LMart, a supermarket chain, has 9 stores in and around Lawrence. They sell imitation Christmas trees sourced from a South American manufacturer. Christmas trees are only sold in the few weeks leading up to Christmas; however, due to the long leadtime, LMart needs to place an order by May.

LMart estimates that the demand at each of the 9 stores will be normally distributed with a mean of 100 and a standard deviation of 30. To reduce demand uncertainty, all the trees will be stored at a central warehouse, and will be shipped from the warehouse to each store as needed. (Assume that the trees can be shipped from the warehouse to the stores instantly.)

It costs LMart $15 to buy each tree from the manufacturer and transport them to the central warehouse. The selling price of a tree is $45 up till Christmas. After Christmas, all remaining trees are sold off at a clearance price of $5 per tree.

1) What is the probability distribution function of the demand faced by the central warehouse?

a. Normal (mean = 900 , standard deviation = 270)

b. Normal (mean = 300 , standard deviation = 90)

c. Normal (mean = 900 , standard deviation = 90)

d. Normal (mean = 300 , standard deviation = 270)

2) What is the critical fractile for LMart? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.

3) What is the optimal number of christmas trees that LMart should order to maximize their profit? If your answer is not an integer, provide at least three decimal places, e.g., 7.500.