Question

A car has a transmission failure rate of 1 in 9 years

a) If you buy one of these cars, what is the probability that the transmission will last more than 9 years?

b) If you buy the 12-year extended warranty, what is the probability that you will have to use it to fix the transmission?

c) How much should the warranty cover if the dealer wants to offer warranty service to no more than 2% of customers?

d) If we take a sample of 45 new cars and the times until the transmission failure are recorded. What is the probability that the average time to failure is less than 8.5 years?

Answer 1

Answer:

Given,

= 1/9

X ~ Exponential

a)

P(X > 9) = e^-x

= e^-1/9*9

= e^-1

= 0.3679

b)

P(X > 12) = e^-(12/9)

= 0.2636

c)

P(X < Xo) = 0.02

1 - e^-xo = 0.02

e^-(xo/9) = 0.98

xo = ln(0.95)*-9

xo = 0.462

d)

Mean = 1/ = 9

Standard deviation = 1/ = 9

P(Xbar < 8.5) = P(z < (8.5 - 9)/(9/sqrt(45)))

= P(z < - 0.37)

= 0.3556912 [since from z table]

= 0.3557