Question

. A normal population has a mean of 35 and a standard deviation of 8. a. What proportion of the population is between 20 and 30? b. What is the probability that a randomly chosen value will be between 30 and 40? solve in R

Answer 1

Answer: a. The proportion of the population is between 20 and 30 =0.2356

b. The  probability that a randomly chosen value will be between 30 and 40 =0.4680

Step-by-step explanation:

Given : Mean : $\mu=35$

Standard deviation : $\sigma = 8$

The formula to calculate z-score :-

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

For x= 20

$z=\dfrac{20-35}{8}=-1.875$

For x= 30

$z=\dfrac{30-35}{8}=-0.625$

For x= 40

$z=\dfrac{40-35}{8}=0.625$

a.

$P(20<x<30)=P(-1.875<z<-0.625)\\= P(-0.625)-P(-1.875)\\=0.2659855-0.0303964=0.2355891\approx0.2356$

b.

$P(30<x<40)=P(-0.625<z<0.625)\\= P(0.625)-P(0.625)\\=0.7340144-0.2659855=0.4680289\approx0.4680$