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A company that records and sells rewritable DVDs wants to compare the reliability of DVD fabricating machines produced by two different manufacturers. They randomly select 500 DVDs produced by each fabricator and find that 484 of the disks produced by the first machine are acceptable and 480 of the disks produced by the second machine are acceptable. Let \( p_1 \) and \( p_2 \) be the proportions of acceptable DVDs produced by the first and second machines, respectively.

Are the conditions for calculating a confidence interval for \( p_1 - p_2 \) met?

- **Random:** met
- **10%:** does not apply
- **Large Counts:** met

- **Random:** met
- **10%:** not met
- **Large Counts:** met

- **Random:** met
- **10%:** does not apply
- **Large Counts:** not met

- **Random:** not met
- **10%:** does not apply
- **Large Counts:** met

- **Random:** met
- **10%:** met
- **Large Counts:** met

Answer :

If 484 of the disks produced by the first machine are acceptable and 480 of the disks produced by the second machine are acceptable, random , 10% and large counts are met. Correct option is E.

To calculate a confidence interval for the difference between the proportions of acceptable DVDs produced by two different fabricating machines, we need to ensure that the following conditions are met:

Random: The sample of DVDs from each machine should be selected randomly, without any bias. This is important to ensure that the sample is representative of the population of all DVDs produced by each machine.

The sample size should be no more than 10% of the population of DVDs produced by each machine. This is important to ensure that the sampling distribution of the sample proportions is approximately normal.

Large counts: The number of acceptable and unacceptable DVDs in each sample should be large enough to satisfy the normal approximation to the sampling distribution of the sample proportions.

Therefore, the correct answer is (e) Random: met, 10%: met, Large Counts: met. All conditions for calculating a confidence interval for the difference between the proportions of acceptable DVDs produced by two different fabricating machines are met in this scenario.

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