A worker$-$machine operation was found to involve $4\mathrm{.}6$ minutes of machine time per cycle in the course of $40$ cycles of stopwatch study. The worker$$s time averaged $3\mathrm{.}1$ minutes per cycle, and the worker was given a rating of $121$ percent $($machine rating is $100$ percent$).$ Midway through the study, the worker took a $10-$minute rest break. Assuming an allowance factor of $13$ percent of work time which is applied only to the worker element $($not the machine element$),$ determine the standard time for this job. $($Do not round intermediate calculations. Round your final answer to $2$ decimal places.$)$

Standard Time: $\_\_\_\_\_\_\_\_\_\_$ minutes

Problem $2(8$ points$)$

In an initial survey designed to estimate the percentage of time air$-$express cargo loaders are idle, an analyst found that loaders were idle in $8$ of the $50$ observations.

What is the estimated percentage of idle time?

Estimated percentage of idle time: $\_\_\_\_\_\_\_\_\_\_\_\_\_\%$

Based on the initial results, approximately how many observations would you require to estimate the actual percentage of idle time to within $7$ percent with a confidence of $99$ percent? $($Do not round intermediate calculations. Round up your final answer to the next whole number.$)$

Number of observations: