Final answer:
The duration of trials is normally distributed with a mean of 21 days and a standard deviation of 7 days. The random variable X represents the number of days a trial takes. Calculations for probabilities and percentiles involving these trials would require understanding and utilizing the properties of normal distribution.
Step-by-step explanation:
The question revolves around calculating the time required to complete a project or trial based on information given about a trial's duration being normally distributed. Specifically, the trial duration has a mean of 21 days and a standard deviation of 7 days. To address the question posed, we need to consider the mathematical concepts of probability and normal distribution.
Looking at the data provided:
- Random variable X represents the duration of the trial in days and follows a normal distribution.
- X follows a normal distribution noted as X~ N(21, 7), where 21 is the mean (μ) and 7 is the standard deviation (σ).
- The probability that a randomly chosen trial lasts at least 24 days falls at 0.3336 on the normal distribution curve.
- For nine such trials, determining the total duration that is at least certain days would involve calculating probabilities given the mean and standard deviations of the sampling distribution of the sample mean.
Based on the information provided:
- The sum of the durations of nine trials, in words, is denoted by ΣX.
- The expectation of ΣX follows the notation EX~ which represents the sum of the expected trial durations.
Specific answers include:
- The probability that the total length of nine trials is at least 225 days.
- 90% of the total of nine of these types of trials would last a certain period, which needs to be calculated using the 90th percentile of the appropriate normal distribution.
- The duration 60% of all trials are completed within can be found by looking at the 60th percentile of the normal distribution of trial durations.