# How large a sample of measurements must be taken to be 95% confident that the error in the estimate of the mean time will not exceed 0.01 seconds?

I have a question regarding confidence intervals as part of a Master's course. The scenario is the following:

In measuring the reaction time of a patient to a certain stimulus, a psychologist estimates the standard deviation as 0.05 seconds. How large a sample of measurements must she take in order to be 95% confident that the error in her estimate of the mean reaction time will not exceed 0.01 seconds?

My thoughts so far are:

Standard deviation = 0.05

Confidence interval = 95%

I am not sure if the NORMSINV() function in Excel could help.

Many thanks in advance.

The confidence interval is $$\bar{X}\pm1.96\frac{\sigma}{\sqrt{n}}$$
Therefore you need to solve for $$n$$ the inequality $$1.96\times\frac{0.05}{\sqrt{n}}<0.01$$
$$\implies\sqrt{n}>\frac{1.96\times0.05}{0.01}=9.8$$ $$\implies n>9.8^2=96.04$$
So the minimum sample size is $$97$$
Note that the number $$1.96$$ corresponds to $$\Phi^{-1}(0.975)$$ for a symmetric $$95\%$$ confidence interval.