A company manufacturers video games with a current defect rate of 0.95%. To make sure as few defective video games are delivered as possible, they are tested before delivery. The test is 98% accurate at determining if a video game is defective. If 100,000 products are manufactured and delivered in a month, Approximately how many defective products are expected to be delivered.
The Correct Answer and Explanation is:
To determine how many defective products are expected to be delivered by the company, we can approach the problem step by step.
Given Information:
- Defect rate = 0.95% (which means 0.0095 of the products are defective).
- Test accuracy = 98% (the test correctly identifies defective products 98% of the time).
- Total products manufactured and delivered = 100,000.
Step 1: Find the total number of defective products.
The defect rate tells us the proportion of products that are defective. Thus, the expected number of defective products is:
[
\text{Defective Products} = 100,000 \times 0.0095 = 950
]
So, the company manufactures 950 defective products.
Step 2: Determine how many defective products pass the test.
Since the test is 98% accurate, it correctly identifies 98% of the defective products. Therefore, the number of defective products that are correctly identified by the test is:
[
\text{Correctly Identified Defective Products} = 950 \times 0.98 = 931
]
Thus, 931 defective products are correctly identified as defective and are likely to be rejected.
Step 3: Determine how many defective products pass the test.
The remaining 2% of defective products will be misidentified as non-defective. This is:
[
\text{Missed Defective Products} = 950 \times 0.02 = 19
]
So, 19 defective products will pass the test and be delivered to customers as non-defective.
Step 4: Determine the number of non-defective products that are incorrectly identified as defective.
Since the test is 98% accurate, it correctly identifies 98% of non-defective products as non-defective. Therefore, 2% of non-defective products will be incorrectly identified as defective. The total number of non-defective products is:
[
\text{Non-defective Products} = 100,000 – 950 = 99,050
]
The number of non-defective products misidentified as defective is:
[
\text{Misidentified Non-defective Products} = 99,050 \times 0.02 = 1,981
]
Step 5: Determine the total number of defective products delivered.
To find the total number of defective products that are delivered, we add the 19 defective products that passed the test (from Step 3) to the 1,981 non-defective products that were misidentified as defective (from Step 4):
[
\text{Total Defective Products Delivered} = 19 + 1,981 = 2,000
]
Conclusion:
Approximately 2,000 defective products are expected to be delivered to customers.
This includes the 19 defective products that pass the test and are delivered as non-defective, along with the 1,981 non-defective products that are incorrectly flagged as defective. This highlights the importance of test accuracy and the resulting impact on product delivery quality.