I'm new to CrossValidated so please excuse any shortcomings in my question.
Suppose I have a sample of 500, for a population of 500,000. I asked my sample of 500 what day of the week they go grocery shopping and assume I do not know the actual distribution of which day of week people go grocery shopping. Because I do not know the distribution, and I cannot assume it is normal, I calculate the 95% confidence intervals via bootstrapping (using the boot package in r). Now I want to know how to minimize the extent of the confidence intervals for this data. I look first to sample size. So I randomly grabbed (without replacement) a subset of sample and calculated the confidence intervals on that subset. I repeated this several times for different size subsets and plotted the (sub)sample size on the x-axis and the extent of the confidence intervals on the y-axis:
I found this plot surprising. I had hoped I would see that as my sample size increased my confidence intervals extent would decrease. Instead I just see randomness. Does anyone have any insights on first, why I am observing this, and second, how I can determine what sample size I need to see a decrease in my 95% confidence interval extent. Please forgive if this is dumb question!