Q.The probability density function of a normal variable $X$ is defined as under $f(x) = \text{constant} \cdot e^{-\frac{1}{2}\left(\frac{x - 25}{10}\right)^2}; \; -\infty < x < \infty$. From this normal distribution estimate the values of the following:

Middle $50\%$ of the standard normal: $-0.675\le Z\le0.675$.

For the standard normal variable $Z$ (mean $0$, SD $1$), the central $50\%$ of the area lies between the first quartile $Q_1$ and the third quartile $Q_3$. Since $Q_1=-0.6745$ and $Q_3=+0.6745$ (each cutting off $25\%$ of the area between it and the mean) …

By the symmetry of the normal distribution, $Q_3-\bar X=\bar X-Q_1$, so $Q_3=11+(11-3)=19$.

In a normal distribution the mean equals the median $Q_2$, and the first and third quartiles are equidistant from the mean:

$\bar X-Q_1=Q_3-\bar X.$ …