The data for a time$-$study observations for an assembly task are presented in the following table:

Observations $($seconds per cycle$)$

Element Performance Rating $1234567$

$1102\%34\mathrm{.}5931\mathrm{.}1931\mathrm{.}8130\mathrm{.}1834\mathrm{.}9631\mathrm{.}7232\mathrm{.}70$

$2113\%40\mathrm{.}2442\mathrm{.}7640\mathrm{.}4541\mathrm{.}7337\mathrm{.}7043\mathrm{.}0438\mathrm{.}72$

$3127\%58\mathrm{.}8253\mathrm{.}3649\mathrm{.}7346\mathrm{.}3257\mathrm{.}6446\mathrm{.}3749\mathrm{.}96$

$4134\%66\mathrm{.}1461\mathrm{.}1183\mathrm{.}8283\mathrm{.}5977\mathrm{.}2380\mathrm{.}0572\mathrm{.}12$

a$.$ Based on the observations, determine the standard time for the task, assuming an allowance of $25\%$ of task time.

b$.$ How many observations would be needed to estimate the mean of each element within $1\mathrm{.}85$ percent of its true value with a $97\mathrm{.}25\%$ confidence?

c$.$ How many observations would be needed to estimate the mean time for the most efficient element $($i$.$e$.,$ the one with the minimum mean time$)$ within $0\mathrm{.}035$ minutes of its true value with a $99\%$ confidence?

d$.$ How many work cycles should be timed to estimate the average cycle time within $2\mathrm{.}9\%$ of the sample mean with a confidence of $98\mathrm{.}75\%$ if a pilot study yielded the time of the least efficient element $($i$.$e$.,$ the one with the maximum mean time$)?$