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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$\mathrm{μ}=10$ months and the sample size is 64.

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, $Xâ‰\yen 0\text{and}m>0$

The exponential distribution's standard deviation is $\mathrm{σ}=\mathrm{μ}=10.$

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

$\stackrel{¯}{X}-N\left({\mathrm{μ}}_X,{\mathrm{σ}}_X/\sqrt{n}\right)$

$\stackrel{¯}{X}~N(10,10/\sqrt{64})$

$\stackrel{¯}{X}~N(10,10/8)$