Question 1168066: A particular fruit's weights are normally distributed, with a mean of 756 grams and a standard deviation of 14 grams.
The heaviest 20% of fruits weigh more than how many grams?
Give your answer to the nearest gram.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the mean is 756 grams.
the standard deviation is 14 grams
look into the z-score table.
you want to find the z-score with an area of .2 to the right of it.
the z-score table only shows the area to the left of the z-score.
subtract .2 from 1 to get .8.
you want to find the z-core with an area of .8 to the left of it.
that will be the same z-score with an area of .2 to the right of it.
the z-score table that i use can be found at https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf
looking into the table, i found:
an area of .79955 is associated with a z-score of .84
an area of .80234 is associated with a z-score of .85
the difference between those areas is .00279 and the difference between those z-scores is .01
take the highest area and subtract .2 from it to get a difference of .00234.
divide that difference from the overall difference of .00279 to get a ratio of .00234 / .00279 = .8387096774.
multiply the difference between the two z-scores of .01 to get .01 * .8387096774 = .0083870968.
subtract .008370968 from the z-score of .85 to get a z-score of .85 - .008370968 = .841612903323.
that's a reasonably close approximation of what the real z-score should be.
you can also use a z-score calculator that will also get you presumably more accurate answer with a lot less headaches.
the z-score calculator on the TI-84 Plus that i have gave me a z-score of .84162122335.
the z-scores are the same out to 4 decimal digits which is usually good enough.
anyway, now that you have the z-score, you can solve for the raw score.
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 or the standard error
in this case, s is the standard deviation.
replace m and s and z with their respective values to solve for x.
you get .841612903323 = (x - 756) / 14.
solve for x to get x = .841612903323 * 14 + 756 = 767.7825806.
80% of the raw scores will be less than that and 20% of the raw scores will be higher than that.
that can be seen through the use of the following calculator that can be found at https://www.omnicalculator.com/statistics/normal-distribution
here's a display of the results of using that calculator.
my inputs were mean of 756 standard deviation of 14 and p(x > X) of .2
the calculator filled in the rest.
it told me that the value of X was 767.783.
that's very close to the raw score i calculated manually of 767.783 rounded to 3 decimal places.
unless you have to use the tables and do the manual interpolation, the recommendation is to use the calculator.
it's much easier.
anyway, round the raw score to the nearest gram to get 768 grams as your answer.
fyi, i used the z-score that i manually calculated through interpolation.
in most cases, i would just use the z-score i got through the use of a calculator.
i only did it manually to show you how the interpolation works.
if you use the tables, you could also just use the z-score associated with the area closest to the one that you want and that will be good enough.
your instructor should be your guide as to how close an answer you need to get.
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