Question

What is the median lifetime of these phones (in years)?

a. $0.1941$

b. $1.3863$

c. $2.0794$

d. $5.5452$

Short answer

The correct answer is option (c).

Step-by-step solution

Given Information

The average lifetime of a certain new cell phone is three years. The manufacturer will replace any cell phone failing within two years of the date of purchase. The lifetime of these cell phones is known to follow an exponential distribution.

Calculation

$\mu =3$

Therefore, the decay rate will be:

$m=\frac{1}{\mu }$

$=\frac{1}{3}$

$=0.3333$

It is known that the cumulative distribution function of exponential distribution is,

$P(X<x)=1-e^{-mx}$.

Explanation

It is also known that median is similar to the ${50}^{\text{th}}$percentile of the data. So, calculate the median as below:

$P(x\le k)=1-e^{-03mk}$

$0.50=1-e^{-a.33}·$

$e^{-0.030x}=0.50$

Take the logarithmic on both the sides. Then,

$\ln \left(e^{-0.3333\times k}\right)=\ln (0.50)$

$-0.3333\times k=-0.6931$

$k=2.0794$