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?

Question

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.

Answer 1

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.