Final answer:
The random variable X represents the number of days a particular type of criminal trial will take. The probability that a trial lasts at least 24 days is approximately 0.3336 or 33.36%. Approximately 60% of all trials of this type are completed within 25.718 days.
Step-by-step explanation:
The number of days a particular type of criminal trial will take can be represented by the random variable X. In this case, X has a normal distribution with a mean of 21 days and a standard deviation of 7 days. Therefore, the random variable X can be denoted as X ~ N(21, 7).
To find the probability that a trial lasts at least 24 days, we can calculate the area under the normal curve to the right of 24 days. This probability can be found using a standard normal distribution table or a calculator, and it is approximately 0.3336 or 33.36%.
To answer the question, we need to find the number of days within which 60% of all trials of this type are completed. This corresponds to the median completion time, which is the same as the 50th percentile of the normal distribution. Using a standard normal distribution table or a calculator, we can find that the z-score corresponding to the 50th percentile is approximately 0.674. We can then use the z-score formula to find the corresponding value of X (number of days):
X = mean + (z-score * standard deviation)
X = 21 + (0.674 * 7)
X ≈ 21 + 4.718
X ≈ 25.718
So, approximately 60% of all trials of this type are completed within 25.718 days.