Answer to Question #150961 in Statistics and Probability for Hay

Question

Question 4
From actual road tests with the tires, Hankook Tires estimated that
the mean tire mileage is 36,500 miles and that the standard
deviation is 5000 miles. Data is normally distributed.
(i) What percentage of the tires can be expected to last more
than 40,000 miles?
(ii) Assume that Hankook Tires is considering a guarantee
that will provide a discount on replacement tires if the
original tires do not provide the guaranteed mileage. What
should the guarantee mileage be if the company wants no
more than 10% of the tires to be eligible

Answer 1

let x denote the number of tires.

x~N(36500,5000²)

z=$\frac{x-\mu}{\sigma}$

a) P(x>40000)

z=$\frac {40000-36500}{5000}$ =0.7

we check the value of p(z>0.7) from the z tables.

=0.24196

24.2% of the tires can be expected to last more than 40000 miles.

b) P(z<0.1)

the value of $\phi ^{-1}(0.1)$ =-1.28

$z=\frac{x-\mu}{\sigma}$

$-1.28=\frac{x-36500}{5000}$

=30100 miles