Chapter 6: Q67E (page 363)

Aluminium cans contaminated by fire. A gigantic warehouselocated in Tampa, Florida, stores approximately 60million empty aluminium beer and soda cans. Recently, a fireoccurred at the warehouse. The smoke from the fire contaminatedmany of the cans with blackspot, rendering them unusable.
A University of South Florida statistician was hiredby the insurance company to estimate p,the true proportionof cans in the warehouse that were contaminated by the fire. How many aluminium cans should be randomly sampled toestimate pto within .02 with 90% confidence?

Short Answer

Expert verified

1692 aluminium cans should be randomly sampled to estimate pto within .02 with 90% confidence

Step by step solution

Given information

Aluminium cans contaminated by fire. A gigantic warehouse located in Tampa, Florida, stores approximately 60 million empty aluminium beer and soda cans.

Recently, a fire occurred at the warehouse. The smoke from the fire contaminated many of the cans with blackspot, rendering them unusable

Finding the sample size

Here the sample proportion is not given. Let’s assume $\hat{p} = 0.50$

$\begin{matrix} \hat{q} = 1 - \hat{p} \\ = 1 - 0.50 \\ = 0.50 \end{matrix}$

Here the standard error is 0.02

The critical value for 90% confidence interval is $z_{α / 2} = z_{0.10 / 2} = z_{0.05} = 1.645$

$\begin{matrix} S E = z_{α / 2} \sqrt{\frac{\hat{p} \hat{q}}{n}} \\ n = {z^{2}}_{α / 2} \frac{\hat{p} \hat{q}}{S E^{2}} \\ n = {1.645}^{2} \times \frac{0.50 \times 0.50}{{0.02}^{2}} \\ n = 1691.266 \\ n \approx 1692 \end{matrix}$