Is this problem related to statistical inference from two population parameters? If so, why does my approach not give the right answer?

There are several ways to do this. So in order to get the exact intended answer, you'd have to know the exact formula used in that book. [There are various formulas for the standard error (pooling E and NE to get a combined estimate of p, or not pooling). Some use a continuity correction, some don't. And so on.] Compare formulas with your friend.

Here is output from prop.test in R, which gives a CI not on your list.

prop.test(c(37,24),c(37+51,24+78))

        2-sample test for equality of proportions 
        with continuity correction

data:  c(37, 24) out of c(37 + 51, 24 + 78)
X-squared = 6.6052, df = 1, p-value = 0.01017
alternative hypothesis: two.sided
95 percent confidence interval:
 0.04261658 0.32770428
sample estimates:
    prop 1    prop 2 
 0.4204545 0.2352941 

Once again, but without the continuity correction, which comes very close to suggested answer (a).

prop.test(c(37,24),c(37+51,24+78), cor=F)

        2-sample test for equality of proportions 
        without continuity correction

data:  c(37, 24) out of c(37 + 51, 24 + 78)
X-squared = 7.4304, df = 1, p-value = 0.006413
alternative hypothesis: two.sided
95 percent confidence interval:
 0.05320035 0.31712050
sample estimates:
    prop 1    prop 2 
 0.4204545 0.2352941