Given that 25% of women have fingers longer than 8.8 cm, the corresponding z-score is \(z = 0.674\) (since 75% is below 8.8 cm).

Using the z-score formula: \(z = \frac{X - \mu}{\sigma}\), we have:

\(0.674 = \frac{8.8 - \mu}{\sigma}\)

\(0.674\sigma = 8.8 - \mu\)

Similarly, for 17.5% having fingers shorter than 7.7 cm, the z-score is \(z = -0.935\).

\(-0.935 = \frac{7.7 - \mu}{\sigma}\)

\(-0.935\sigma = 7.7 - \mu\)

We now have two equations:

1. \(0.674\sigma = 8.8 - \mu\)
2. \(-0.935\sigma = 7.7 - \mu\)

Subtract the second equation from the first:

\(0.674\sigma + 0.935\sigma = 8.8 - 7.7\)

\(1.609\sigma = 1.1\)

\(\sigma = \frac{1.1}{1.609} \approx 0.684\)

Substitute \(\sigma = 0.684\) back into the first equation:

\(0.674 \times 0.684 = 8.8 - \mu\)

\(0.461 = 8.8 - \mu\)

\(\mu = 8.8 - 0.461\)

\(\mu = 8.34\)