rainfall in the state of new york has a mean of 46.3 inches with a standard deviation of 5.6 inches.
you want to know the probability that the mean of rainfall in new york would be greater than 48 inches, given that the mean is 46.3 inches with a standard deviation of 5.6 inches.
your sample size is 45.
you would use the x-score formula of z = (x - m) / s.
z is the z-score.
x is the raw score you are testing against the mean.
m is the mean.
s is the standard error, otherwise known as the standard deviation of the distribution of sample means.
the formula for standard error is s = standard deviation of the population divided by the square root of the sample size.
that makes s = 5.6 / sqrt(45) = .8347987116.
your z-score formula becomes z = (48 - 46.3) / .83447987116 = 2.036419051.
the probability of getting a z-score greater than 2.036419051 is equal to .0208540785.
based on this, you would conclude that the probability for the average rainfall in new york in any given year being greater than 48 inches is about .0209 or 2.09%.
here's what the distribution would look like.
once you find the z-score, you have to find the area under the normal distribution curve that is to the right of it.
the online calculator i used does the job for me.
without that, you would have to find the area to the left of the z-score and then translate to the area to the right of the z-score by taking that figure and subtracting it from 1.
you can also use the z-score tables which are not quite as accurate but will get you in the ball park.
when using the tables, you would round your z-score to 2 decimal digits.
2.036 would round to 2.04.
the area to the left of a z-score of 2.04 in the z-score table would be equal to .97932.
that's the area to the left of the z-score.
the area to the right of z-score would be 1 - .97932 = .02068.
that's pretty close to the .0208540785 i got using the calculator.
here's the table i used.
https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf
here's the online calculator that i used.
http://davidmlane.com/hyperstat/z_table.html
the basic procedure for this problem would be:
get the value of the standard error which is equal to standard deviation divided by square root of sample size.
ust that standard error in the formula of z = (x - m) / s to find the z-score.
use a calculator or the z-score table to find the area to the right of that z-score.
any questions about any of this, let me know and i'll try to explain further in more detail.