Question 1202778: In New York State, the mean salary for high school teachers in 2017 was $105,410 with a standard deviation of $9,680. Only Alaska’s mean salary was higher! Assume New York’s state salaries follow a normal distribution.
a. What percent of New York’s state high school teachers earn between $90,000 and $95,000?
b. What percent of New York’s state high school teachers earn between $95,000 and $110,000?
c. What percent of New York’s state high school teachers earn less than $80,000?
Answer by ikleyn(52867)   (Show Source):
You can put this solution on YOUR website! .
In New York State, the mean salary for high school teachers in 2017 was $105,410 with a standard deviation of $9,680.
Only Alaska’s mean salary was higher! Assume New York’s state salaries follow a normal distribution.
(a) What percent of New York’s state high school teachers earn between $90,000 and $95,000?
(b) What percent of New York’s state high school teachers earn between $95,000 and $110,000?
(c) What percent of New York’s state high school teachers earn less than $80,000?
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The major starting point in solving this and many other similar problems is that
the wording description only DISTRACTS your attention.
All you need to know are the parameters of the appropriate/relevant normal curve:
its mean value and its standard deviation.
In your case, the mean is 105410 dollars; the standard deviation is 9680 dollars.
The probabilities they want from you, are the areas under this normal curve
in the designed diapasons.
Now see how it works.
(a) In part (a), they want you determine the area under the normal curve
between the raw mark of 90000 and 95000 dollars.
In part (a), same as in other parts (b) and (c), the mean is 105410; the standard deviation is 9680.
You may use the standard function "normalcdf" in your calculator TI-83 or TI-84.
z1 z2 mean SD <<<---=== formatting pattern
P = normalcfd(90000, 95000, 105410, 9680)
The calculator gives the answer P = 0.0854, or 8.54%.
(b) In part (b), they want you determine the area under the normal curve
between the raw mark of 95000 and 11000 dollars (very similar to part (a)).
In part (b), same as in other parts (a) and (c), the mean is 105410; the standard deviation is 9680.
Again, you may use the standard function "normalcdf" in your calculator TI-83 or TI-84.
z1 z2 mean SD <<<---=== formatting pattern
P = normalcfd(95000, 110000, 105410, 9680)
The calculator gives the answer P = 0.5412, or 54.12%.
(c) In part (c), they want you determine the area under the normal curve
on the left of the raw mark of 80000 dollars.
In part (c), same as in other parts (a) and (b), the mean is 105410; the standard deviation is 9680.
Again, you may use the standard function "normalcdf" in your calculator TI-83 or TI-84.
z1 z2 mean SD <<<---=== formatting pattern
P = normalcfd(-9999, 80000, 105410, 9680)
The calculator gives the answer P = 0.0043, or 0.43%.
At this point, the problem is just solved.
But I want present you an alternative way to solve this and similar problems.
Go to website https://onlinestatbook.com/2/calculators/normal_dist.html
and use free of charge calculator there.
It allows to do all these calculations, and every time it shows you
the area of interest under the normal curve.
Its interface is very friendly, very simple and very convenient.
It will help you to learn the subject in 5 - 10 minutes.
You may play with this calculator as long as you need, until everything in the subject
becomes clear to you.
After that, you may return to your regular hand calculator TI-83 / TI-84
enriched with full understanding of the subject.
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