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%.
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.)
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