Question

assume the random variable x is normally distributed with mean =82 and standard deviation =5. Find the probability P(x < 75)

Answer 1

The following parameters are provided in the question:

$\begin{gathered} x=75 \\ \mu=82 \\ \sigma=5 \end{gathered}$

First, we will calculate the z-score:

$z=\frac{x-\mu}{\sigma}$

Substituting, we have:

$\begin{gathered} z=\frac{75-82}{5}=\frac{-7}{5} \\ z=-1.4 \end{gathered}$

The distribution curve and the area that represents the probability is shown below:

Therefore, the probability is calculated to be:

$P(x<75)=Pr(z<-1.4)=Pr(Z<0)-Pr(0Using the area under the normal curve calculator, we have that:[tex]\begin{gathered} Pr(Z<0)=0.5 \\ Pr(0Therefore, the probability will be:[tex]\begin{gathered} P(x<75)=0.5-0.4192 \\ \therefore \\ P(x<75)=0.0808 \end{gathered}$