(i) The probability that a person buys something is 0.66 (since 34% do not buy anything). We model this situation with a binomial distribution: \(X \sim B(15, 0.66)\).

The probability that at least 14 people buy something is \(P(X \geq 14) = P(X = 14) + P(X = 15)\).

Using the binomial probability formula:

\(P(X = 14) = \binom{15}{14} (0.66)^{14} (0.34)^1\)

\(P(X = 15) = \binom{15}{15} (0.66)^{15}\)

Calculating these gives:

\(P(X = 14) = 15 \times (0.66)^{14} \times 0.34\)

\(P(X = 15) = (0.66)^{15}\)

Adding these probabilities: \(P(X \geq 14) = 0.0171\).

(ii) The probability that a person spends less than $50 is 0.87. We want \((0.87)^n < 0.04\).

Taking logarithms: \(n \log(0.87) < \log(0.04)\).

Solving for \(n\):

\(n > \frac{\log(0.04)}{\log(0.87)} \approx 23.5\).

The smallest integer \(n\) is 24.