Question

1. A insurance office keeps track of the number of car insurance claims filed each day. Based on the data collected, it determines that the following probability distribution applies:
Number of Claims Probability 0 .05 1 .15 2 .25 3 .45 4 .10 a. What is the expected number of new claims filed each day?
b. If a claim pays out on average $5000, what is the average cost per day?
c. If the ofice is open 250 days a year, what is the total cost of claims in a year?
d. If there are 10,000 policy holders, what does each policy holder need to pay each year for the compayn to break even?

2.  Assume that a set of test scores in an Introduction to Finance class is normally distributed with a mean of 75 and a standard deviation of 8. Use the Standard Normal Tables to find the percentage of scores less than 86.
Use the Standard Normal Tables to find the percentage of scores between 64 and 86.
What score would be needed to be in the top 10%?

Answer 1

1)

a)

X	P(X)	X*P(X)
0	0.05	0
1	0.15	0.15
2	0.25	0.5
3	0.45	1.35
4	0.10	0.40

expected number of new claims filed each day = mean = E[X] = Σx*P(X) =            2.40

b) the average cost per day = $5000*2.40 = $12000

c)  total cost of claims in a year = 250*120000 = $3000000

2)

a)

µ =    75          
σ =    8          
              
P( X ≤    86   ) = P( (X-µ)/σ ≤ (86-75) /8)      
=P(Z ≤   1.375   ) =   0.9154 or 91.54% (answer)

b)

P (   64   < X <   86   )                      
=P( (64-75)/8 < (X-µ)/σ < (86-75)/8 )                                      
                                      
P (    -1.375   < Z <    1.375   )                       
= P ( Z <    1.375   ) - P ( Z <   -1.375   ) =    0.9154   -    0.0846   =    0.8309 or 83.09% (answer)

c)

µ=   75                  
σ =    8                  
proportion=   0.9                  
                      
Z value at    0.9   =   1.282   (excel formula =NORMSINV(   0.9   ) )
z=(x-µ)/σ                      
so, X=zσ+µ=   1.282   *   8   +   75  
X   =   85.25 (answer)