According to my calculations, the average height of the unit hyper-hemisphere in $\mathbb{R}^n$ is given by the formula

$h_n:=\frac{S_{n-2}}{V_{n-2}} \int_{r=0}^1 r^{n-2} \sqrt{1-r^2} dr$,

where $S_{n-2}$, $V_{n-2}$ is respectively the surface and volume of the hypersphere in $\mathbb{R}^{n-1}$.

Question 1: How can we understand the quantity $h_n$? In particular, is $h_n$ increasing? Does it tend to $1$ as $n$ goes to infinity?

Question 2: Let $x$ be a random variable uniformly distributed on the unit sphere $S^{n-1}$ and let $b$ be any fixed vector on the sphere. According to my understanding the expected value of the vector $\operatorname{sign}(b^\top x) x$ is equal to $h_n \cdot b$. Is this correct?