Question 1121793: The accuracy of a cutting machine is such that the mean length is the length it is set to and the standard deviation is 2.2mm. The lengths follow a normal distribution. The engineer wants to make sure that no more than 20% of the lengths cut are longer than 400mm. (It is less of a problem if the lengths are a little short.) What should she set the mean length to be?
Hint: Find the value of z for which Pr(Z>z)=0.20 then use the z-score formula to calculate the mean.
Do not give the units in your answer; just give the number.
Give the answer to two decimal places.
Thank you so much in advance
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if you want 20% to the right of the critical z-score, you can use the following online calculator to give you the resulting z-score.
that's the online z-score calculator that i used.
http://davidmlane.com/hyperstat/z_table.html
the z-score table i used can be found at http://www.z-table.com/
use of this calculator will tell you that the critical z-score is .841.
my straight line interpolation from the use of the z-score table gets me a z-score of .8417857143 which equals .842 rounded to 3 decimal places.
my TI-84-Plus calculator tells me the z-score is .8416212335 which equals .842 rounded to 3 decimal places.
what this points out is that you might get a different critical z-score depending on the method or calculator used to get that critical z-score.
either way, you will get a reasonably accurate result.
i went with critical z-scores of .841 and .842 and .84 to see what the difference would be.
the formula used to find the mean is derived from the z-score formula shown below.
z = (x - m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard deviation.
solve for the mean in this formula as follows:
start with z = (x - m) / s
multiply both sides by s to get z * s = x - m
add m to both sides of this formula and subtract z * s from both sides of this formula to get:
m = x - z * s
m = x - z * s is the formula to find the mean given the critical z-score and the critical raw score and the standard deviation.
the critical raw score used in this formula is 400 and the standard deviation used is 2.2.
when i used .841 as the critical z-score, this formula became:
m = 400 - .841 * 2.2 = 398.1498
when i used .842 as the critical z-score, this formula became:
m = 400 - .842 * 2.2 = 398.1476.
when i used .84 as the critical z-score, this formula became:
m = 400 - .84 * 2.2 = 398.152.
when z = .841, this is the result that i got:
when z = .842, this is the result that i got:
when z = .84, this is the result that i got:
you can see that:
a z-score of .841 gave me a mean of 398.1498 with an area of 20.02% to the right of the critical raw score of 400.
a z-score of .842 gave me a mean 398.1476 with an area of 19.99% to the right of the critical raw score of 400.
a z-score of .84 gave me a mean of 398.152 withy an area of 20.05% to the right of the critical raw score of 400.
the z-score of .842 was the better z-score because it resulted in an area to the right of it as something slightly less than 20%.
however, any one of these z-scores would have given me an answer that was well within 1% of the required area of 20% to the right of the critical z-score.
the point is that you will get different answers depending on the method used to derive the critical z-score, but those answers will more then likely be well within a reasonably acceptable range of possible answers.
since they want your answer to 2 decimal places, it appears that all of these will give you the same answer.
that answer will be a mean of 398.15 rounded to 2 decimal places.
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