Answer:
a) We are within 2 deviations from the mean since 30 -2*5 = 20 and 30 + 2*5 = 40. So our value for k = 2 and we can find the % like this:
![\% = (1- \frac{1}{2^2}) *100 = 75\%](https://tex.z-dn.net/?f=%5C%25%20%3D%20%281-%20%5Cfrac%7B1%7D%7B2%5E2%7D%29%20%2A100%20%3D%2075%5C%25)
b) We are within 3 deviations from the mean since 30 -3*5 = 20 and 30 + 3*5 = 45. So our value for k = 3 and we can find the % like this:
![\% = (1- \frac{1}{3^2}) *100 = 88.88\% \approx 89\%](https://tex.z-dn.net/?f=%5C%25%20%3D%20%281-%20%5Cfrac%7B1%7D%7B3%5E2%7D%29%20%2A100%20%3D%2088.88%5C%25%20%5Capprox%2089%5C%25)
c) We are within 1.4 deviations from the mean since 30 -1.4*5 = 23 and 30 + 1.4*5 = 37. So our value for k = 1.4 and we can find the % like this:
![\% = (1- \frac{1}{1.4^2}) *100 = 48.97\% \approx 49\%](https://tex.z-dn.net/?f=%5C%25%20%3D%20%281-%20%5Cfrac%7B1%7D%7B1.4%5E2%7D%29%20%2A100%20%3D%2048.97%5C%25%20%5Capprox%2049%5C%25)
d) We are within 2.6 deviations from the mean since 30 -2.6*5 = 17 and 30 + 2.6*5 = 43. So our value for k = 2.6 and we can find the % like this:
![\% = (1- \frac{1}{2.6^2}) *100 = 85.2\% \approx 85\%](https://tex.z-dn.net/?f=%5C%25%20%3D%20%281-%20%5Cfrac%7B1%7D%7B2.6%5E2%7D%29%20%2A100%20%3D%2085.2%5C%25%20%5Capprox%2085%5C%25)
e) We are within 3.4 deviations from the mean since 30 -3.4*5 =13 and 30 + 3.4*5 = 47. So our value for k = 3.4 and we can find the % like this:
![\% = (1- \frac{1}{3.4^2}) *100 = 91.35\% \approx 91\%](https://tex.z-dn.net/?f=%5C%25%20%3D%20%281-%20%5Cfrac%7B1%7D%7B3.4%5E2%7D%29%20%2A100%20%3D%2091.35%5C%25%20%5Capprox%2091%5C%25)
Step-by-step explanation:
For this case we need to remember what says the Chebysev theorem, and it says that we have at least
of the data liying within k deviations from the mean on the interval ![\bar X \pm k s](https://tex.z-dn.net/?f=%20%5Cbar%20X%20%5Cpm%20k%20s)
We know that ![\bar X= 30, s=5](https://tex.z-dn.net/?f=%5Cbar%20X%3D%2030%2C%20s%3D5)
Part a
For this case we want the values between 20 and 40
So we are within 2 deviations from the mean since 30 -2*5 = 20 and 30 + 2*5 = 40. So our value for k = 2 and we can find the % like this:
![\% = (1- \frac{1}{2^2}) *100 = 75\%](https://tex.z-dn.net/?f=%5C%25%20%3D%20%281-%20%5Cfrac%7B1%7D%7B2%5E2%7D%29%20%2A100%20%3D%2075%5C%25)
Part b
For this case we want the values between 15 and 45
So we are within 3 deviations from the mean since 30 -3*5 = 20 and 30 + 3*5 = 45. So our value for k = 3 and we can find the % like this:
![\% = (1- \frac{1}{3^2}) *100 = 88.88\% \approx 89\%](https://tex.z-dn.net/?f=%5C%25%20%3D%20%281-%20%5Cfrac%7B1%7D%7B3%5E2%7D%29%20%2A100%20%3D%2088.88%5C%25%20%5Capprox%2089%5C%25)
Part c
For this case we want the values between 23 and 37
So we are within 1.4 deviations from the mean since 30 -1.4*5 = 23 and 30 + 1.4*5 = 37. So our value for k = 1.4 and we can find the % like this:
![\% = (1- \frac{1}{1.4^2}) *100 = 48.97\% \approx 49\%](https://tex.z-dn.net/?f=%5C%25%20%3D%20%281-%20%5Cfrac%7B1%7D%7B1.4%5E2%7D%29%20%2A100%20%3D%2048.97%5C%25%20%5Capprox%2049%5C%25)
Part d
For this case we want the values between 17 and 43
So we are within 2.6 deviations from the mean since 30 -2.6*5 = 17 and 30 + 2.6*5 = 43. So our value for k = 2.6 and we can find the % like this:
![\% = (1- \frac{1}{2.6^2}) *100 = 85.2\% \approx 85\%](https://tex.z-dn.net/?f=%5C%25%20%3D%20%281-%20%5Cfrac%7B1%7D%7B2.6%5E2%7D%29%20%2A100%20%3D%2085.2%5C%25%20%5Capprox%2085%5C%25)
Part e
For this case we want the values between 13 and 47
So we are within 3.4 deviations from the mean since 30 -3.4*5 =13 and 30 + 3.4*5 = 47. So our value for k = 3.4 and we can find the % like this:
![\% = (1- \frac{1}{3.4^2}) *100 = 91.35\% \approx 91\%](https://tex.z-dn.net/?f=%5C%25%20%3D%20%281-%20%5Cfrac%7B1%7D%7B3.4%5E2%7D%29%20%2A100%20%3D%2091.35%5C%25%20%5Capprox%2091%5C%25)