Percent of Data Values in 215-305 Range | Chebyshev's Theorem

[anita010963]

a distribution has a mean of 260 and a standard deviation of 18 What is the percentage of data values that will fall in the range of 215 to 305 please simple or explain

 

[John Creighto]

I'm going to answer your question but I'm making no assumptions while your teacher probably wanted you to assume a normal distribution.

So 260 is exactly in the middle of the interval. Parts of the distribution outside the interval will have the least effect on the standard deviation if they are at the end points. Similarly points inside the distribution will have the least effect if they are at the mean.

Therefore, Assume the distribution has either the weight at the endpoints or at the mean. Let $$w_m$$ be the weight at the mean and $$w_e$$ be the weight at an end points. Then $$w_m+2*w_e=1$$.

Also

$$2* \left({305-215 \over 2} \right)^2w_e=18$$

$$45w_e=18 <=> w_e=0.4$$

and therefore:

$$w_m=1-2*0.4=0.2$$

Therefore at least 20% of the distribution lies between the endpoints.

 

[John Creighto]

Hmmm...I wonder how my answer compares to this:

http://en.wikipedia.org/wiki/Chebyshev's_inequality#Probabilistic_statement

Where are you stuck in applying the theorem?