# If $X$ a normal variate with mean $80$ and standard deviation $10$ compute the following probabilites by standardizing. $P(70$$<X) ## 1 Answer Toolbox: • Standard normal distribution: • In a standard normal distribution \mu=0,\sigma ^2=1 • The random variable X can be converted to the standard normal variable Z by the transformation • Z=\large\frac{X-\mu}{\sigma} • The probability density function Z is \phi(z)=\large\frac{1}{\sqrt{2\pi}}$$e^{-\Large\frac{1}{2}z^2};-\infty < Z < \infty$
• $Z\sim N(0,1)$
Step 1:
$X\sim N(80,10^2)$
Let $Z=\large\frac{X-\mu}{\sigma}$
The probability density function $Z$ is $\phi(z)=\large\frac{1}{\sqrt{2\pi}}$$e^{-\Large\frac{1}{2}z^2};-\infty < Z < \infty Step 2: P(70 < X) When X=70 Z=\large\frac{70-80}{10}$$=-1$
$P(70 < X)=P(-1 < Z)$
$\qquad\qquad=P(0 < Z < 1)+0.5$
$\qquad\qquad=0.3413+0.5$
$\qquad\qquad=0.8413$