Online Test Scenario:
A random sample of $1000$ is taken from a much larger population and the $95\%$ confidence interval for the population mean is calculated as $24.1 \pm 4.2$.
Question:
If instead an independent random sample of only $500$ had been taken from the same population which of the following is the most plausible $95\%$ confidence interval that would have been obtained?
- $24.1 \pm 3.0$
- $24.1 \pm 4.2$
- $24.1 \pm 5.9$
- $24.1 \pm 8.4$
My Attempt:
I am not sure how to go about this as i haven't covered it in my studies, an explanation as to why one of the 4 is correct and how to calculate it would help greatly.
A 95% confidence interval is constructed as $\mu \pm 2\sigma_x$, where $\sigma_x$ is the standard error given by $\sigma_x = \sigma/\sqrt{n}$. Thus, because the sample of $1000$ is $2$ times as large as $500$, we expect our standard error for $1000$ elements in our sample to be $1/\sqrt{2}$ times that for $500$ elements in our sample, so our confidence interval for the smaller interval should be $24.1\pm4.2\sqrt{2} = 24.1 \pm 5.9$.
To have a more intuitive answer, note that as we take larger samples, we can be more confident that the observed mean approximates the population mean well, so our error will become smaller. This eliminates the first two choices; deciding between $3$ and $4$ requires you to know about the $\sqrt{N}$ dependence.