$13/05/2024$@$17$:$00$

Duration: $30$ minutes

Haynes is the distributor of Burton Snowboards with most of the sales being made in the winter season in Australia. Haynes makes a one$-$time purchase of the boards prior to each winter season at a cost of $$250$ each and sells each board for $$380.$ Any boards unsold at the end of the winter season are marked down to $$225$ and sold in a special spring sale. Virtually all marked$-$down snowboards are sold. The following is the number of sales of snowboards during the past $9$ winter seasons: $30,50,30,60,16,40,30,30,20.$

a$.$ Estimate the mean and the variance of the demand for boards each winter.

b$.$ Assume that the demand for boards each winter follows a normal distribution, with mean and variance given by what you obtained in part $($a$).$ Determine the optimal number of boards for Haynes to buy prior to each winter season. Show the safety stock quantity in your solution.