Question 1199053
34% is the probability that the cartons contain less than 961 ml of juice.
that is also the area to the left of of the z-score represented by that value.


the z-score formula is z = (x - m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard deviation


the z-score that has .34 of the area to the left of it is equal to -.412463135, according to the online calculator at <a href = "https://davidmlane.com/hyperstat/z_table.html" target = "_blalnk">https://davidmlane.com/hyperstat/z_table.html</a>


this is what the results of that calculator look like:


<img src = "http://theo.x10hosting.com/2022/120901.jpg">


since you are working with the z-score, the mean is made equal to 0 and the standard deviation is made equal to 1.


the z-score that has .34 of the area under the normal distribution curve is equal to -.412, according to that calculator.


to find the mean, you would use the z-score formula.


in this problem, that formula becomes:
-.412 = (961 - m) / 120
-.412 is the z-score
961 is the raw score]
m is the mean
120 is the standard deviation.


multiply both sides of that formul by 120 to get:
-.412 * 120 = 961 - m
add m to both sides of that equation and add .412 * 120 to both sides of that equation to get:
m = 961 + .412 * 120
solve for m to get m = 1010.44
that's your mean.
round to the nearest integer to get 1010.


here's what it looks like on that calculator.


<img src = "http://theo.x10hosting.com/2022/120901.jpg">


since you are working with the raw score, the mean is made 1010.44 and the standard deviation is made 120.


let me know if you have any questions.
theo