Question

A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured in pounds, how many books should be tested to estimate with \(95 \%\) confidence to within \(0.1 \mathrm{lb}\), the average force required to break the binding? Assume that \(\sigma\) is known to be \(0.8 \mathrm{lb}\).

Short answer

The minimum number of books to be tested is obtained after the calculations in step 3, final value in step 5, which is the rounded up value of \(n\).

Step-by-step solution

Understand the Requirements

You need to identify the information provided in the task and what is asked. The given information includes the standard deviation (σ=0.8 lb), the desired half-width of the confidence interval (E=0.1 lb), and the confidence level (95%). You're asked to calculate the minimum required number of books to be tested.

Determining the Z-score

In correspondence to the 95% confidence interval, the Z-score is quick to find on a Z-table, or it can be calculated using a calculator providing probability distributions. For a 95% confidence interval the Z-score is typically \(Z=1.96\).

Applying the Formula for Sample Size

The formula for the sample size (n) in estimating a mean with a certain level of confidence is given by \(n = (Z \cdot σ / E)^2\). Now you can substitute the given or derived values into the equation: \(n = (1.96 \cdot 0.8 / 0.1)^2\).

Computing the Sample Size

After substituting in the formula calculate the value of \(n\), then this number must be rounded up because the number of books tested cannot be a decimal.

Final result

The final result is the minimum n value obtained after calculation and rounding up, which is the minimum number of books needed to be tested to estimate with 95% confidence to within 0.1 lb, the average force required to break the binding.