Question

Lisa has collected data to find that the number of pages per book on a book shelf has a normal distribution. What is the probability that a randomly selected book has fewer than 168 pages if the mean (μ) is 190 pages and the standard deviation (σ) is 22 pages? Use the empirical rule.Enter your answer as a percent rounded to two decimal places if necessary.

Provide your answer below:

Answer 1

To find the probability that a randomly selected book has fewer than 168 pages, we need to use the empirical rule, which is a guideline for how data is distributed in a normal distribution.

The empirical rule states that (approximately):

• 68% of the data points will fall within one standard deviation of the mean.

• 95% of the data points will fall within two standard deviations of the mean.

• 99.7% of the data points will fall within three standard deviations of the mean.

In this case, we have:

• Mean (μ) = 190 pages

• Standard deviation (σ) = 22 pages

• Lower bound (x) = 168 pages

We can calculate how many standard deviations away from the mean x is by using this formula:

• z = (x - μ) / σ

Plugging in our values, we get:

• z = (168 - 190) / 22
• z = -1

This means that x is one standard deviation below the mean.

So, we are looking for the probability that a randomly selected book has a value less than -1 standard deviations from the mean. Using the empirical rule, we know that approximately 68% of the data falls within one standard deviation of the mean. Therefore, approximately 34% of the data falls between the mean and -1 standard deviation.

To find the area under the normal distribution curve to the left of -1 standard deviation, we can use a standard normal distribution table (z-table) or calculator. The area to the left of -1 standard deviation is approximately 15.87%.

Therefore, the probability that a randomly selected book has fewer than 168 pages is approximately 15.87%.

// I hope this helps! //