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.
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.
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.
To find the number of books a library patron must read to be eligible for the reward, we need to determine the z-score corresponding to the top 25% of patrons and then convert it back to the original scale using the mean and standard deviation.
First, we need to find the z-score using the cumulative distribution function (CDF) of the standard normal distribution. The z-score corresponding to the top 25% is the z-score that leaves 75% below it.
Using a calculator or a statistical table, we can find that the z-score for the top 25% is approximately 0.674.
Next, we need to convert the z-score back to the original scale using the formula:
X = (z * σ) + μ
where X is the number of books a patron must read, z is the z-score, σ is the standard deviation, and μ is the mean.
Given that the average number of books read is 42 with a standard deviation of 10, we can substitute these values into the formula:
X = (0.674 * 10) + 42
Calculating this expression gives us:
X ≈ 6.74 + 42
Rounding up to the nearest whole number:
X ≈ 49
Therefore, a library patron must read at least 49 books to be eligible for the reward if the library is examining 100 patrons.