Question

select the correct answer. the numbers of pages in the books in a library follow a normal distribution. if the mean number of pages is 180 and the standard deviation is 30 pages, what can you conclude? a. about 60% of the books have fewer than 150 pages. b. about 16% of the books have fewer than 150 pages. c. about 95% of the books have more than 150 pages. d. about 16% of the books have more than 150 pages.

Explanation:

Step1: Calculate the z - score

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x = 150$, $\mu=180$, and $\sigma = 30$. So $z=\frac{150 - 180}{30}=\frac{- 30}{30}=-1$.

Step2: Use the properties of the normal distribution

In a normal distribution, about 68% of the data lies within 1 standard - deviation of the mean ($\mu\pm\sigma$), which means 34% of the data lies between $\mu-\sigma$ and $\mu$, and 34% lies between $\mu$ and $\mu + \sigma$. The total area under the normal curve is 100%. The area to the left of $\mu-\sigma$ is $\frac{100 - 68}{2}=16\%$. Since $x = 150$ is 1 standard - deviation below the mean ($\mu-\sigma$ with $\mu = 180$ and $\sigma=30$), about 16% of the books have fewer than 150 pages.

Answer:

B. About 16% of the books have fewer than 150 pages.