I have this statement:
The weights of a school are distributed in a normal way $\sim N(85, > 8)$ If a sampling is done. what should be the size of this sample, so that the probability of the mean of this sample is less than $87$, is $0.977$?
My attempt:
I need a sample $K$ of the weights, of size $n$, such that $P(\overline K<87) = 0.977$.
So $n$ need to be $> 30$(according to central limit theorem) and thus $\overline K \sim N(85, \frac{8}{\sqrt{n}})$.
Now I standardize the normal distribution. $P(\overline K<87) = P(Z < \frac{\sqrt{n}}{4})$, where $Z$ is a random var $\sim N(0,1)$.
So i have: $P(Z < \frac{\sqrt{n}}{4}) = 0.977$
But from here, I do not know how to calculate $n$. Any hint is appreciated.
PD: I must not use the formula or calculus, and the answer must be $64$
