# We have data come from a normally distributed population with standard deviation 2.6, what sample size is needed to make sure that with 99% probability, the mean of the sample will be in error by at most 0.25?

We have data come from a normally distributed population with standard deviation 2.6, what sample size is needed to make sure that with 99% probability, the mean of the sample will be in error by at most 0.25?
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dilettato5t1
Step 1
It seems you are talking about the margin of error for or of a confidence interval for the mean. The general formula is
$\overline{X}±{z}^{\ast }\cdot \frac{\sigma }{\sqrt{n}}$
Where $\overline{X}$ is the sample mean, $\sigma$ is the population standard deviation, z∗ is the critical value from the standard normal distribution and depends on the size of the confidence interval, and n is the sample size.
We want the term on the right, the margin of error (denoted ME below) to be less than or equal to a fixed size. Thsu we need
${z}^{\ast }\cdot \frac{\sigma }{\sqrt{n}}\le ME$
Rearranging to solve for n gives
$n\ge {\left(\frac{{z}^{\ast }\sigma }{ME}\right)}^{2}$
The critical value for a 99% confidence interval is ${z}^{\ast }=2.5758$ (the 99.5th percentile of a Standard Normal distribution to 4 decimal places). Then using your values of $\sigma =2.6$ and $ME=0.25$ gives
$n\ge {\left(\frac{\left(2.5758\right)\left(2.6\right)}{0.25}\right)}^{2}\approx 717.6$
Thus $n=718$ will do as we wish. The way you stated the problem seemed to suggest that 2.6 was the population standard deviation. If it was actually the sample standard deviation then the formula is a bit different. In that case the confidence interval is given by
$\overline{X}±{t}_{df}^{\ast }\cdot \frac{s}{\sqrt{n}}$
where the degrees of freedom, $df=n-1$. The main issue here is that the critical t value depends not only on the confidence level but also on the sample size. In any case, for a given confidence level, ${t}_{df}^{\ast }>{z}^{\ast }$ so the value we used ${z}^{\ast }=2.5758$ will underestimate our sample size but not by much. If we use $n=718$ as an estimate for the sample size then $df=718-1=717$ and then for a 99% confidence interval ${t}_{717}^{\ast }=2.5827$ which then gives $n\ge 722$ as our sample size and another iteration has no effect on the result.
Step 2
The other issue is that s also depends on sample size. This should just result in a better estimate for $\sigma$, the population standard deviation, but it can still affect the sample size to achieve the desired ME. In general, except in the case of very small samples, neither of these sources of inaccuracy will be significant. Use the sample size used to get s to determine the critical t value and your estimate for required sample size should at least be large enough as the value of ${t}_{df}^{\ast }$ decreases as n and thus df increases.