The average library patron reads 42 books each year, with a standard deviation of 10 books. The library would like to reward its top 25% of patrons, as judged by the number of books read each year.

Question

Use a calculator to find how many books must a library patron read to be eligible for the reward if the library is only examining 100 patrons.

Round x¯ and σx¯ up to the nearest whole number.

Answer 1

Step 1: Find the z-score for the top 25% of patrons.

Since we want to find the top 25% of patrons, we need to find the z-score that corresponds to the 75th percentile. Using a standard normal distribution table, we find that the z-score for the 75th percentile is approximately 0.67.

Step 2: Use the z-score formula to solve for the number of books a patron must read to be in the top 25%.

z = (x - μ) / σ

Where:
x = number of books read
μ = population mean = 42
σ = population standard deviation = 10
z = 0.67 (from step 1)

0.67 = (x - 42) / 10

6.7 = x - 42

x = 48.7

Step 3: Round up x to the nearest whole number.

Since we cannot have a fractional number of books, we round up to the nearest whole number. This gives us:

x = 49

Therefore, a library patron must read at least 49 books to be eligible for the reward if the library is only examining 100 patrons.