Question

Depending upon the usage of the cell phone, the useful life of a cell phone is normally distributed with a mean of 2 years and a standard deviation of 4 months.

a. What is the probability that a random person would find the cell phone of no use after 4 months?

b. You took a random sample of 25 cell phone users.  What is the probability that the sample mean life would lie between 2.1 and 2.2 years?      

Answer 1

Let the useful life of phone be denoted by 'X'

z-score = = (x - 24) / 4

Here the mean = 2 years which is 2*12 =24 months. We convert to months because all the other values are given in months

a. What is the probability that a random person would find the cell phone of no use after 4 months?

No use after 4 months that means the useful life will be less than 4 months

P( X< 4) = P( Z < -5)

= 1 - P( Z < 5)

= 1 - 1 ..............using normal distribution tables

P( X < 4) = 0

It means that there is no chance that the phone won't be of use after 4 months.

b. You took a random sample of 25 cell phone users.  What is the probability that the sample mean life would lie between 2.1 and 2.2 years?

Here we have been given a random as well as normally distributed sample. So we can use the centrl limit theorem to find the distribution for the ,mean life.

n = 25

z-score = ( - 24) / 0.8

Between 2.1 and 2.2 years. Converting to months. Lie between 25.2 and 26.4 months

P( 25.2 < < 26.4) = P( < 26.4) - P( < 25.2)

= P( Z < 3) - P( Z < 1.5)

=0.99865 - 0.93319   ..............using normal distribution tables

P(2.1 < < 2.2 ) = 0.0654

Although both values individually have high probability but since we are given a very small interval probabiltiy reduces.