A common question is estimate some quantity $X$, and give a 95% confidence interval for it. The estimation part, I understand how to think about and attack, but I'm always lost when it asks for a confidence interval for the estimate. I know that when it says 95% confidence interval, if my estimate is $\hat{\mu}$, then I give the confidence interval $[\hat{\mu}-2\sigma,\hat{\mu}+2\sigma]$, but how would I estimate the std? I demonstrate my thought process below for an example problem, if possible, please let me know where I can come up with $\sigma$ from this.
A question recently asked me the distance from SF to NYC. I live in Washington, and figured that it takes me 3 hours to go from South Washington to Seattle, and that's driving 60MPH, and so the width is approximately $\frac{4}{3}$ of that, so around $250$ miles. I think 4 WA can fit width wise in the US, and the US is approximately $3$ times longer than the width, and so I figure the length is approximately $3000$ and the length is $1000$. Then we have that the diagonal is approximately $\sqrt{3000^2 + 1000^2} \approx 3200$ miles. The real answer is actually pretty close to this, but in this calculation, how would I calculate $\sigma$? Where is the variance coming in to play?