Suppose that the duration of a particular type of criminal trial is known to be normally distributed with a mean of 21 days and a standard deviation of seven days.

a. In words, define the random variable X.

b. X ~ _____(_____,_____)

c. If one of the trials is randomly chosen, find the probability that it lasted at least 24 days. Sketch the graph and write the probability statement.

d. Sixty percent of all trials of this type are completed within how many days?

the duration of a particular type of criminal trial is known to be normally distributed with a mean of 21 days and a standard deviation of seven days.

We have to define the random variable$X$.

A random variable is a variable that reflects the probability in a random experiment. Upper case alphabets such as $X,Y,$and so on are used to represent it.

The given case entails keeping track of the length of a specific type of criminal trial. Every trial at a particular interval has a different duration. As a result, it can be thought of as a random variable with the following definition:

$X:$The number of days that a certain sort of criminal trial lasts (in days).

$X$seems to be a continuous random variable because it can take an endless number of values in a given interval.

We have to fill$X~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

$X$is a random variable with a mean of $21$and a standard deviation of $7$that represents the length (in days) of a specific sort of criminal trial. If a random variable $X$has a normal distribution with a mean of $\mu$and a standard deviation of s, it is denoted as $X~N(\mu ,s)$. As a result, the random variable $X$in this situation is denoted as: $X~N(21,7)$

We need to figure out what the chances are that a randomly picked trial will last at least 24 days.

The probability that a randomly selected trial will last at least 24 days is calculated as follows:

$$\begin{gathered} P(x\ge 24)=P\left(\frac{x-\mu }{\sigma }\ge \frac{24-21}{7}\right) \\ =P(z\ge 0.4286) \end{gathered}$$

The shaded zone in the graph below depicts the needed probability. As a result, it can be written:

$$\begin{gathered} P(z\ge 0.4286)=0.5-P(0\le z\le 0.4286) \\ =0.5-0.1659 \end{gathered}$$

From normal table : $0.3341$

We must determine the number of days required to complete 60% of criminal cases.

The needed probability is calculated as follows:

$P(x<k)=P\left(\frac{x-\mu }{\sigma }<\frac{k-21}{7}\right)=P\left(z<z_1\right)=0.6$

We get the following results from the normal distribution tables:

$z_1=0.2533$

From the equation $x=\mu +z\sigma$, we have to replace $x$with $k$and $z$with $z_1$

$$\begin{gathered} k=21+0.2533*7 \\ =22.77 \end{gathered}$$