Question 1026699: A muffler company advertises that you will receive a rebate if it takes longer than 30 minutes to replace a muffler. Experience has shown that the time taken to replace a muffler is approximately normally distributed with a mean of 25 minutes and a standard deviation of 2.5 minutes.
a) What proportion of mufflers take between 22 and 26 minutes to replace?
b) What should be the rebate time of 30 minutes be changed to if the company wishes to provide only 1% of customers with a rebate?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! mean is 25.
standard deviation is 2.5.
use z-score to normalize.
z = (x-m)/s
x is the raw score.
m is the mean
s is the standard deviation.
x = 22
m = 25
s = 2.5
z = (22 - 25) / 2.5 = -1.2
x = 26
m = 25
s = 2.5
z = (26-25)/2.5 = 1/2.5 = .4
look in the z-score table for area under the distribution curve to the left of those z-scores.
area to the left for z-score of -1.2 = .0548
area to the left for z-score of .4 = .6554
area in between = .6554 - .0548 = .6006
.6006 of mufflers take between 22 and 26 minutes to replace.
in order to have only 1% of the customers get a rebate, look in the z-score table for the z-score that has 99% of the area under the normal distribution curve to the left of it.
that area will have a z-score of 2.33 which has .9901 area to the left of it.
that's just a little bit less than .01 to the right of it, because 1 - .9901 = .0099 which is a touch under .01.
to find the raw score associated with that, use the z-score formula again.
z = 2.33
x = what you want to find.
m = 25
s = 2.5
z = (x-m)/s becomes 2.33 = (x-25)/2.5
solve for x to get x = 2.33 * 2.5 + 25 which is equal to 30.825.
customers would get a rebate if it takes more than 30.825 minutes.
that would be approximately 30 minutes and 50 seconds.
that's pretty close to 31, so they would probably round that off to 31 minutes.
at the current existing rate, you would calculate the proportion using the z-score formula again.
m = 25
s = 2.5
x = 30
z = (30 - 25) / 2.5 = 2.
with a z-score of 2, the area to the left of the z-score is .9772, and so the area to the right side of the z-score is 1 - .9772 = .0228.
that's roughly 2.3% that get the rebate.
that's the percent that's currently getting the rebate.
increase to 31 minutes and z=(x-m)/2.5 becomes z = 2.4 which gets an area to the left of that z-score equal to .9918 which gets an area of 1 - .9918 = .0082 to the right of it.
that's .8% who would then get the rebate.
the z-score table i used can be found here:
http://www.stat.ufl.edu/~athienit/Tables/Ztable.pdf
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