I'm currently studying probability and statistics and had a question. Right now I'm learning about parameter estimation and the following passage is confusing me a bit:
Let us say that we are told that $\theta$ is approximately normal and with $90\%$ confidence, $\theta$ lies between $5$ and $9$, symmetrically around $7$. Then we can write $P(\theta)$ to be normal with mean $7$ because:
$$\begin{align} & P\left(-1.64 \lt \frac{\theta - \mu}{\sigma} \lt 1.64 \right) = 0.9 \\ & P\left( \mu - 1.64\sigma \lt \theta \lt \mu + 1.64\sigma\right) = 0.9\end{align}$$
we take $1.64\sigma = 2$ and use $\sigma = \dfrac{2}{1.64}$. We can thus assume:
$$P(\theta) \sim \mathcal{N}(7,\ (2/1.64)^2)$$
Since the passage said that our distribution lies "symmetrically around $7$", I can understand that $\mu = 7$. I can also understand that $1.64\sigma = 2$ because $\theta$ lies between $5$ and $9$, which are both two standard deviations away from the mean (i.e. $\mu$).
What does the $1.64$ mean here, and where did it come from?
Thanks.