Answer to Question #290410 in Statistics and Probability for lusientoo

Question

Given a standard normal distribution with mean =78 and standard deviation = 28, find P(85 < X < 102)

Answer 1

Given: $\mu=78,\sigma=28$

$P(85 < X < 102)=P(X<102)-P(X<85)$

Now,

$\begin{aligned} & P[x <102] \\ =& P\left[z <\frac{x-\mu}{\sigma}\right] \\ =& P\left[z <\frac{102-78}{28}\right] \\ =& P[z <0.857] \\ =& 0.80428 \\ \end{aligned}$

and

$\begin{aligned} & P[x <85] \\ =& P\left[z <\frac{x-\mu}{\sigma}\right] \\ =& P\left[z <\frac{85-78}{28}\right] \\ =& P[z <0.25] \\ =& 0.59871 \\ \end{aligned}$

So,

$P(85 < X < 102)=P(X<102)-P(X<85)=0.80428-0.59871=0.20557$