Statistical Analysis Question: How many measurements to reduce uncertainty?

1. Sep 22, 2009

katastrophe

1. The problem statement, all variables and given/known data
a. Mike makes six independent measurements of the diameter D of a leap year detector that made its way into the lab, obtaining D = 4.64, 4.78, 4.82, 4.68, 4.80, and 4.95m. What would result would he report in a lab writeup?
b. Alice does the same experiment as Mike, but makes only one measurement. Based on Mike's results, what would you expect for the uncertainty in Alice's single measurement of D?
c. How many measurements would have to be made altogether to reduce the uncertainty in the mean to +/- 0.003 m?

2. Relevant equations
D = $$\Sigma$$x$$_{i}$$/N
$$\sigma$$D = $$\delta$$D
$$\sigma$$D=$$\sqrt{(1/(N-1))*\Sigma(xi-\overline{x})^{2}}$$

3. The attempt at a solution

So, my real question doesn't come until part C, but I figure it definitely wouldn't hurt to make sure I have the first two parts right to base my work upon:

A. D = $$\Sigma$$x$$_{i}$$/N = (4.64+4.78+4.82+4.68+4.80+4.95)/6 = 4.78 m

B. $$\sigma$$D=$$\sqrt{(1/N-1)*\Sigma(xi-\overline{x})^{2}}$$ = $$\sqrt{(1/6-1)*((4.64-4.78)^{2}+(4.78-4.78)^{2}+(4.82-4.78)^{2}+(4.68-4.78)^{2}+(4.80-4.78)^{2}+(4.95-4.78)^{2})}$$
= 0.11m

C. So here's where my question is. I assume that we would use the same formula as in B, but how can we know what the new x$$_{i}$$ and $$\overline{x}$$ would be? Do you use a different formula to determine how to decrease the uncertainty? I know that each measurement induces a sqrt2 decrease in uncertainty, is this how you calculate it? Thanks so much!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution