Question

The distribution of the number of occurrences of the letter "a" on the pages of a book is found to be a normal distribution with a mean of 60 and a standard deviation of 15. If there are 600 pages in the book, what percentage of the time is "a" found 45 times or more on a page?
a. 68% of the time
b. 50% of the time
c. 84% of the time
d. 34% of the time

Answer 1

Notice that 45 is 1 standard deviation below the mean. Recall that, for a normally distributed random variable $X$,


$\mathbb P(|X-\mu|\le\sigma)=\mathbb P(\mu-\sigma\le X\le\mu+\sigma)\approx0.68$

$\mathbb P(60-15\le X\le60+15)\approx0.68$

Since the normal distribution is symmetric about its mean, we have

$\mathbb P(60-15\le X\le60)\approx0.34$

Also by virtue of symmetry, we have

$\mathbb P(X\ge60)=0.50$

So,

$\mathbb P(X\ge45)=\mathbb P(45\le X\le60)+\mathbb P(X\ge60)\approx0.84$

making the answer, C.