Galaxy Co$.$ sells virtual reality $($VR$)$ goggles, particularly targeting 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 $160$ units and a standard deviation of $40$ units. At the

beginning of each month, Galaxy orders enough goggles from its supplier to bring the inventory level up to $140$

goggles. If the monthly demand is less than $140,$ Galaxy pays $$20$ per pair of goggles that remain in inventory at the

end of the month. If the monthly demand exceeds $140,$ Galaxy sells only the $140$ 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.

What is the average monthly profit resulting from its policy of stocking $140$ pairs of goggles at the beginning of each

month? Round your answer to the nearest dollar.

What is the proportion of months in which demand is completely satisfied?

Use the simulation model to compare the profitability of monthly replenishment levels of $140$ and $160$ pairs of

goggles. Use a $95\%$ confidence interval on the difference between the average profit that each replenishment level

generates to make your comparison.