Question

A perfume bottle is designed to have a capacity of 15 ounces. There is variation in the bottle manufacturing process. Based on historical data, let’s suppose that the bottle capacity can be reasonably modeled by a normal distribution with a mean of 15 ounces and a standard deviation of 0.2 ounces. What proportion of these bottles will have a capacity between 14.4 and 15.6 ounces?

Answer 1

Answer:

0.9974 proportion of these bottles will have a capacity between 14.4 and 15.6 ounces

Step-by-step explanation:

The bottle capacity can be reasonably modeled by a normal distribution with a mean of 15 ounces and a standard deviation of 0.2 ounces.

Mean = 15 ounces

Standard deviation = 0.2 ounces

We are supposed to find proportion of these bottles will have a capacity between 14.4 and 15.6 ounces i.e.P(14.4<x<15.6)

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

At x = 15.6

$z=\frac{15.6-15}{0.2}$

z=3

P(z<3)=0.9987

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

At x = 14.4

$z=\frac{14.4-15}{0.2}$

z=-3

P(z<-3)=.0013

P(14.4<x<15.6)=P(-3<z<3)=P(z<3)-P(z<-3)=0.9987-0.0013=0.9974

Hence 0.9974 proportion of these bottles will have a capacity between 14.4 and 15.6 ounces