Answer:
You need to have 538 people in your study.
Step-by-step explanation:
We have that to find our
level, that is the subtraction of 1 by the confidence interval divided by 2. So:

Now, we have to find z in the Ztable as such z has a pvalue of
.
That is z with a pvalue of
, so Z = 1.88.
Now, find the margin of error M as such

In which
is the standard deviation of the population and n is the size of the sample.
You practiced on the people at work and found a standard deviation of about 0.37 seconds.
This means that 
You want to get a 94% confidence interval that is only 0.06 in width.
This means that 
How many people do you need to have in your study?
This is n for which M = 0.03. So






Rounding up:
You need to have 538 people in your study.