False signals. A procedure has been designed that will detect 99% of the defective assemblies in a manufacturing process. However, the probability that the procedure will declare a good assembly defective is 0.05. If 2% of all assemblies are defective, what percentage of the assemblies signaled as defective by the process are really good?
To find the percentage of assemblies signaled as defective by the process that are really good, we need to first calculate the false positive rate.
Let's assume we have a total of 1000 assemblies (since percentages are given), and 2% of them are actually defective. So there would be 0.02 * 1000 = 20 defective assemblies.
The procedure is designed to detect 99% of defective assemblies, so it will correctly identify 0.99 * 20 = 19.8 defective assemblies (approximated as 20).
However, the probability that the procedure will incorrectly declare a good assembly as defective is 0.05. So out of the 980 good assemblies (1000 total - 20 defective), the procedure will incorrectly identify 0.05 * 980 = 49 good assemblies as defective.
Now, to find the percentage of assemblies signaled as defective by the process that are really good, we divide the number of good assemblies signaled as defective (49) by the total number of assemblies signaled as defective (49 + 20) and multiply by 100:
Percentage of good assemblies signaled as defective = (49 / (49 + 20)) * 100 = 71.43%
Therefore, approximately 71.43% of the assemblies signaled as defective by the process are really good.