# How big should be a sample size?

Question:

Assuming the population standard deviation σ = 3, how large should a sample be to estimate the population mean µ with a margin of error not exceeding 0.5?

SE<= 0.5=3/√n ---> n=36

But probably the correct answer is:

So, can anyone explain more about the second solution?

• Hint: consider where the "$1.96$" came in and notice that your answer doesn't use this factor at all. Why not?
– whuber
Oct 31, 2018 at 18:53
• The difference come from the explanation of "a margin of error". You explain the a margin of error as SE (standard error). The second explanation of a margin of error is the half of the width of the 95% CI (confidence interval) which is 1.96*SE. Oct 31, 2018 at 18:55

A 95% confidence interval for normal population $$\mu,$$ where the SD is $$\sigma,$$ based on a sample of size $$n$$ is of the form $$\bar X \pm 1.96\sigma/\sqrt{n}.$$ As in @statistician's Comment, the 'standard error' of $$\bar X$$ is $$SD(\bar X) = \sigma/\sqrt{n}.$$ And the 'margin of error' of the confidence interval is $$\Delta = 1.96\sigma/\sqrt{n}.$$ You know $$\sigma = 3$$ and you want to find $$n$$ so that $$\Delta = 0.5$$ (or perhaps a bit smaller).
So solve $$\Delta = 1.96\sigma/\sqrt{n}$$ for $$n$$ in terms of $$\Delta$$ and $$\sigma.$$ That gives the equation $$n = (1.96\sigma/\Delta)^2,$$ to which you refer in your question. So $$n \approx 11.76^2 = 138.4.$$ Rounding up to the nearest integer, you get $$n = 139$$ as the sample size that gives $$\Delta$$ just a bit smaller than the required 0.5.
Note: Your solution $$n = 36$$ gives SE$$=3/\sqrt{36} = 0.5,$$ equivalent to $$\Delta = 1.96(0.5),$$ but not to $$\Delta = 0.5.$$ Roughly speaking, to cut $$\Delta$$ in half requires a four-fold increase in $$n,$$ and dividing by 1.96 is almost cutting $$\Delta$$ in half, which explains why $$4(36) = 144$$ is not far from the correct answer $$n = 139.$$
• Because $\sigma$ is known, the width of the CI is determined by $\alpha$ and $n,$ but of course not 100% of the CIs will cover $\mu.$ Oct 31, 2018 at 23:35