$($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 $210$ units and a standard deviation of $52\mathrm{.}5$ units. At the

beginning of each month, Galaxy orders enough goggles from its supplier to bring the inventory level up to $190$ goggles. If the monthly demand is less than $190,$ Galaxy

pays $$20$ per pair of goggles that remains in inventory at the end of the month. If the monthly demand exceeds $190,$ Galaxy sells only the $190$ 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 $190$ pairs of goggles at the beginning of each month? Round your answer to the nearest dollar.

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$($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 $190$ and $210$ 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 $210$ versus a replenishment level of $190$ is $

It means that monthly

replenishment level of

increases profitability.