Question

For the following questions, define each of the random variables you use.

The life of the batteries produced by a manufacturer is distributed according to a normal law with an average duration of 110 hours and a standard deviation of 10 hours.

a) What proportion of the batteries produced will last more than 125 hours.

b) The warranty accompanying the batteries states that if a battery lasts less than X hours the manufacturer agrees to replace it. What should be the value of X if the manufacturer does not want to replace more than 5% of the batteries sold.

Answer 1

Solution :

Given that ,

mean = = 110

standard deviation = = 10

(a)

P(x > 125) = 1 - P(x < 125)

= 1 - P((x - ) / < (125 - 110) / 10)

= 1 - P(z < 1.5)

= 1 - 0.9332   

= 0.0668

P(x > 125) = 0.0668

Proportion = 0.0668

(b)

P(Z > z) = 5%

1 - P(Z < z) = 0.05

P(Z < z) = 1 - 0.05 = 0.95

P(Z < 1.645) = 0.95

z = 1.645

Using z-score formula,

x = z * +

x = 1.645 * 10 + 110 = 126.45

X = 125.45