Final answer:
To find the probability of nine criminal trials lasting at least 225 days, one must calculate the mean and standard deviation of the total, convert to a z-score, and use a standard normal distribution table. Similarly, to determine the minimum duration for ninety percent of the trials, one finds the appropriate z-score and solves using the mean and standard deviation.
Step-by-step explanation:
To calculate the probability that the total length of the nine trials is at least 225 days, using the given mean of 21 days and standard deviation of seven days, we can utilize the Central Limit Theorem. The Central Limit Theorem states that the distribution of sample means will approach a normal distribution as the sample size increases.
- First calculate the expected value (mean) of the total length:
- The mean of each trial is 21 days, so for nine trials, the total mean is 9 × 21 = 189 days.
- Then calculate the standard deviation of the total:
- The standard deviation for one trial is 7 days, for nine independent trials it's the standard deviation times the square root of the number of trials, which is 7 × √9 = 7 × 3 = 21 days.
- Lastly, to find the probability that the sum of the durations is at least 225 days, we convert this to a z-score and then use the standard normal distribution table for probability.
- The z-score is calculated by taking the difference between the target sum (225) and the mean sum (189), and dividing by the standard deviation of the sum (21), which gives us (225 - 189) / 21 = 36 / 21 ≈ 1.71.
- Looking up the z-score of 1.71 on the standard normal distribution table gives us the probability to the left of the z-score. To find the probability that the total is at least this value, we subtract this value from 1.
To find the duration for which ninety percent of the total of nine trials will last at least that long, we look up the z-score associated with the 90th percentile and then solve for the trial duration using the mean and standard deviation.