Question

on average, a library patron reads 42 books each year, with a standard deviation of 12 books. the library would like to reward its top 25% of patrons, as judged by the number of books read each year. how many books must a library patron read to be eligible for the reward if the library is only examining 100 patrons? use the z-table below: z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.3 0.618 0.622 0.626 0.629 0.633 0.637 0.641 0.644 0.648 0.652 0.4 0.655 0.659 0.663 0.666 0.670 0.674 0.677 0.681 0.684 0.688 0.5 0.691 0.695 0.698 0.702 0.705 0.709 0.712 0.716 0.719 0.722 0.6 0.726 0.729 0.732 0.736 0.739 0.742 0.745 0.749 0.752 0.755 0.7 0.758 0.761 0.764 0.767 0.770 0.773 0.776 0.779 0.782 0.785 0.8 0.788 0.791 0.794 0.797 0.800 0.802 0.805 0.808 0.811 0.813 0.9 0.816 0.819 0.821 0.824 0.826 0.829 0.831 0.834 0.836 0.839

Answer 1

A library patron must read at least 51 books to be eligible for the reward if the library is only examining 100 patrons.

How to determine the number of books

To determine the number of books a library patron must read to be eligible for the reward, find the z-score corresponding to the top 25% of patrons. The z-score represents how many standard deviations a value is from the mean.

First, find the z-score corresponding to the top 25% of the distribution. Since the distribution is normal and symmetric, use the z-table to find the z-score.

Looking at the z-table provided, the closest value to 0.25 is 0.244. Corresponding to this value, the z-score is approximately 0.674.

Now, use the z-score formula to find the raw score (number of books) that corresponds to this z-score:

z = (X - μ) / σ

Where:

z is the z-score,

X is the raw score,

μ is the mean, and

σ is the standard deviation.

Rearranging the formula to solve for X:

X = z * σ + μ

Substituting the given values:

z = 0.674

σ = 12 (standard deviation)

μ = 42 (mean)

X = 0.674 * 12 + 42

X ≈ 50.088

Since the number of books must be a whole number, we can round up to the nearest whole number.

Answer 2

A library patron must read at least 51 books to be eligible for the reward if the library is only examining 100 patrons.

How to determine the number of books

To determine the number of books a library patron must read to be eligible for the reward, find the z-score corresponding to the top 25% of patrons. The z-score represents how many standard deviations a value is from the mean.

First, find the z-score corresponding to the top 25% of the distribution. Since the distribution is normal and symmetric, use the z-table to find the z-score.

Looking at the z-table provided, the closest value to 0.25 is 0.244. Corresponding to this value, the z-score is approximately 0.674.

Now, use the z-score formula to find the raw score (number of books) that corresponds to this z-score:

z = (X - μ) / σ

Where:

z is the z-score,

X is the raw score,

μ is the mean, and

σ is the standard deviation.

Rearranging the formula to solve for X:

X = z * σ + μ

Substituting the given values:

z = 0.674

σ = 12 (standard deviation)

μ = 42 (mean)

X = 0.674 * 12 + 42

X ≈ 50.088

Since the number of books must be a whole number, we can round up to the nearest whole number.

Therefore, a library patron must read at least 51 books to be eligible for the reward if the library is only examining 100 patrons.