Question

The volumes of soda in quart soda bottles are normally distributed with a mean of 32 ounces and a standard deviation of 1.2 ounces. what percentage of soda bottles will have a volume less than 31.46 ounces

Answer 1

The answer for this question is 70%

Answer 2

32.64%

Explanation:

Let's define the random variable X in the following way:

X: the volume of a soda in a quart soda bottle. We know that X is normally distributed with a mean of 32 ounces and a standard deviation of 1.2 ounces, i.e.,

`\mu` = 32 ounces

`\sigma` = 1.2 ounces

The normal density function for a random variable with a mean of 32 and a standard deviation of 1.2 is given by

`f(x)=(1)/(√(2\pi )1.2) \exp[-((x-32)^(2) )/(2(1.2)^2) ]`

and we need to calculate the following probability

`P(X\leq 31.46)`, this probability is given by

`P(X\leq 31.46) =\int\limits^(31.46)_(-\infty) {(1)/(√(2\pi )1.2) \exp[-((x-32)^(2) )/(2(1.2)^2) ]} \, dx` = 0.3263552

you can use a computer to calculate this probability or a table from a book. You can use the following instruction in the R statistical programming language for example

pnorm(31.46, mean = 32, sd = 1.2) with give us 0.3263552, then,

the percentage we are looking for is (0.3263552)(100)=32.64.