Question

Variable life insurance return rates. With a variable life insurance policy, the rate of return on the investment (i.e., the death benefit) varies from year to year. A study of these variable return rates was published in International Journal of Statistical Distributions (Vol. 1, 2015). A transformedratio of the return rates (x) for two consecutive years was shown to have a normal distribution, with $\mathit{\mu }\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{5}$ and role="math" localid="1660283206727" $\mathit{\sigma }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2}$. Use the standard normal table or statistical software to find the following probabilities.

a.$\mathbf{P}\mathbf{(}\mathbf{1}\mathbf{.}\mathbf{3}\mathbf{<}\mathit{x}\mathbf{<}\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{)}$

b. $\mathbf{P}\mathbf{(}\mathit{x}\mathbf{>}\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{)}$

c. $\mathbf{P}\mathbf{(}\mathit{x}\mathbf{<}\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{)}$

Short answer

1. $P\left(1.3<x<1.6\right)=0.5328$
2. role="math" localid="1660283402755" $P\left(x>1.4\right)=0.6915$

c. $P\left(x<1.5\right)=0.50$

Step-by-step solution

Given information

Variable life insurance return rates followa normal distribution with$\mu =1.5$ and$\sigma =0.2$

Finding the z values

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{x-1.5}{0.2}\end{array}$

For, x = 1.6

Then,

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{1.6-1.5}{0.2}\\ =0.5\end{array}$

For, x=1.3

Then,

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{1.3-1.5}{0.2}\\ =-1\end{array}$

For, x=1.4

Then,

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{1.4-1.5}{0.2}\\ =-0.5\end{array}$

For, x=1.5

Then,

$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ =\frac{1.5-1.5}{0.2}\\ =0\end{array}$

Probability calculation when P(1.3<x<1.6)

a.

$$\begin{gathered} P\left(1.3<x<1.6\right)=P\left(x<1.6\right)-P\left(x<1.3\right) \\ =P\left(z<0.5\right)-P\left(z<-1\right) \\ =P\left(z<0.5\right)-\left(1-P\left(z<-1\right)\right) \\ =0.6915-\left(1-0.8413\right) \\ =0.6915-1+0.8413 \\ =0.5328 \\ P\left(1.3<x<1.6\right)=0.5328 \end{gathered}$$

Therefore, the value of P(1.3<x<1.6) is 0.5328.

Probability calculation when P(x>1.4)

$$\begin{gathered} P\left(x>1.4\right)=1-P\left(x<1.4\right) \\ =1-P\left(z<-0.5\right) \\ =1-\left(1-P\left(z<-0.5\right)\right) \\ =0.6915 \\ P\left(x>1.4\right)=0.6915 \end{gathered}$$

Therefore, the value of$P\left(x>1.4\right)$ is 0.6915.

Probability calculation when P(x<1.5)

b.

$$\begin{gathered} P(x<1.5)=P(z<0) \\ =0.50 \\ P(x<1.5)=0.50 \end{gathered}$$

Therefore, the value of P(x<1.5) is 0.50.