1. The time until recharge for a battery in a laptop computer under common conditions is normally distributed with a mean of 260 minutes and a standard deviation of 50 minutes. a. What is the probability that a battery lasts more than four hours? b. What are the quartiles (the 25% and 75% values) of battery life? c. What value of life in minutes is exceeded with 95% probability?

$\mu=260\\\sigma=50$

a) For this problem, calculate a Z score and use the normal probability Z table or technology to get the probability.

$Z = {(x - \mu)\over\sigma}$

x is 4 hours in minutes, which is 240 minutes

So,

$Z = {(240 - 260)\over50}\\ Z = -0.4\\ P(Z \gt -0.4) = P(Z \lt 0.4) = 0.6554$

b) To find the quartiles,

Find Z values for the lower 25% and the upper 25% of the data, which is -0.67 and 0.67

$\mu +( Z\times\sigma)$

The lower quartile(25%) is,

260 + (-0.67)(50) = 226.50

The upper quartile(75%) is,

260 + (0.67)(50) = 293.50

c) To find what value is exceeded with 95% probability, find the Z value for bottom 5%, so look up .0500 on the Z table, which is -1.65

$\mu + (Z\times\sigma)$

260 + (-1.65)(50) = 177.50

The value exceeded with 95% probability is 177.50