Monday, September 14, 2015

For the first part, suppose we want to know approximately how much data falls within 2 standard deviations from the mean using Chebyshev's theorem. This is how we do it. We square the standard deviations first to get 4. Then we take 1/4. Now simple subtract 1/4 from 1 to get 3/4. Therefore at least 75% of the data will fall within 2 st dev from the mean. In this problem we don't know the standard deviations, but we know it has to be at least 70%. So we basically are working in reverse now.
That means that 1 - .3 = .7........... therefore .3 = 1/(standard deviations)^2
.3(standard deviations)^2 = 1
divide by .3 to get (standard deviations)^2 = 3.33, therefore standard deviations = 1.83
We know that at least 70% of the data falls within 1.83 standard deviations from the mean.
If you don't understand, please let me know!
For the second part, we simply get the mean and standard deviation of the data set. I did it on my calculator and get mean = 5 and st dev = 1.777
Therefore to see the range of values that are within 1.83 st deviations, take 5 +/- 1.83(1.777)
That gives us 1.75 and 8.25. Look at the data and you see the lowest value is 2.22 and the highest is 8.11, so ALL of the data, 100% fall within the range.

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