SOLUTION: While this question was answered, it wasn't helpful for me. Q.The volumes of apple juice in cartons are normally distributed with a standard deviation of 120 mL. Find the mean, t

Algebra ->  Probability-and-statistics -> SOLUTION: While this question was answered, it wasn't helpful for me. Q.The volumes of apple juice in cartons are normally distributed with a standard deviation of 120 mL. Find the mean, t      Log On


   

Question 1199053: While this question was answered, it wasn't helpful for me.
Q.The volumes of apple juice in cartons are normally distributed with a standard deviation of 120 mL. Find the mean, to the nearest whole number, if 34% of these cartons contain less that 961 mL of juice
A: I thought that 0.34 could be the probability of 'success' but I'm not sure of it or of where to go for this question. Much appreciated.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
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 https://davidmlane.com/hyperstat/z_table.html

this is what the results of that calculator look like:



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.



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