Question

The Gruver Glass Company produces thermal panel glass windows in three states. The Ohio plant produces 50% of the output, the Kentucky plant produces 30%, and the Missouri plant produces 20%. Of the windows produced in Ohio, 1% are defective, in Kentucky, 3%; and in Missouri 2%. a. What proportions of all windows are defective? b. Given that a window is defective, find the probability that it was produced at the Kentucky plant.

Answer 1

The overall proportion of defective windows is 1.4%, and the probability that a defective window was produced at the Kentucky plant is 64.29%.

Step-by-step explanation:

The student's question involves calculating the combined defective rate of windows produced by the Gruver Glass Company across three plants and determining the probability that a defective window came from the Kentucky plant. To answer this:

1. The total proportion of defective windows is calculated by multiplying the proportion of total production from each plant by the defect rate at that plant and summing these products.
2. To find the probability that a defective window came from the Kentucky plant, we use Bayes' theorem. We calculate the probability that a window is defective and is from Kentucky divided by the probability that a window is defective.

The combined defective proportion is:
(0.50 * 0.01) + (0.30 * 0.03) + (0.20 * 0.02) = 0.014 or 1.4%.

The probability that a defective window came from the Kentucky plant is:
((0.30 * 0.03) / 0.014) = 0.6429 or 64.29%.