Answer:
a) ![1- \frac{1}{2.5^2} = 0.84](https://tex.z-dn.net/?f=%201-%20%5Cfrac%7B1%7D%7B2.5%5E2%7D%20%3D%200.84)
And that represent 84% of the data within 2.5 deviations from the mean
b) For this case we can assume that the limits between 39 and 59 are given by:
![39 =\mu -\sigma= 49-\sigma](https://tex.z-dn.net/?f=%2039%20%3D%5Cmu%20-%5Csigma%3D%2049-%5Csigma)
![59 =\mu +\sigma= 49+\sigma](https://tex.z-dn.net/?f=%2059%20%3D%5Cmu%20%2B%5Csigma%3D%2049%2B%5Csigma)
Because within one deviation from the mena we have at least 68% of the data.
And we can solve for the deviation and we got:
![\sigma = 49-39 = 10](https://tex.z-dn.net/?f=%20%5Csigma%20%3D%2049-39%20%3D%2010)
![\sigma= 59-49 = 10](https://tex.z-dn.net/?f=%5Csigma%3D%2059-49%20%3D%2010)
Step-by-step explanation:
Part a
Data given
reprsent the population mean
represent the population standard deviation
The Chebyshev's Theorem states that for any dataset
• We have at least 75% of all the data within two deviations from the mean.
• We have at least 88.9% of all the data within three deviations from the mean.
• We have at least 93.8% of all the data within four deviations from the mean.
Or in general words "For any set of data (either population or sample) and for any constant k greater than 1, the proportion of the data that must lie within k standard deviations on either side of the mean is at least:
And if we use the value of k=2.5 we got:
![1- \frac{1}{2.5^2} = 0.84](https://tex.z-dn.net/?f=%201-%20%5Cfrac%7B1%7D%7B2.5%5E2%7D%20%3D%200.84)
And that represent 84% of the data within 2.5 deviations from the mean
Part b
For this case we can assume that the limits between 39 and 59 are given by:
![39 =\mu -\sigma= 49-\sigma](https://tex.z-dn.net/?f=%2039%20%3D%5Cmu%20-%5Csigma%3D%2049-%5Csigma)
![59 =\mu +\sigma= 49+\sigma](https://tex.z-dn.net/?f=%2059%20%3D%5Cmu%20%2B%5Csigma%3D%2049%2B%5Csigma)
Because within one deviation from the mena we have at least 68% of the data.
And we can solve for the deviation and we got:
![\sigma = 49-39 = 10](https://tex.z-dn.net/?f=%20%5Csigma%20%3D%2049-39%20%3D%2010)
![\sigma= 59-49 = 10](https://tex.z-dn.net/?f=%5Csigma%3D%2059-49%20%3D%2010)