Finding the mean and standard deviation on a z-score This is a problem that I don't know what to do. Our teacher didn't even teach us how to do this so I'm quite lost on how to explain this.
Here is the given question
In a given word problem, it is customary that all values are given. In the case of a word problem with normal distribution, it is common that the values of the score, the mean, and the standard deviation are given to compute for the value z. Likewise, it is also common that the value of z is already given. But what if in a given word problem concerning normal distribution, the values of the score (X) and the mean are missing and are required to be solved, how will you solve it? What if the word problem requires a percentage? How should you solve it?
 A: The usual statement for a standard score is $z=\frac{x-\mu}{\sigma}$ and you can solve for any one of the four given the other three with

*

*$z=\frac{x-\mu}{\sigma}$ (assuming $\sigma > 0$)

*$\sigma=\frac{x-\mu}{z}$  (if $z=0$ then $x=\mu$ and you cannot solve for $\sigma$; otherwise $z$ and $x-\mu$ must have the same sign)

*$x = \mu+z \sigma $

*$\mu = x- z \sigma$
If you also have a probability such as $p=\mathbb P(Z\le z)$ and a normal distribution then you can use the standard normal cumulative distribution function and its inverse to say $p = \Phi(z)$ or $z=\Phi^{-1}(p)$.  So knowing $p$ or $z$ implies you know the other.
You can then extend this for example to $\mathbb P(X \le x)  = \Phi\left(\frac{x-\mu}{\sigma}\right) $ or $x = \mu+\Phi^{-1}(p) \sigma $ or similar results.
In many problems, not knowing two of $x$ or $\mu$ or $\sigma$ or both of $z$ and $p$ could cause difficulties in being able to determine them.
Probability percentages may be easier to handle rewritten as decimal fractions: for example $95\% = 0.95$.
