# Answer to Question #3945 in Statistics and Probability for susan

Question #3945
A new Firestone tire is guaranteed to last for 40,000 miles. The actual mean life of the tires is 47,000 miles with a standard deviation of 4,000 miles. a) What is the probability that a tire will not last for 40,000 miles? b) The Ford Motor Company uses Firestone Tires and is concerned about the performance of the tires. They conduct a study and find that out of the 120 tires they tested, the mean life was only 46,500 miles. Should the Ford Motor Company be concerned about Firestone&#039;s honesty about its tires? Provide a statistical justification for your response.
1
2011-08-15T06:41:26-0400
a) Let X is random variable that response for tire life. X is normally distributed with mean 47000 and with standard deviation 4000. Consider a random variable Y, normally distributed with mean 0 and standard deviation 1. X=4000Y+47000
We need to find probability
P(X<40000)=P(4000Y+47000<40000)=P(Y<-1.75)=(using the table)=0.04

b) Ford company tested 120 tires, and released that sum of all tire lives were 46500*120. Consider a random variable Z, that is the sum of 120 tire lives. Each of addons is normally distributed, the sum of normally distributed values is also normally distributed with parameters 120*47000 and 120*4000. We need to find probability that P(Z<120*46500)=P(120*4000Y+47000*4000<120*46500)=P(Y<-1/8)=0.45
So, the probability that average tire life is less than 46500 is 0.45. So, it is very possibly that the Ford company made mistake.

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