Estimating location of lower tail

Say I have a sample, assumed from a Normal Distribution. I want to find a confidence interval for this parameter:

\(L = \mu - \alpha * \sigma\)

where \(\alpha\) is a known constant.

I guess I should use the estimator:

\(\bar{x} - \alpha*s\)

But what standard error should I use?

Just for context - the situation is that I have measured the size of 8 manufactured objects, and want to figure out a reasonable value for the minimum size that they might ever take.

Cheers! This is my first post, so I hope it makes sense, and the maths typesetting works.


TS Contributor
this would be a nice occasion to use the bootstrap method. Just generate a bunch ( >100000)samples with replacement from your data and create the distribution of your value.