Specimen Paper SPEC/5/TZ0 P2 Q8

The length, \(X\) mm, of a certain species of seashell is normally distributed with mean \(25\) and variance, \(\sigma^2\).
The probability that \(X\) is less than \(24.15\) is \(0.1446\).

1. Find \(\mathrm{P}( 24.15 < X < 25)\).

   [2]

2. 1. Find \(\sigma\), the standard deviation of \(X\).

   2. Hence, find the probability that a seashell selected at random has a length greater than \(26\) mm.

      [5]

A random sample of \(10\) seashells is collected on a beach. Let \(Y\) represent the number of seashells with lengths greater than \(26\) mm.

3. Find \(\mathrm{E}(Y)\).

   [3]

4. Find the probability that exactly three of these seashells have a length greater than \(26\) mm.

   [2]

A seashell selected at random has a length less than \(26\) mm.

5. Find the probability that its length is between \(24.15\) mm and \(25\) mm.

   [3]