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Question 953348: The average beginning annual salary for teachers in California is $41,181. Assume that this is a normal distribution with a standard deviation of $ 725. One quarter of starting teachers in California will make how " at least" much money per year? (Round to the nearest dollar)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! average beginning salary is 41,181.
standard deviation is 725.
you want to know at least how much money 25% of the starting teachers in California will make.
I'm presuming that you are talking about the top 25% of the starting teachers.
I'll assume that since it makes the most sense and there is no information indicating otherwise.
If these teachers are in the top 25%, that means that 75% of the teachers are in the bottom 75%.
If you look in the z-score table, you will find that the z-score that has 75% of the area under the normal distribution curve is somewhere between .67 and .68.
z-score of .67 has 74.86% of the area under the normal distribution curve to the left of it.
z-score of .68 has 75.17% of the area under the normal distribution curve to the left of it.
The actual z-score that has 75% of the area under the normal distribution curve to the left of it is somewhere in between.
You can interpolate, or you can use a z-score calculator, or you can use the z-score that has an area closest to 75%, or you can use both z-score figures and say that the answer is somewhere between them.
I will split the difference and say that the z-score is .675.
This is close enough for government work, except maybe landing on the moon.
So the z-score is determined to be .675.
Now you want to find the raw score associated with that.
the formula for z-score is:
z = (x - m) / s
z is the z-score.
x is the raw score.
m is the mean score.
s is the standard deviation.
when z = .675 and m = 41181 and s = 725, the formula becomes:
.675 = (x - 41181) / 725.
solve for x to get:
x = .675 * 725 + 41181 which makes x = 41670.375 which you can round off to 41,670.
The top 25% of the starting teachers will make a salary of at least 41,670 give or take a few dollars.
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