Question

The length of time a particular smartphone's battery lasts follows an exponential distribution with a mean of ten months. A sample of 64 of these smartphones is taken.

Find the 80th percentile for the total length of time 64 batteries last .

Short answer

The 80th percentile for the total length of time 64 batteries last is approximately

$11.05$

Step-by-step solution

Given Information

According to the given details, the length of time a particular smartphone's battery lasts follows the exponential distribution. The mean of the distribution is$\mu =10$ months and the sample size is 64.

Explanation

The exponential distribution is used to determine how long a smartphone's battery lasts. The exponential distribution's probability density function $X~Exp\left(m\right)$, where $m$is the decay parameter is given as:

$f\left(X\right)=m{e}^{(-mx)}$

Where, $X\ge 0\text{and}m>0$

The exponential distribution's standard deviation is $\sigma =\mu =10.$

Distribution Of Mean

If, $\overline{X}$is the average length of time that 64 batteries last, the distribution of mean length for 64 batteries will be normal. The distribution of $\overline{X}$is the mean length time of 64 batteries last is given as below:

$\overline{X}-N\left({\mu }_X,{\sigma }_X/\sqrt{n}\right)$

$\overline{X}~N(10,10/\sqrt{64})$

$\overline{X}~N(10,10/8)$