# Tchebysheff's Theorem

• Oct 31st 2009, 11:24 PM
Robb
Tchebysheff's Theorem
A manufacturer of tires wants to advertise a mileage interval that excludes no more than 10% of the milage on tires he sells. All he knows is that, for a large number of tires tested, the mean mileage was 25000 miles, and the standard deviation was 4000 miles. What interval would you suggest?

Just not sure on how to set this up, is it:
$\displaystyle P(|Y-25000|<k\sigma )\geq 0.1=1-\frac{1}{k^2}$
• Oct 31st 2009, 11:52 PM
mr fantastic
Quote:

Originally Posted by Robb
A manufacturer of tires wants to advertise a mileage interval that excludes no more than 10% of the milage on tires he sells. All he knows is that, for a large number of tires tested, the mean mileage was 25000 miles, and the standard deviation was 4000 miles. What interval would you suggest?

Just not sure on how to set this up, is it:
$\displaystyle P(|Y-25000|<k\sigma )\geq 0.1=1-\frac{1}{k^2}$

Do you have to use Tchebysheff's Theorem? If so, then it will be

$\displaystyle \Pr(|Y-25000| \geq k\sigma )\leq \frac{1}{k^2} = \frac{1}{10}$.

However, you can get much tighter bounds if you use the Central Limit Theorem: Y ~ Normal$\displaystyle (\mu = 25000, \sigma = 4000)$ and find the value of $\displaystyle y^*$ such that $\displaystyle \Pr(|Y-25000| \geq y^*) = \frac{1}{10}$.
• Nov 1st 2009, 01:00 AM
Robb
Thanks!
Yeah, the question is from the section on Tchebysheff's theorem :p