Problem $7-12($Algo$)$

How many work cycles should be timed to estimate the average cycle time to within $5$ percent of the sample mean with a confidence of $98\mathrm{.}0$ percent if a pilot study yielded these times $($minutes$)$: $4\mathrm{.}1,4\mathrm{.}6,5\mathrm{.}1,4\mathrm{.}9,5\mathrm{.}9,$ and $4\mathrm{.}4?$ The standard deviation is $0\mathrm{.}631$ minutes per cycle. $($Use the "mean time" value to $2$ decimal places and other values to $3$ decimal places for intermediate calculations. Round up your final answer to the next whole number.$)$

Number of work cycles

$230$