Question

2. (a) Consider an insurer that offers 2 types of policy: home insurance and car insurance.
20% of all customers have a home insurance policy, and 92% of all customers have
a car insurance policy. Every customer has at least one of the two types of policies.
Calculate the probability that a randomly selected customer
(i) has home insurance, given that he has car insurance, (3)
(ii) does not have car insurance, given that he has home insurance. (3)
(b) Consider the number of days to maturity of 10 short-term investments below:
85 77 88 78 65 78 57 81 36 90
(i) Make a stem-and-leaf plot for the days-to-maturity data. (3)
(ii) Hence, or otherwise, compute the interquartile range of the data. (4)
(iii) Are there any outliers in the data. (3)
(c) An insurance company has a portfolio of 12,000 policies. Based on past data, the
company estimates that the probability of a claim on any one policy in a year is
0.0045. It assumes no policy will generate more than one claim in a year.
(i) Determine the approximate probability of more than 64 claims from the portfolio of 12,000 policies in a year. (5)
(ii) Determine an approximate equal-tailed interval into which the number of claims
per year will fall with probability 0.95. (2)
(iii) In practice 70 claims were received in a particular year. A Director of the company complains about the range of estimates in part (ii) being wrong. Comment
on the Director’s complaint.

Answer 1

(i)

Stem-and-Leaf Display: No. of days

Stem-and-leaf of No. of days N = 10
Leaf Unit = 1.0

   3 6
   4
   5 7
6 5
7 788
8 158
9 0

(ii) N=total no. of observations=10. (N+1)/4=2.75, 3(N+1)/4=8.25

Q1=First Quartile=2nd observation+0.75(Third observation-2nd observation)=57+0.75*(65-57)=63

Q3=Third Quartile=8th observation+0.25(9th observation-8th observation)=85+0.25*(88-85)=85.75

Inter-quartile range(IQR)=Q3-Q1=85.75-63=22.75

(iii) Lower whisker=Q1-1.5*IQR=63-1.5*22.75=28.875

Upper whisker=Q3+1.5*IQR=85.75+1.5*22.75=119.875

Any value that lies outside the interval (28.875, 119.875) is treated as an outlier. Hence there is no outlier.