Answer to Question #324246 in Statistics and Probability for zzzzz

Question

Given a normal distribution with μ = 180 and σ = 20, find

a. The area above 230

b. The area below 215

c. The area between 185 and 225

Answer 1

We have a normal distribution, $\mu=180, \sigma=20.$

Let's convert it to the standard normal distribution,

$z=\cfrac{x-\mu}{\sigma}.$

$\text{a. } z_1=\cfrac{230-180}{20}=2.50;\\ P(X>230)=P(Z>2. 50)=\\ =1-P(Z<2.50)=$

$=1-0.9938=0.0062$ ​(from z-table).

$\text{b. } z_2=\cfrac{215-180}{20}=1.75;\\ P(X<215)=P(Z<1.75)=0.9599$

 ​(from z-table).

$\text{c. } z_3=\cfrac{185-180}{20}=0.25;\\ z_4=\cfrac{225-180}{20}=2.25;\\ P(185<X<225)=P(0.25<Z<2.25)=\\ P(Z<2.25)-P(Z<0.25)=\\ =0.9878-0.5987=0.3891.$

 ​(from z-table).