The length, $X$ mm, of a certain species of seashell is normally distributed with mean $25$ and

Question

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 $0.1446$. Find $\mathrm{P}( 24.15 < X < 25)$. [2] [5] Find $\sigma$, the standard deviation of $X$. Hence, find the probability that a seashell selected at random has a length greater than $26$ mm. A random sample of $10$ seashells is collected on a beach. Let $Y$ represent the number of seashells with lengths greater than $26$ mm. Find $\mathrm{E}(Y)$. [3] 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. Find the probability that its length is between $24.15$ mm and $25$ mm. [3]

Accepted Answer

(a) $0.3554$ (b)(i) $\sigma = 0.802$ (b)(ii) $\mathrm{P}(X > 26) = 0.106$ (c) $\mathrm{E}(Y) = 1.06$ (d) $\mathrm{P}(Y = 3) = 0.0655$ (e) $0.398$