Final answer:
The duration of criminal trials is normally distributed with a mean and standard deviation given. The random variable X is used to represent the trial duration. To find probabilities or percentiles, we calculate z-scores or check the corresponding values in a standard normal distribution.
Step-by-step explanation:
Understanding Criminal Trial Durations
The question deals with the duration of criminal trials which is modeled by a normal distribution. This is a typical question under the subject of statistics, a branch of mathematics.
79. Duration of a Criminal Trial:
a. The random variable X represents the duration of a criminal trial in days.
b. The distribution of X can be expressed as X~N(21, 49), where N stands for the normal distribution, '21' is the mean, and '49' is the variance (since the standard deviation is 7 and variance is the square of the standard deviation).
c. To find the probability of a trial lasting at least 24 days, we calculate the z-score and use standard normal distribution tables or technology. The probability statement is P(X ≥ 24). As for the graph, it's the area under the normal curve to the right of the value 24.
Sixty percent of trials are completed within the time frame we find by looking at the corresponding percentile of the normal distribution, which would be the 60th percentile.
73. Sample of Nine Trials:
a. The sum of durations for the nine trials is represented by ΣX.
b. The expected mean duration for the sample of nine trials can be represented by EX~, which is equal to the population mean since the expectation of the sample mean is equal to the population mean.