Normal distribution of the serum cholesterol

Question

I'm trying to solve this problem with normal distribution.

The serum cholesterol level X in 14-year-old boys has approximately a normal distribution with mean 170 and standard deviation 30. In a middle school there are 300 14-year-old boys. Find the probability that at least 8 boys have a serum cholesterol level that exceeds 230.

Data from text:

$\mu = 170 $

$\sigma= 30 $

$n=8$

$z=\frac{\overline{X}-\mu}{\frac{\sigma}{\sqrt{n}}}=5.66$

$P(\overline{X}>230)=P(Z>z)=P(Z>5.66)=1-P(Z<5.66)=1-1=0$

probability of $0 \cdot 300 = 0$

In the solutions it says that probability is $0.3974$

Could you please help me and tell me where did I make mistake?

Answer 1

1.$ p = Pr(\{X \geq 230\}) = ... = 1 - \phi(\frac{230 -170}{30}) = 1- \phi(2)= \ \ ...$

2. Laplace's Integral de Moivre Theorem with probability $ p $ and $ n=300.$

$ Pr(\{ Y \geq 8\}) = 1 - Pr(\{Y <8\}) = \ \ ... $