A tire manufacturer states that a certain type of tire has a

A tire manufacturer states that a certain type of tire has a mean lifetime of 60,000 miles. Suppose lifetimes are normally distributed with standard deviation =3,500 miles.

a) Find the probability that if you buy one such tire, it will last only 57,000 or fewer miles. If you had this experience, is it particularly strong evidence that the tire is not as good as claimed?

b) A consumer group buys five such tires and tests them. Find the probability that average lifetime of the five tires will be 57,000 miles or less. If the mean is so low, is that particularly strong evidence that the tire is not as good as claimed?

Solution

Normal Distribution
Mean ( u ) =60000
Standard Deviation ( sd )=3500
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X > 57000) = (57000-60000)/3500
= -3000/3500 = -0.8571
= P ( Z >-0.857) From Standard Normal Table
= 0.8043                  
P(X < = 57000) = (1 - P(X > 57000)
= 1 - 0.8043 = 0.1957                  

No, the reason we have mean of 57000 or low is higher rate, mean rate is stated to 60000 miles

b)
P(X > 57000) = (57000-60000)/3500/ Sqrt ( 5 )
= -3000/1565.248= -1.9166
= P ( Z >-1.9166) From Standard Normal Table
= 0.9724                  
P(X < = 57000) = (1 - P(X > 57000)
= 1 - 0.9724 = 0.0276                  

No, we don\'t have strong evidence that the tire is not as good as claimed