Answer to Question #118625 in Statistics and Probability for Michael

Question

The mean life of a tire is 30,000 km. The standard deviation is 2000 km.

a. 68% of all tires will have a life between ______ km and _____ km
b. 95% of all tires will have a life between ______ km and ______ km
c. What percent of the tires will have a life that exceeds 26,000 km ?
d. If a company purchased 2000 tires , how many tires would you expect to last more than 28,000 km ?

Answer 1

Given  μ=30000 km,σ=2000 km

(a) According to Empirical rule, the area covered between mean minus one standard deviation and mean plus one standard deviation of a normal distribution is 68%.

P(μ−σ<X<μ+σ)=68%

P(30000−2000<X<30000+2000)=68%

P(28000<X<32000)=68%

68% of all tires will have a life between 28000 km and 32000 km. 

(b) According to Empirical rule, the area covered between mean minus two standard deviations and mean plus two standard deviations of a normal distribution is 95%.

P(μ−2σ<X<μ+2σ)=95%

P(30000−2(2000)<X<30000+2(2000))=95%

P(26000<X<34000)=95%

95% of all tires will have a life between 26000 km and 34000 km.

(c)

P(X>26000)=1−P(X≤26000)=1−(1−P(26000<X<34000))/2=1-(1−0.95)/2​=0.975=97.5%

(d)

P(X>28000)=1−P(X≤28000)=1-(1−P(28000<X<32000))/2=1-(1-0.68)/2=0.84

N=0.84*2000=1680