Is this problem related to statistical inference from two population parameters? If so, why does my approach not give the right answer? 
What I tried was use the statistic
z = pbar1 - pbar2 +- z of a/2 * SQRT[{pbar1*(1-pbar1)}/n1   + {pbar2*(1-pbar1)}/n2 ]
pbar1= 37/88 = .235
n1 = 88
n2=102
pbar2=.235
z = .42045-.235 +- 1.96 * SQRT[{.42(1-.42)}/88  + {.235(1-.235}/102]
which gives me
.185 +- .127
Which is roughly .058 to .312
So roughly option (a) below.
A friend tells me it is option (c), but I do not see a mathematical reason why.
Ami I right, or is he right?
 A: 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 

