Problem 17 - Algebra 2 and Trigonometry

Question

A librarian estimates that the average number of books checked out by a library patron is 4 with a standard deviation of 2 books. If the number of books checked out each day approximates a normal distribution, what percent of the library patrons checked out more than 7 books yesterday?

Step-by-Step Solution

Answer

Approximately 6.68% of patrons checked out more than 7 books.

Step 1: Understand the Problem

We need to find the percentage of library patrons who checked out more than 7 books yesterday. We're given that the average number of books checked out is 4, and the standard deviation is 2. The distribution is normal.

Step 2: Calculate the Z-Score

To find the percentage of patrons who checked out more than 7 books, we first calculate the Z-score for 7 books using the formula: \[ Z = \frac{X - \mu}{\sigma} \]where \( X = 7 \), \( \mu = 4 \), and \( \sigma = 2 \).

Key Concepts

Understanding Z-score

The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. The Z-score can help determine how unusual or usual an observation is in a normal distribution. It's calculated using the formula: \[ Z = \frac{X - \mu}{\sigma} \]where:

• \( X \) is the value you're examining (in this case, 7 books).
• \( \mu \) is the mean or average value (4 books here).
• \( \sigma \) is the standard deviation (2 books).

The Z-score tells you how many standard deviations the number 7 is away from the mean of 4. A higher absolute Z-score means the value is further from the mean. A Z-score of 0 means the value is exactly at the mean. To apply this knowledge, we use the Z-score formula to see if checking out more than 7 books is typical or not.

The Role of Standard Deviation

Standard deviation is a measure that tells us how spread out the numbers in a data set are. In essence, it shows the amount of variation or dispersion in a set of values. A smaller standard deviation indicates that the values tend to be close to the mean. In our library example, the standard deviation is 2 books, meaning most library patrons check out books within 2 books of the average of 4 books. If most values cluster closely around the mean, the standard deviation will be small; if they are more spread out, the deviation is larger. Using standard deviation along with the mean helps us to estimate the likelihood of different observations. Since our data supposedly follows a normal distribution, understanding the standard deviation allows us to calculate probabilities for different book-checkout scenarios.

Library Statistics in Normal Distribution

When library data follows a normal distribution, it means that most patrons check out books near the average, with fewer patrons checking out either very few or many books. This distribution is bell-shaped and symmetric around the mean. Given a normal distribution as in our library statistics, we are interested in determining what percentage of patrons checked out more than 7 books. To do this, we utilize the Z-score we calculated earlier. By finding the Z-score corresponding to 7 books and referring to standard normal distribution tables, we can identify the proportion of patrons who lie beyond this threshold. Understanding library statistics within this framework allows us to make informed guesses about borrowing habits. It uses both average checkouts and standard deviation as guidance, providing a clear picture of patron behavior patterns.