Finding probability of Sample standard deviation given population is normally distributed For a random sample of size $n,$ if the values are taken from the $N(a, b)$ population, what is the probability that $S$ (where $S^2$ is sample variance) will exceed a particular value?
I had 2 approaches in mind here. Please let me know which one is correct

*

*Since $(n-1) S^2/\sigma^2$ is chi square distributed, can I say the square root of term on LHS is normally distributed?


*Use $x-\mu/(S/\sqrt{n})$ which is t distributed to find the answer
 A: *

*I didn't get which LHS you meant, but the square root of $\chi^2(n)$ is not normal (or maybe it's not what you meant). Also you almost found the credible interval you were looking for. Since $\cfrac{(n - 1) S^2}{\sigma^2}$ is $H = \chi^2(n-1)$ you may just say $\mathbb{P} \Bigg( \chi^2_{\frac{\alpha}{2}, n-1} \leq H \leq \chi^2_{1 - \frac{\alpha}{2}, n-1} \Bigg) = 1 - \alpha$, multiply the insides by $\sigma^2$ and get $\mathbb{P} \Bigg( \cfrac{(n - 1) S^2}{\chi^2_{\frac{\alpha}{2}, n-1}} \leq \sigma^2 \leq \cfrac{(n - 1) S^2}{\chi^2_{1 - \frac{\alpha}{2}, n-1}} \Bigg) = 1 - \alpha$. For exactly $\sigma$ and not $\sigma^2$ you can take a square root then. Also notice $S^2$ is the unbiased sample variance.

A: Here is your problem for a specific case:
Suppose $n = 10, \sigma = 15,$ and your desired
cutoff value $c$ has $c^2 = 200.$
Then, $\frac{(n-1)S^2}{\sigma^2} \sim \mathsf{Chisq}(\nu=n-1).$ So
$$P(S > c) =P(S^2 > c^2) = P\left(\frac{9S^2}{225} > \frac{9(200)}{225} = 8\right)\\ = P(Q > 8) = 1 - P(Q\le 8) = 0.5341,$$
where $Q\sim\mathsf{Chisq}(\nu=9).$ You can evaluate this
probability (at least approximately) from printed tables
of chi-squared distributions or (exactly) by using software. Using R, where pchisq is a chi-squared CDF, the answer is obtained as follows:
1 - pchisq(8, 9)
[1] 0.5341462

In the figure below, you want the area under the density curve to the right of the vertical dotted line.

curve(dchisq(x, 9), 0, 30, lwd=2, ylab="PDF", xlab="q", main="")
abline(h=0, col="green2");  abline(v=0, col="green2")
abline(v = 8, col="red", lwd=2, lty="dotted")

