Getting standard deviation given probability

Hi all
If you have a mean of 2 and you want to get standard deviation with probability of 90% that the new mean will lie between 1.5 and 2.5. I tried to solve out this question by considering the inverse of Z score
Probability of 90% correspond to z =1.281 then I don't know how to solve out the following equation to get t0 (standard deviation)

Could you help me to solve this equation


Omega Contributor
I don't follow your question or what you are explicitly asking. If this is a problem for a book or course, can you post the original question.

This is a problem in a course and it is about Byesian analysis. The problem is asking about estimating a priori estimate in the population. The mean measurement in the population is around 2, and we will assume there is a 90% probability of it being between 1.5 and 2.5. So the priori estimate follow normal distribution with a mean of 2 and we need to estimate standard deviation


TS Contributor
Ok so what you are saying - just to make sure we understand you - is that you have a population mean \( \mu\) which i assume to be normally distributed \( \mu \sim \mathcal N(\mu_0,\tau^2_0)\) where \( \mu_0=2\) and \( \tau_0^2\) is undecided but youre beliefs are that \( Pr(1.5\leq\mu \leq 2.5)=0.9\) - not 1.2815 as in youre picture because how can the probability be more than 1? - anyway you want to use this information in order to decide which prior to use and therefore need to find the \( \tau_0^2\) such that \( Pr\left(\frac{1.5-\mu_0}{\tau_0}\leq \frac{\mu-\mu_0}{\tau_0}\leq \frac{2.5-\mu_0}{\tau_0}\right)=0.9\)?