Let the probability that a person is a man be \(p = 0.34\). The number of men in a sample of 14 people follows a binomial distribution \(B(14, 0.34)\).

We need to find \(P(X \leq 2)\), where \(X\) is the number of men.

\(P(X = 0) = \binom{14}{0} (0.34)^0 (0.66)^{14} = (0.66)^{14}\)

\(P(X = 1) = \binom{14}{1} (0.34)^1 (0.66)^{13} = 14 \times 0.34 \times (0.66)^{13}\)

\(P(X = 2) = \binom{14}{2} (0.34)^2 (0.66)^{12} = \frac{14 \times 13}{2} \times (0.34)^2 \times (0.66)^{12}\)

Calculating each term:

\(P(X = 0) = 0.0029758\)

\(P(X = 1) = 0.02146239\)

\(P(X = 2) = 0.071866\)

Adding these probabilities gives:

\(P(X \leq 2) = 0.0029758 + 0.02146239 + 0.071866 = 0.0963\)