Question

Statewide Auto Insurance believes that for every trip longer than 10 minutes that a teenager drives, there is a 1 in 1,000 chance that the drive will results in an auto accident. Assume that the cost of an accident can be modeled with a beta distribution with an alpha parameter of 1.5, a beta parameter of 3, a minimum value of $500, and a maximum value of $20,000.

USING EXCEL - Construct a simulation model to answer the following questions.

a. If a teenager drives 500 trips longer than 10 minutes, what is the average cost resulting from accidents? Provide a 95% confidence interval on this mean.
b. If a teenager drives 500 trips longer than 10 minutes, what is the probability that the total cost from accidents will exceed $8,000? Provide a 95% confidence interval on this proportion.

Answer 1

SOLUTION:-

The probability that a random trip longer that 10 minutes that a teenager drives will result in an auto accident is 1/000=0.001

Let X be the number of trips, out of 500 trips longer that 10 minutes that a teenager drives, that result in an auto accident.

X has a Binomial distribution with parameters, number of trials n=500 and probability of success(the probability of having an accident) p=0.001.

We will use excel function =BINOM.INV(500,0.001,RAND()) to simulate X, the number of accidents

Let Y be the cost of an accident. Y has a beta distribution with parameters alpha=1.5 and beta=3 and a minimum value of $500 and a maximum value of $20000

We will use the excel function =BETA.INV(RAND(),1.5,3,500,20000) to generate the cost of an accident

The total cost from X accidents is C=X*Y

Prepare the following sheet

copy the rows to make 1000 trials. Paste the Number of accidents (X) and Cost of accident (Y) as values to avoid changes

Get the following

a) Get the average cost of 500 trips by taking the average of 1000 simulated values of the total cost

Let be a random variable indicating the the sample average cost of a randomly selected set of 1000 trials.

Due to CLT, we know that has a normal distribution with mean = and

standard error      where s is the standard deviation of the total cost

The critical value of z is obtained for 95% confidence interval using

.

We get , using the standard normal tables or excel function =NORM.INV(1-0.025,0,1)

95% confidence interval of mean cost is

Prepare the following

Get this

ans: Average Cost $ 3,539

Provide a 95% confidence interval on this mean

ans:

Lower Bound: $ 3,164

Upper Bound: $ 3,914

b) The probability that the total cost from accidents will exceed $8,000 is

Using CLT we can say that the distribution of proportion can be approximated by a normal distribution.

The standard error of proportion is

The critical value of z is obtained for 95% confidence interval using

.

We get , using the standard normal tables or excel function =NORM.INV(1-0.025,0,1)

95% confidence interval of mean cost is

Prepare the following

get the following

ans: Probability(Accident Cost>$8,000) = 19.2%

Provide a 95% confidence interval on this proportion

ans:

Lower Bound: 16.8%

Upper Bound: 21.6%