Answer to Question #333508 in Statistics and Probability for sullyoon

Question

2. The mean height of Grade 11 students at a certain senior high school is 164 cm. and the standard deviation is 10 cm. Assuming that the heights are normally distributed, what percent of the height is

a. greater the 168 cm?

b. less than 150 cm

c. between 150 and 168

Answer 1

We have a normal distribution, $\mu=164, \sigma=10.$

Let's convert it to the standard normal distribution,

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

$\text{a. }z_1=\cfrac{168-164}{10}=0.40; \\P(X>168)=P(Z>0.40)=\\ =1-P(Z<0.40)=\\ =1-0.6554=0.3446=34.46\%\text{ (from z-table)}.$

$\text{b. }z_2=\cfrac{150-164}{10}=-1.40; \\P(X<150)=P(Z<-1.40)=\\ =0.0808=8.08\%\text{ (from z-table)}.$

$\text{c. }P(150<X<168)=\\ =P(-1.40<Z<0.40)=\\ =P(Z<0.40)-P(Z<-1.40)=\\ =0.6554-0.0808=0.5746=57.46\%.$