What information can be retrieved from the slope in linear regression Here is the problem:

The question is weird in the fact that we're using categorical variables, but I see it this way:
Since one extra year of education has 4 times the effect on income as going from single to married or vice versa, it "matters" more as the effect is larger. However, marital status is just a categorical value so you can't just have a value of .5 or something in there, so the question just seems off to me.
Same logic for part b.
The question just seems worded weirdly to me, so I'm just looking to see how you guys would answer this question.
 A: I agree these questions are strange (and, arguably, unsound to use as test items).  But perhaps they can be answered if we adopt some reasonable assumptions.
First, what does "matters more" mean?  There are several ways this can be understood, even when variables are not commensurable (and we can hardly conceive of how a change in gender could be equated with some number of years of education!).


*

*We could look at the standardized coefficients, which express how much the fit varies (on a standardized scale of income) when an independent variable is changed by one unit (on a standardized scale).

*For binary explanatory variables, which are naturally quantized regardless of how they are coded, we could compare the effects on income of switching their values.
(Please note that neither understanding of "matters" is intended in a causal sense: they only address the degree of association as estimated with this linear fit, accounting for these particular variables.)
The latter sense does not enable us to draw any conclusions about question (1), since education is not binary, but it immediately implies that gender "matters more" than marital status, because a change in gender makes $1.5$ times the difference in predicted income of a change in marital status, ceteris paribus.
The first sense of "matters" is more problematic, because we aren't given the information needed to estimate the standardized coefficients.  If we assume the study is a typical large Western adult population, though, then


*

*Years of education will vary from less than 12 to more than 16, most likely with a standard deviation of several years.  (However, if the study focused on one particular kind of job where almost all workers have comparable educations, then the SD of years of education could be well below $1$.)

*There will be approximately equal numbers of each gender.  Their standard deviation therefore will be quite close to $1/2$.  (In some cases--such as a study of a male-dominated field--it nevertheless is possible for this standard deviation to be much lower.)

*There might be approximately equal numbers of married and unmarried people--or maybe not.  If not, then the standard deviation might be a little less than $1/2$--or even a lot less if one status is relatively rare in the sample.
I have listed these from the likely largest standard deviation to smallest.  The larger the SD is, the larger the standardized coefficient will be (since it is proportional to the SD of the explanatory variable).  Under these assumptions we would have a basis to conclude that years of education matter the most (by far) and that gender still matters more than marital status.  If we worry about imbalances in gender and marital status, then we can no longer conclude anything about which "matters" more, but it remains the case that education matters more than either provided there is a sufficient range exhibited within the data.
Consequently, we have developed reasonable arguments why the answer to (a) is "yes" (education matters more than marital status) but that the answer to (b) is either "yes" or "maybe" (gender matters more than marital status).  Most importantly, we have also uncovered possible situations where these answers could be completely reversed.  What our analysis really has accomplished is to characterize the situations where the reversals could happen.

In comments, @Buckminster brings up the point that even though each estimated coefficient is significant, perhaps we should not be quick to assume that one of the true coefficients is greater than another.  How reliable is that assumption?
When an estimate $b$ of a coefficient $\beta$ is "significant," that means a two-sided test of the hypothesis $\beta=0$ is rejected.  We are told the significance level is at least $99\%$.  This implies the standard error of $b$ is less than $b / Z_{1-\alpha/2}$ for $\alpha = 1 - 0.99 = 0.01$ and, usually, $Z$ is a quantile of the standard Normal distribution.  Consequently, if two estimates $b_0$ and $b_1$ are uncorrelated (or negatively correlated), the standard error of their difference cannot exceed
$$ \sqrt{\left(\frac{b_0}{Z_{1-\alpha/2}}\right)^2 + \left(\frac{b_1}{Z_{1-\alpha/2}}\right)^2}.$$
In the case $b_1 = 1.5 b_0$ this gives an upper bound of $0.7 b_0$, implying $b_1-b_0$ is $0.5/0.7 = 0.72$ standard errors away from $0$.  We are concerned only whether $\beta_1 \le \beta_0$: this is a one-sided test with p-value $1 - \Phi(0.72) = 0.24$.  Although that's not terribly low, it is some evidence that we can act as if it truly is the case that $\beta_1 \gt \beta_0$.  But because the evidence is weak, @Buckminster is well justified in calling this issue to our attention.
A: It seems you have two questions: 
1) What do slopes for (dummy-coded) categorical predictors tell us? 
The parameter estimate for a dummy-coded categorical variable provides an estimate of the difference between the groups, controlling for the effects of the covariates. In the context of this question, the slope for gender would represent the estimated mean difference in income between men and women after accounting for education and marital status.
The fact that you are controlling for covariates is important. Imagine if for whatever reason, on average, women had 5 more years of education than men. A simple model regressing income on gender might prove significant, but that wouldn't mean that gender itself is responsible for differences in income. The significance test of the slope for gender in the context of the more complex model tests us that gender significantly predicts income beyond education and marital status. Still, we must keep in mind that gender (or indeed any of our predictors) may simply be redundant with other unknown variables and we should therefore be cautious about making conclusions about what "matters."
2) What conclusions can we draw from comparing slopes in the context of a multiple regression model.
Not much. The significance tests on each of your predictors are asking if those values from zero. For example, the fact that your slope for gender is significant just means that in the context of the following model, $\beta_1 \neq 0$:
$$Income_i = \beta_0 + \beta_1 Gender_i + \beta_2 MaritalStatus_i + \beta_3 Education_i +\varepsilon_i$$
These significance tests, however, do not test whether these slopes differ from one another. Additionally, because the scale of your various slopes are different (e.g. $\beta_1$ is the predicted difference in income between men and women whereas $\beta_3$ is the predicted increase in income per additional year of education), comparing the magnitudes of your slope estimates is not particularly enlightening.
