A manufacturer of tires wants to advertise a mileage

Step 1 of 2:

A manufacturer of tires wants to advertise a mileage interval that excludes no more than  of the mileage on tires he sells.

A large number of tires is tested, the mean mileage was  miles, and the standard deviation was  miles.

We need to suggest the interval.

Tchebysheff’s Theorem Let \(Y\) be a random variable with mean and finite variance \(\sigma^{2}\). Then, for any constant \(k>0\),

\(P(|Y-\mu| \leq k \sigma) \geq 1-\frac{1}{k^{2}} \text { or } P(|Y-\mu| \geq k \sigma) \leq \frac{1}{k^{2}}\) ……….. (1)

This result applies for any probability distribution, whether the probability histogram is the normal shape or not.