How do you.....?

Question

Find the 95% confidence interval for the STANDARD DEVIATION, not the mean (which is =mean -/+ z (std.deviation/square root n))? I can't find it in my text, please help!

Answer 1

A parametric confidence interval for a population variance involves the chi-square distribution. For a given sample size n and sample variance...

(n-1)s^2 / X^2_L < sigma^2 < (n-1)s^2 / X^2_U, where

X^2_L is the 2.5 percentile of a chi-square with n-1 degrees of freedom and

X^2_U is the 97.5 percentile of a chi-square with n-1 degrees of freedom.

Of course this is for a population variance. Although it's not totally kosher to do this, taking the square root of the lower and upper limits could give an approximate 95% confidence interval. In other words, for a 95% confidence interval for the population sd, use

( sqrt{(n-1)s^2 / X^2_L}, sqrt{(n-1)s^2 / X^2_U} ).

Now, this confidence interval can only really be used if the population is normal. You would need to have a pretty large sample before the Central Limit Theorem would kick in for this one. If you are taking a class in elementary statistics, the above would be good enough. However, if not, I would suggest taking another approach. Using a nonparameteric bootstrap would probably be a better way to go, especially if the sample is looking to be different from the normal.

Answer 2

Here's a page that will calculate it for you, but let me see if I can find a formula or table or something where you can caluculate it for yourself...
http://home.ubalt.edu/ntsbarsh/Business-stat/otherapplets/Esteem.htm
see also
http://wind.cc.whecn.edu/~pwildman/statnew/confidence_interval_about_a_variance_or_standard_deviation.htm

The closest I could find to something like you are asking for, is finding confidence intervals for mean when standard deviation is unknown. This uses the t distribution instead of the z distribution.
http://www.andrews.edu/~calkins/math/edrm611/edrm09.htm
You could probably solve for a confidence interval of sigma from this, but it wouldn't be any simple formula, it would be a relation between the z and t formulas, which are both integral defined functions. Also I found a program in Statistica for this.
http://www.statsoft.com/support/faq5/stbasic/interval.html

Answer 3

$1.98