Question

A "KRISPY CREAM DONUT" has a normal distributed lifetime with Mean = 20 hours and a Standard Deviation = 8.5 hours

a) What is the probability that having the donut survived 10 hours, it will become bad within the following 10 hours?

b) What would be the failure rate (Hazard function) at 15 hours?

c) How often should the donut be thrown away, to maintain an accumulated failure of less than 10%?

Answer 1

Solution :

Given that ,

a) mean = = 20

standard deviation = = 8.5

within 10 = 20 ± 10 = 10, 30

a) P(110 < x < 30 )  

= P[(10 - 20) / 8.5 < (x - ) /   < (30 - 20 ) / 8.5 )]

= P(-1.18 < Z < 1.18)

= P(Z < 1.18) - P(Z < -1.18)

Using z table,  

= 0.8810 - 0.1190

=0.7620

Probability = 0.7620

b ) X = 15

Using z-score formula,

z = X - /  

= ( 15 - 20 ) / 8.5

= - 5 / 8.5

= -0.59

The failure rate = -0.59

c ) P(Z < z) = 10%

P(Z < z) = 0.10

P(Z < - 0.61) = 0.10

z = - 0.61

Using z-score formula,

x = z * +

x = -1.28 * 8.5 + 20

x = 9.12