if i have an unknown mean, a standard deviation of 4, and P(X < 8 ) = 0.3085, how do I calculate the mean somehow using the standard normal distribution and it's cummulative function?
I know that P(x<8 ) = 0.3085 is the same as Phi((8 - mean)/4) = 0.3085, and now I want to isolate the mean. I can firstly remove phi on the left side by using it's inverse function phi^-1, but how do I take the phi^-1(0.3085)?
I have no idea how to do it on a calculator, software, no nothing....
If you look at a Standard Normal table, e.g., http://www.mathsisfun.com/data/standard-normal-distribution-table.html , you can see that the probability, i.e., p-value, corresponds to , approximately, $Z=-0.5$. Then , using $X=8$, $\sigma=4$, you have: $$\frac{8-\mu}{4}=-0.5$$, so that $\mu=10$.