Final Answers:
1.8%: This is the base rate of defects, calculated by dividing the expected number of defects (10) by the total number of parts (550) and multiplying by 100%.
45: This is the expected number of positive tests, calculated by multiplying the base rate of defects (1.8%) by the test accuracy (90%) and the total number of parts (550).
10.8%: This is the approximate probability of a truly defective part given a positive test result, also known as the Positive Predictive Value (PPV). It is calculated using the formula: PPV = (true positives) / (all positives) ≈ (10 * 0.9) / (45).
Step-by-step explanation:
1. Base Rate:
The base rate is the overall prevalence of defects in the entire population, in this case, the total number of parts.
Simply dividing the expected number of defects by the total number of parts and multiplying by 100% gives the base rate: 10 / 550 * 100% ≈ 1.8%.
2. Expected Positive Tests:
The test accuracy tells us the proportion of times the test correctly identifies a defective part.
Multiplying the base rate by the test accuracy and the total number of parts gives the expected number of positive tests: 1.8% * 90% * 550 ≈ 45.
3. Positive Predictive Value (PPV):
PPV tells us the probability that a part is truly defective given a positive test result.
We can estimate this using the formula: PPV = (true positives) / (all positives).
Assuming all positive tests are true positives (which is an oversimplification, hence the "approximate" qualifier), we get: PPV ≈ (10 * 0.9) / (45) ≈ 10.8%.