Answer:
Around 3082 work cycles should be timed to estimate the average cycle time to within 1 percent of the sample mean with a confidence of 99%
Explanation:
From the pilot study, we calculate the standard deviation,
Pilot Study Mean = M = (4.3+4.3+4.8+4.7+4.8+4.7)/6
M = 27.6/6
M = 4.6
With this, we calculate the standard deviation,

Now, N = 6, and we use N = 6 and not N - 1, because we are only looking at the sample mean and not the whole population,
then, we get S,
S = 0.216
(if you need to form an opinion for the whole population, then S = 0.237)
We need an to be within 1 percent of the sample mean so at most, the margin of error can be 1%
so, margin of error = 0.01,
Now, for a confidence level of 99%, we get and alpha value of,
alpha = 0.01/2 = 0.005
which gives a z value of (-2.57)
z = -2.57,
So, we need to calculate how many work cycles n should be timed,
Using the formula,

n = 3081.58
n = 3082
So, 3082 work cycles should be timed to estimate the average cycle time to within 1 percent of the sample mean with a confidence of 99%