Final answer:
The median number of pages that can be printed is 3440. The probabilities for being less than 4000, more than 2500, and in between 2500 and 4000 depend on z-scores and the standard normal distribution. The Central Limit Theorem is used to find the distribution of the sample mean of pages from 25 toner cartridges, with the same mean but a smaller standard deviation.
Step-by-step explanation:
The median number of pages that can be printed from a laser printer's toner cartridge in this scenario would be equal to the mean, since the distribution is normal. Therefore, the median number of pages is 3440 pages.
For the probabilities, we use the properties of the normal distribution:
- The probability that the number of pages will be less than 4000 is obtained by calculating the z-score for 4000 and then finding the cumulative probability from the z-table or using a calculator — this gives a cumulative probability less than 4000.
- The probability that the number of pages will be more than 2500 is found by calculating the z-score for 2500 and then subtracting this cumulative probability from 1.
- The probability that the number of pages will be between 2500 and 4000 is the difference between the probability of being less than 4000 and the probability of being less than 2500.
Regarding the 95% chance that a toner cartridge won't last more than a certain number of pages, this can be found by calculating the upper bound of a 95% confidence interval around the mean.
According to the Central Limit Theorem, the standard deviation of the average number of pages from 25 toner cartridges would be the population standard deviation divided by the square root of 25 (the sample size), which is 780/√25. The mean would remain 3440 pages. We then calculate the probability that the average number of pages printed by the 25 toner cartridges is more than 3500 by finding the z-score for 3500 and looking up the corresponding cumulative probability.