Question

In a city library, the mean number of pages in a novel is 525 with a standard deviation of 200. Approximately 30% of the novels have fewer than 400 pages. Suppose that you randomly select 50 novels from the library. What is the probability that the average number of pages in the sample is less than 500?

Answer 1

Using the normal distribution and the central limit theorem, it is found that there is a 0.1894 = 18.94% probability that the average number of pages in the sample is less than 500.

Normal Probability Distribution

In a normal distribution with mean $\mu$ and standard deviation $\sigma$, the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

• It measures how many standard deviations the measure is from the mean.

• After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.

• By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation $s = \frac{\sigma}{\sqrt{n}}$.

In this problem:

• The mean is 525, hence $\mu = 525$.

• The standard deviation is 200, hence $\sigma = 200$.

• A sample of 50 is taken, hence $n = 50, s = \frac{200}{\sqrt{50}}$.

The probability that the average number of pages in the sample is less than 500 is the p-value of Z when X = 500, hence:

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{500 - 525}{\frac{200}{\sqrt{50}}}$

$Z = -0.88$

$Z = -0.88$ has a p-value of 0.1894.

0.1894 = 18.94% probability that the average number of pages in the sample is less than 500.

To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213