A variable has a mean of 1,120 and a standard deviation of 120.
a. Using Chebyshev's theorem, what percentage of the observations fall between 760 and 1,480? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)
b. Using Chebyshev’s theorem, what percentage of the observations fall between 880 and 1,360? (Do not round intermediate calculations. Round your answer to the nearest whole percent.)
a. Chebyshev’s theorem states that for any given number k greater than 1, at least 1 - 1/k^2 of the observations fall within k standard deviations of the mean.
In this case, we want to find the percentage of observations that fall between 760 and 1,480.
First, we need to find the number of standard deviations away from the mean these values are:
760 - 1,120 = -360
1,480 - 1,120 = 360
We can see that both values are 3 standard deviations away from the mean (since the standard deviation is 120).
Using Chebyshev’s theorem with k=3, we can say that at least 1 - 1/3^2 = 1 - 1/9 = 8/9 = 0.8889 or 88.89% of the observations fall within 3 standard deviations of the mean.
b. Similar to part a, we want to find the percentage of observations that fall between 880 and 1,360.
The number of standard deviations away from the mean these values are:
880 - 1,120 = -240
1,360 - 1,120 = 240
Both values are 2 standard deviations away from the mean.
Using Chebyshev’s theorem with k=2, we can say that at least 1 - 1/2^2 = 1 - 1/4 = 3/4 = 0.75 or 75% of the observations fall within 2 standard deviations of the mean.
Therefore, the percentage of observations that fall between 880 and 1,360 is approximately 75%.
To determine the percentage of observations that fall between two values using Chebyshev's theorem, we can use the formula:
Percentage = (1 - 1 / k^2) * 100
where k is the number of standard deviations from the mean.
a. For the range 760 to 1,480:
First, we need to determine the number of standard deviations away from the mean these values are. Since the mean is 1,120 and the standard deviation is 120, we can calculate the number of standard deviations for each value:
For 760:
(z-score) = (760 - 1120) / 120 = -2
For 1480:
(z-score) = (1480 - 1120) / 120 = 3
Now we can calculate the percentage of observations within this range using Chebyshev's theorem:
Percentage = (1 - 1 / k^2) * 100
Percentage = (1 - 1 / (3^2)) * 100
Percentage = (1 - 1/9) * 100
Percentage = (8/9) * 100
Percentage = 88.9%
Therefore, approximately 88.9% of the observations fall between 760 and 1,480.
b. For the range 880 to 1,360:
We can follow the same steps as above to calculate the percentage using Chebyshev's theorem.
For 880:
(z-score) = (880 - 1120) / 120 = -2
For 1360:
(z-score) = (1360 - 1120) / 120 = 2
Percentage = (1 - 1 / (2^2)) * 100
Percentage = (1 - 1/4) * 100
Percentage = (3/4) * 100
Percentage = 75%
Therefore, approximately 75% of the observations fall between 880 and 1,360.