(i) To find the probability that a straw fits into the hole, we need to calculate the probability that the diameter is less than 3 mm. This is given by:

\(P(X < 3.0) = P\left( Z < \frac{3.0 - 2.6}{0.25} \right) = P(Z < 1.6)\)

Using the standard normal distribution table, \(P(Z < 1.6) = 0.945\).

(ii) For 500 straws, we use a normal approximation to the binomial distribution. Let \(X \sim B(500, 0.9452)\). The normal approximation is \(X \sim N(472.6, 25.898)\).

We need \(P(X \geq 480)\), which is approximated by:

\(P\left( Z > \frac{479.5 - 472.6}{\sqrt{25.898}} \right) = P(Z > 1.3558)\)

Using the standard normal distribution table, \(P(Z > 1.3558) = 1 - 0.9125 = 0.0875\).

(iii) The approximation is justified because both \(500 \times 0.9452\) and \(500 \times (1 - 0.9452)\) are greater than 5, satisfying the conditions for using the normal approximation to the binomial distribution.