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 IQR for the mean amount of time 64 batteries last.

Short answer

The IQR for the mean amount of time 64 batteries last is1.69 months

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$\sigma =\mu =10$

Step 3:  Distribution of Mean

If ,$\overline{X}$is the average length of time that 64 batteries endure, then the distribution of the average length of 64 batteries will be normal. The following shows the distribution of$\overline{X}$which is the average length of time that 64 batteries last:

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

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

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

Calculation

To calculate the Interquartile range, calculate ${75}^{\text{th}}$and the${25}^{\text{th}}$percentile, use the T1-83 calculator. For this, click on $2^{\text{nd}}$, then DISTR, and then scroll down to the invNorm option and enter the provided details. After this, click on ENTER button of the calculator to get the result. The screenshot is given as below:

Therefore, the interquartile range (|QR) for the total length of time of 64 batteries last Is given as:

$\left(\mathrm{IQR}\right)=Q_1-Q_1$

$=10.8431-9.1568$

=1.6862

=1.69 Months