Question

1. if a product contains 550 critical parts for a month's production and you expect that 10 of those parts will be defective, what is the approximate base rate for defects? 1% 1.8% 2.2% 3.1% -select- 2. if the test used in question 1 is 90% accurate, how many positive tests of defects can you expect out of the 550 parts? 45 54 63 97 -select- 3. what is the approximate probability of a genuinely defective part given a positive test result: p(truly positive case | positive test result)? 7.5% 10.8% 12.5% 14.3%

Answer 1

The approximate base rate for defects is 1.8%. Out of 550 parts, you can expect around 10 positive tests of defects. The probability of a genuinely defective part given a positive test result cannot be calculated without additional information.

Step-by-step explanation:

To find the approximate base rate for defects, divide the number of defective parts by the total number of critical parts and multiply by 100 to get the percentage. In this case, 10 defective parts out of 550 critical parts will give an approximate base rate for defects of (10/550) x 100 = 1.8%.

To calculate the number of positive tests of defects expected out of the 550 parts, multiply the expected base rate for defects by the total number of parts. In this case, 1.8% of 550 parts will give approximately 9.9, so we can expect around 10 positive tests of defects.

The probability of a genuinely defective part given a positive test result can be found using Bayes' theorem. However, the necessary information, such as the false positive rate, is not provided in the question. Therefore, we cannot calculate the approximate probability in this case.

Answer 2

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%.