Asoft drink machine outputs a mean of 27 ounces per cup. the machine's output is normally distributed with a standard deviation of 3 ounces. what is the probability of filling a cup between 24 and 32 ounces? round your answer to four decimal places.

The probability will be the area under the normal distribution  from 24 to 32.

We normalize these so we can use the standard normal distribution





Typically we'd just use a computer to tell us the integral of the standard normal from -1 to 5/3.  In the old days we'd use the z table.  It tells us the normal integral from 0 to positive z.

We know the integral of the Gaussian from -1 to 1 standard deviation is 68%; from the z table we see the probability from 0 to 1 is half that, 0.34134.  That will also be the integral from -1 to 0. 

The z table says the integral from 0 to 5/3, 1.67, is 0.45254.

So our total probability, integral from -1 to 5/3, is

 

That's really from z=-1 to z=+1.67 so likely a bit off in the last couple digits.

What am i doing here. add?