Tchebysheff's Theorem

mathlete2 · #1 $Old$ February 22nd, 2008, 12:10 PM

Here's my question : The U.S Mint produces dimes with an average diameter of .5 inch and standard deviation .01. Using Tchebysheff's theorem, find a lower bound for the number of coins in a lot of 400 coins that are expected to have a diameter between .48 and .52.

-I just really am lost on this problem, anyone able to help?

CaptainBlack · #2 $Old$ February 22nd, 2008, 12:39 PM

Quote:

Originally Posted by mathlete2 $View Post$

Here's my question : The U.S Mint produces dimes with an average diameter of .5 inch and standard deviation .01. Using Tchebysheff's theorem, find a lower bound for the number of coins in a lot of 400 coins that are expected to have a diameter between .48 and .52.

-I just really am lost on this problem, anyone able to help?

Tchebysheff's or Chebyshev's inequality is:

$P(|X-\mu|>k\sigma )\le \frac{1}{k^2}$

In the case of the dimes we are being asked for bound on:

$P(|X-\mu|\le 2\sigma )=1-P(|X-\mu|> 2\sigma) \ge 1-\frac{1}{4}$

RonL

mathpenitent · #3 $Old$ April 14th, 2008, 07:21 PM

the answer to ur question is '300'. application of tchebysheff's formula. Sigma+-2(standard deviation) is the range of 0.48 to 0.52 - so going by tchebyscheff's theorem atleast 3/4 of the values(coins minted) will lie within the range provided. 3/4 of 400 = 300.
guess you may have solved it yourself by now i just came upon this theorem and thought i'd look it up when i came across this bit...interesting but i wonder how long i'd retain it. all the best dude.

Quote:

Originally Posted by mathlete2 $View Post$

Here's my question : The U.S Mint produces dimes with an average diameter of .5 inch and standard deviation .01. Using Tchebysheff's theorem, find a lower bound for the number of coins in a lot of 400 coins that are expected to have a diameter between .48 and .52.

-I just really am lost on this problem, anyone able to help?