achieverh3

2021-12-04

Great Insurance company has collected the data regarding the duration (how long it takes) to settle insurance claims. The data collected indicate that the duration follows a normal distribution with mean 28 days and standard deviation 8 days.
a) What proportion of the duration is between 20 and 40 days old?
b) What proportion of the duration is less than 30 days old?
c) What is the number of days in which 75% of all claims are above?

pseudoenergy34

Step 1
Let us assume $X=$ the duration (how long it takes) to settle insurance claims.
Given that X follows a normal distribution with $mean=28$ and standard deviation $=8$
Step 2
a) The proportion of the duration is between 20 and 40 days old is:
$P\left(20
$=P\left(\frac{20-28}{8}
$=P\left(-1
$=0.7745$
Step 3
b) The proportion of the duration is less than 30 days old is:
$P\left(X<30\right)$
$=P\left(Z<\frac{30-28}{8}\right)$
$=P\left(Z<0.25\right)$
$=0.5987$
c) Let the number of days $=K$ in which 75% of all claims are above
$P\left(X>K\right)=0.75$
$P\left(Z>\frac{K-28}{8}\right)=0.75$
$\frac{K-28}{8}=-0.674$
$K=28+8\left(-0.674\right)$
$K=22.608$