Question

The number of pages in the books in a library follow a normal distribution. The mean number of pages in a book is 150 with a standard deviation of 30. if the library has 500 books, how many of the books have less than 180 pages.

Answer 1

As the mean is E[X] =150 and $\sigma$ = 30, using the normal distribution to get the answer is just using Z. So, what we need is P(X<180). 

$Z= \frac{x-E[X]}{\sigma } = \frac{180-150}{30} = \frac{30}{30} = 1$

P(x<180) = P(z< 1) = $\phi$(1) = 0.8413. 

Then, just multiplying the amount of books you have, which is 500, with the probability would give how many books are less than 180 pages, being:
$0.8413 * 500 = 420.65$