The Capital Bank Marketing Department has recently conducted a study
of a sample of the bank's customers. At issue is whether there is a
difference between the mean credit card balance between female and
male customers. If they find that the two groups differ, they will
target the lower group with a marketing campaign designed to increase
their use of the credit card.

The calculated mean and variance of the credit card balances for the
male customers and the mean and variance of the credit card balances
for the female customers are as follows:

232 males  with a mean of ( $ 746.51) and a Variance of (86743.26)

68 females with a mean of ( $ 778.13) and a Variance of (87152.24)

Question: Based on just these two calculations, would you guess that
there is a significant difference between the two means? Between the
two variances?

Given the way you've worded your question ("would you guess"): look at
the two matching sets of numbers. The difference between the two means
is $31.62, which is about 4% of the higher number. The difference
between the two variances is 409, about 1/200 of 1%.  My "guess" would
be is that there is no "signifcant difference" between the two.

The uncertainty in the mean is the standard deviation (the sqare root
of the variance) divided by the square root of the number of samples. 
So the males have a mean of 746.51 +/- 19.34, and the females a mean
of 778.13 +/- 35.80.  The difference in the two means is 31.62, and
the uncertainty in the difference is 40.69 (that's the square root of
the sum of the squares of the uncertainty in each of the two groups). 
It is not possible to reject the hypothesis that the two groups have
the same mean.  The two variances are nearly identical--the standard
deviations differ by less than a dollar.  It is certainly not possible
to reject the hypothesis that the variances are identical.