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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:

Extent of Bootstrapped Confidence Intervals vs Sample Size

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!

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    $\begingroup$ I see a few questions here. Could you define the key variables and specify the questions more e.g. $subset$ used, what exactly is $extent$ of confidence interval etc? $\endgroup$ – Delyan Savchev Jan 26 '15 at 20:22
  • $\begingroup$ Your outcome is the day they went shopping? Are you treating this as categorical? Maybe there's something very weird about your distribution. Try it again, but with a normally distributed variable (which you can generate using rnorm(500000). $\endgroup$ – Jeremy Miles Jan 26 '15 at 20:54
  • $\begingroup$ "What day of the week you go shopping" is categorical, so the distribution across categories cannot possibly be normal. It doesn't even make sense to discuss normality of something that exists only in discrete categories. There's not enough information here to identify for sure the cause of what's happening in your plot. $\endgroup$ – Glen_b Jan 27 '15 at 1:08
  • $\begingroup$ The setup for your image link got damaged when you put it in, so it didn't work. I have fixed it. $\endgroup$ – Glen_b Jan 27 '15 at 1:08

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