SOLUTION: the weight of a shipment of canned food are normally distributed. if 0.15% of them are lighter than 347 g and 2.5% of them are heavier than 377 g, find mean and the standard deviat

Algebra ->  Statistics  -> Density-curves-and-normal-distributions -> SOLUTION: the weight of a shipment of canned food are normally distributed. if 0.15% of them are lighter than 347 g and 2.5% of them are heavier than 377 g, find mean and the standard deviat      Log On


   

Question 1073285: the weight of a shipment of canned food are normally distributed. if 0.15% of them are lighter than 347 g and 2.5% of them are heavier than 377 g, find mean and the standard deviation of the weights of the canned food
Found 2 solutions by Theo, Boreal:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
alpha on the left is .0015
alpha on the right is .025

z-score on the left is -2.968
z-score on the right is 1.96

z-score for the mean is 0.

the difference between the high z-score and the low z-score is equal to 2.968 + 1.96 = 4.928

the difference between the high raw score and the low raw score is equal to 377 - 347 = 30.

the standard deviation of the raw score should be 30 / 3.928 = 6.087662338

if that's true, than you can solve for the mean as follows:

z = (x-m)/sd

solve for m in this formula as follows:

start with z = (x-m)/sd

multiply both sides by sd to get z * sd = x - m

subtract x from both sides to get z * sd - x = -m

multiply both sides by -1 to get -z * sd + x = m

rearrange the terms to get x - z * sd = m

commute the equation to get m = x - z * sd

with the low raw score and the low z-score and the calculated sd, this formula becomes:

m = 347 - (-2.968 * 6.087662338 = 365.0681818

with the high raw score and the high z-score and the calculated sd, this formulabecomes:

m = 377 - 1.968 * 6.087662338 = 365.0681818.

that's your answer.

the mean is 365.0681818

the standard deviation is 6.087662338

the z-score calculator i used can be found at this link:

http://davidmlane.com/hyperstat/z_table.html

the following pictures show the calculations.

the first 2 pictures calculate the z-score from the percentages.

the second 2 pictures calculate the percentages from the z-score.

the third 2 pictures calculate the percentages from the raw score.

the percentages used and shown are the decimal equivalent of the percentages.

.15% = .0015
2.5% = .025

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Answer by Boreal(15235) About Me  (Show Source):
You can put this solution on YOUR website!
x1=347; x2=377
0.15% is 2.967 sd less than the mean.
2.5% is 1.96 sd greater than the mean.
z=(x-mean)/sd
1.96*sd=377-mean;mean+1.96 sd=377
-2.967*sd=347-mean; mean-2.967=347
therefore, 377-1.96sd=347+2.967sd
30=4.927 sd
sd=6.089
mean=347+2.967(6.089)=347+18.1=365.1
mean=377-1.96(6.089)=377-11.93=365.1
mean is 365.1
sd is 6.09