Question

Many people consider their smart phone to be essential! Communication, news, Internet, entertainment, photos, and just keeping current are all conveniently possible with a smart phone. However, the battery better be charged or the phone is useless. Battery life of course depends on the frequency, duration, and type of use. One study involving heavy use of the phones showed the mean of the battery life to be 9.75 hours with a standard deviation of 2.3 hours. Then the battery needs to be recharged. Assume the battery life between charges is normally distributed. (a) Find the probability that with heavy use, the battery life exceeds 10 hours. (Round your answer to four decimal places.) (b) You are planning your recharging schedule so that the probability your phone will die is no more than 5%. After how many hours should you plan to recharge your phone? (Round your answer to the nearest tenth of an hour.)

Answer 1

a) 0.4567

b) 6 hours

Explanation:

We are given the following information the question:

The battery life follows a normal distribution with

`\mu = 9.75\\\sigma = 2.3`

Formula:

`z = \displaystyle(x-\mu)/(\sigma)`

a) We have to find probability such that battery life exceeds 10 hours, that is,

P(X>10)

`P(z > \displaystyle(10 - 9.75)/(2.3)) = P(z>0.1086)\\\\= 1 - P(z \leq 0.1086)\\= 1- 0.5433\\=0.4567`

b) We have to find battery life such that

`P(X \leq x) = 0.05\\\\P(z \leq \displaystyle(x-9.75)/(2.3)) = 0.05\\\\\displaystyle(x-9.75)/(2.3) = -1.64 \\\\x = (-1.64)(2.3) + 9.75\\x = 5.978`

Here, we calculated the value of z from the normal distribution table.

So, after approximately 6 hours one should plan to recharge the phone.