math skills, reading skills, and SES (Socio-economic status) at age 7, and then measured their income at age 42.

They found that

How much money the people made at midlife
was predicted by math ability at age seven. The other factors may have helped them on the
path to success, but even when those were controlled for, the
association between basic math and reading skills and future
socioeconomic status remained, and remained
significant…

Do you notice how this description uses the key phrase, "even when those were controlled for…"?  This should signal to you that the researchers used multiple regression in their design.

a) What might the regression table have looked like in the study? What would the DV have been? What would the predictor variables have been?  Estimate what you think the beta might have been for the predictor, "Math skills at age 7" (that is, is it positive or negative? Significant or not?).

b) Suppose a critic reads this article and says, "I don't think that it's math skills–I think it's IQ. Smarter people just earn more money and they did better at math as kids–that's all." What should you say in response?  In this study, is IQ a potential third variable that could explain the association between math skills and future income?

c) Now suppose that a critic suggests that school quality might be a third variable. Kids who go to higher-quality schools had better math skills and also made more money. In this study, is school quality a potential third variable that could have explained the association between math skills and future income?

d) One part of the story says,

…How much money the people made at midlife
was predicted, for girls only, by
early reading ability.

Is this sentence describing a mediator? A moderator? Or a third variable?