Question

Please answer the following question:

Truck tire life is normally distributed with a mean of 60,000 miles and a standard deviation of 4,000 miles. a) What is the probability that a tire will last 72,000 miles or more? b) For a set of four tires, what is the probability that the average tire life is less than 55,000 miles? c) For a set of four tires, what is the probability that the average tire life is between 57,000 and 63,000 miles?

Answer 1

Solution :

a)

P(x $\geq$ 72000) = 1 - P(x  $\leq$ 72000)

= 1 - P[(x - $\mu$ ) / $\sigma$ $\leq$ (72000 - 60000) / 4000]

= 1 -  P(z $\leq$ 3)   

= 1 - 0.9987

= 0.0013

Probability = 0.0013

b)

P(x < 55000) = P[(x - $\mu$ ) / $\sigma$ < (55000 - 60000) / 4000]

= P(z < -1.25)

= 0.1056

Probability = 0.1056

c)

P(57000 < x < 63000) = P[(57000 - 60000)/ 4000) < (x - $\mu$ ) /$\sigma$  < (63000 - 60000) / 4000) ]

= P(-0.75 < z < 0.75)

= P(z < 0.75) - P(z < -0.75)

= 0.7734 - 0.2266

= 0.5468

Probability = 0.5468