An electronics manufacturer is monitoring the production of specialized micro-drills. Each batch contains 50 drills, and a drill is classified as defective if it fails a precision test. A quality control inspector takes a random sample of 200 batches and records the number of defective drills, xxx, in each batch. The results are summarized in the table below:
| Number of defective drills (xxx) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ≥10\ge 10≥10 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Frequency (fff) | 2 | 10 | 20 | 35 | 45 | 38 | 25 | 15 | 7 | 3 | 0 |
Calculate the mean and the variance of these data.
Explain why the results in part (a) suggest that a Binomial distribution may be a suitable model for the number of defective drills per batch.
The manufacturer uses a Binomial distribution X∼B(50,0.1)X \sim B(50, 0.1)X∼B(50,0.1) to model the number of defective drills per batch.
For a randomly selected batch find, using this model, the probability that there are (i) at most 2 defective drills, (ii) at least 3 but no more than 6 defective drills.
A large-scale production run consists of 200 drills.
Using the manufacturer's model and a suitable approximation, show that the probability that there are more than 25 defective drills in a production run is 0.10 to 2 decimal places. Show your working clearly. (Solutions relying on calculator technology are not acceptable.)
A period of 15 production runs is selected at random.
Find the probability that in this period there are exactly 3 production runs that have more than 25 defective drills. Show your working clearly.