A bookstore is interested in the average length of books in its adult fiction section. Find the minimum sample size required to create a $99\%$ confidence interval with a margin of error no more than $30$ pages if...
(A) The standard deviation is known to be about $96$ pages.

• A. $2039$
• B. $8$
• C. $68$
• D. $45$

Answer: (C)

Verified step by step guidance

1. Identify the confidence level and corresponding z-score. For a 99% confidence interval, find the z-value (critical value) from the standard normal distribution table that corresponds to the middle 99% of the distribution.
2. Note the margin of error (E) given, which is the maximum allowable error in the estimate. Here, E = 30 pages.
3. Use the known population standard deviation (\sigma = 96 pages) since it is given and the sample size formula for estimating a mean with a known standard deviation applies.
4. Apply the formula for the minimum sample size \( n \) required to achieve the desired margin of error at the specified confidence level: \[ n = \left( \frac{z \times \sigma}{E} \right)^2 \]
5. Calculate the value inside the parentheses first (\( \frac{z \times \sigma}{E} \)), then square the result to find \( n \). Since sample size must be a whole number, round up to the next whole number if necessary.