$($All answers were generated using $1,000$ trials and native Excel functionality.$)$

Galaxy Co$.$ sells virtual reality $($VR$)$ goggles targeted to customers who like to play video games. Galaxy procures each pair of goggles for $$150$ from its supplier

and sells each pair of goggles for $$300.$ Monthly demand for the VR goggles is a normal random variable with a mean of $120$ units and a standard deviation of

$30$ units. At the beginning of each month, Galaxy orders enough goggles from its supplier to bring the inventory level up to $100$ goggles. If the monthly demand

is less than $100,$ Galaxy pays $$20$ per pair of goggles that remains in inventory at the end of the month. If the monthly demand exceeds $100,$ Galaxy sells only

the $100$ pairs of goggles in stock. Galaxy assigns a shortage cost of $$40$ for each unit of demand that is unsatisfied to represent a loss$-$of$-$goodwill among its

customers. Management would like to use a simulation model to analyze this situation.

$($a$)$ What is the average monthly profit resulting from its policy of stocking $100$ pairs of goggles at the beginning of each month? Round your answer to the

nearest dollar.

$

$($b$)$ What is the proportion of months in which demand is completely satisfied? Round your answer to the nearest whole number.

$\%$

$($c$)$ Use the simulation model to compare the profitability of monthly replenishment levels of $100$ and $120$ pairs of goggles. Use a $95\%$ confidence interval on

the difference between the average profit that each replenishment level generates to make your comparison. Round your answer to the nearest dollar.

The average difference between the net profit generated by a replenishment level of $120$ versus a replenishment level of $100$ is $

$.$ It means that

monthly replenishment level of

increases profitability.