## Create a histogram of the distribution of total cost values, Business Management

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Synthetic Auto Insurance is trying to decide how much money to keep in liquid assets to cover insurance claims. In the past, the company held some of the premiums it received in interest bearing checking accounts and put the rest into less liquid investments that generate a higher return.

It has also determined that the number of repair claims filed each week is a random variable which can take values 1, 2, 4, 5, 6, 7, 8, 9. The probability of having 1 or 9 claims in a week is equal. The probability that there are 2, 6, 7 or 8 claims in a week is twice the probability of there being a single claim. The probability that there are 4 claims in a week is twice the probability that there are 2 claims in a week. The probability that there are 5 claims in a week is 6 times the probability that there is a single claim in a week. The dollar amount per claim fits a normal distribution with a mean claim amount of \$2000 and a standard deviation of \$400.

In addition to repair claims, the company also receives claims for cars that have been "totaled" and cannot be repaired. There is a 20% chance in any week of receiving this type of claim. The claims for "totaled" cars cost has a uniform distribution, in the range \$10000 to \$35000. Not all repair claims are legitimate: 1% of the repair claims filed are rejected. Of the "totaled" claims filed, 0.5% of them are rejected.

a. Create a spreadsheet model of the total claims cost incurred by the company for 10 weeks. Calculate the average payout per week. If the company decides to keep \$20,000 cash on hand to pay claims, in how many of the weeks is this amount of money insufficient to pay the claims? [Print out your excel file to support your answer].

b. Create a new spreadsheet model of the total claims cost incurred by the company in any week using crystal ball. Include information in this model which will help the company determine the probability that \$20,000 would be inadequate to pay the claims. Replicate the model 5000 times. [Print out your crystal ball worksheet and the histogram and summary statistics output to support your answer as an appendix]

i. Create a histogram of the distribution of total cost values that were generated.

ii. Determine the percentage of the time that \$20,000 is inadequate to pay the claims based on your crystal ball output.

iii. Create a 95% confidence interval for the true probability that claims exceed \$20,000 in a given week based on your crystal ball output.

iv. Determine the average amount that the company pays out in claims in any week and the 80% confidence interval for the average claim paid in any given week based on the crystal ball output.

c. Change the seed value from your initial seed value. Assume 10 different seed values and for each seed value replicate the model 5000 times. Do not use consecutive numbers for seed values. Tabulate the average claim and the probability that \$20,000 will be inadequate to pay the total amount of claims in a new worksheet. [Print out the statistics from each replication. Also create a table with the results of all 10 replications and develop summary statistics.]

i. In how many cases does the average claim fall within the confidence interval developed in part b iv.? If there are situations when the average claim does not fall within the confidence interval, explain why.

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