Question

A gigantic warehouse stores approximately 40 million empty aluminum 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. A statistician was hired by the insurance company to estimate​ p, the true proportion of cans in the warehouse that were contaminated by the fire. How many aluminum cans should be randomly sampled to estimate p to within 0.08 with 90​% ​confidence? The statistician should sample nothing cans to estimate the proportion that were contaminated by the fire to within 0.08 with 90​% confidence. ​(Round up to the nearest​ can.)

Answer 1

Answer: 106

Step-by-step explanation:

Let p be the true population proportion of cans in the warehouse that were contaminated by the fire.

The formula to find the required sample size when the prior populatio proportion is unknown:

$n=0.25(\dfrac{z^*}{E})^2$ , where z* = Critical z-value related to confidence interval and E = Margin of error.

As per given , we have

E= 0.08

Critical z-value for 90% confidence level = 1.645

Then, substitute all values in formula , we get

$n=0.25(\dfrac{1.645}{0.08})^2\\\\=0.25(422.81640625)\\\\=105.704101563\approx106$

Hence, the statistician should sample 106 cans to estimate the proportion that were contaminated by the fire to within 0.08 with 90​% confidence.