Please be gentle with me, I'm not a statistician, but I am struggling to understand the following problems which crop up in my work. I'm clearly missing something or possibly just misunderstanding what's going on. Sorry for going on a bit.

There is a US bit of software called VSP (Visual Sampling Plan) which tells you how many soil samples you need to take to make decisions about the state of land with specified degrees of confidence. For example, you want to know if the average concentration of a contaminant in a field is above or below a safe level, referred to as the Action Level (AL). You attempt to find the average concentration by sampling. But how many samples do you need to get your desired confidence levels with your sample mean? VSP will tell you.

If you decide on the basis of sampling that a clean site (true mean concentration < AL) is dirty (because your sample mean > AL), this is a beta error. If you decide a dirty site (true mean > AL) is clean (because your sample mean < AL), this is an alpha error. In VSP you specify alpha, beta and something called the gray region. Now, see Figure 4.2 in this document:

http://vsp.pnl.gov/docs/vsp05f.pdf

My questions are these:

1. Surely if the true mean is at the action level (10 on the figure), your sample mean would be more than the action level 50% of the time, and this would always be the case, since the distribution of samples means would have its median at the true mean. Increasing the sample size would not change that. Why does Figure 4.2 not show this?

2. It also seems to me that once you have defined the gray area and alpha (or beta), then beta (or alpha) is determined and can not be specified separately. At least if I'm right in thinking that the curve in Figure 4.2 is the same shape as the (some) cumulative distribution function. And yet VSP asks you to specify alpha, beta and the width of the gray region. Why?

3. What exactly is the interpretation of the gray region?

4. How and why does the central limit theorem apply here? What does increasing n do exactly?

Basically, what's going on? Someone suggested to me that it is how it is because you don't compare the sample mean to the AL but rather an upper estimate of the sample mean (e.g. 95th percentile from its distribution). Does that make sense?

Here is another link to VSP if it helps:

www.pnl.gov/main/publications/external/technical_reports/PNNL13490.pdf

Thanks!