I would like to understand in the third snippet below in the context of statistical hypotheses testing how was obtained the formula (namely the numbers $1-,22.5$ and $-3\theta$) in the example 8.2.3 originated from the example 8.2.1:
$$Q(\theta)=1-\Psi(22.5-3\theta)$$
It actually says $Q(\theta)=1-\Phi(22.5-3\theta)$ where $\Phi$ is the cumulative distribution function of a standard normal distribution.
Here the underlying population is $\mathcal N(\theta, 1)$ so the average of a sample size $9$ has mean $\theta$ and standard deviation $1/3$.
So the probability the sample average is less than $7.5$ is $\Phi\left(\dfrac{7.5 - \theta}{1/3}\right) = \Phi\left({22.5 - 3\theta}\right)$
and the probability the sample average is greater than $7.5$ is $1-\Phi(22.5 - 3\theta)$.
This is the power of a test that rejects $H_0$ iff $\bar X > 7.5$