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Suppose that the duration of a particular type of criminal trial is known to have a mean of $21$days and a standard deviation of seven days. We randomly sample nine trials.

definition for $\underset{}{∑}X$: the total length of time for nine criminal trials

The sum of random variables in normal distribution is given by, $\underset{}{∑}X=N(n\mathrm{μ},\sqrt{n}\mathrm{σ})$

Plugging all the values in the above equation, we get,

$$\begin{gathered} \underset{}{∑}X=N(9×21,\sqrt{9}×7) \\ \underset{}{∑}X=N(189,21) \end{gathered}$$

the probability that the total length of the nine trials is at least $225$days.

role="math" localid="1651602155511" $$\begin{gathered} P(\underset{}{∑}X≤225)=normalcdf(lower,upper,n\mathrm{μ},\sqrt{n}\mathrm{σ}) \\ P(\underset{}{∑}X≤225)=normalcdf(225,1E99,189,21)=0.0432 \end{gathered}$$

Let$k=the90thprecentile.$

Find $k$, where $P(\underset{}{∑}x<k)=0.90.$

$\text{invNorm}\left(0.90,189,21\right)=162.09$

ninety percent of the total nine trials of this type will last$162$days or more.